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How do you solve 2x+7= -17?
Mar 4, 2018
$x = - 12$
Explanation:
$2 x + 7 = - 17$
$2 x = - 17 - 7$
$2 x = - 24$
(Divide both sides by $2$)
$x = - 12$
Mar 4, 2018
$x = - 12$
Explanation:
$2 x + 7 = - 17$
Start by subtracting $7$ from both sides
$2 x + 7 - 7 = - 17 - 7$
$2 x = - 24$
Then divide both sides by $2$
$\frac{\cancel{2} x}{\cancel{2}} = \frac{- 24}{2}$
$x = - 12$<|endoftext|>
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Hot water and hypoxia: 'The Great Dying's' greatest killers
Increased marine temperatures and reduced oxygen availability were the specific environmental features responsible for the extinctions of vast swaths of ancient ocean life — nearly 96% of all marine species — during the catastrophic end-Permian mass extinction event, a new study finds. The study further predicts a novel pattern of extinction risk during this time, with lower rates of extinction in the tropics than in high latitudes, highlighting the potential for this pattern in future extinction events triggered by similar environmental changes, some of which are already underway. In a related Perspective, Lee Kump writes: “As our understanding of the drivers and consequences of end-Permian climate change and mass extinction improves, the lessons for the future become clear.” Nearly 252 million years ago, intense volcanic activity belched massive volumes of greenhouse gasses into the atmosphere and triggered rapid changes to the climate, which resulted in “the Great Dying” — the largest mass extinction event in Earth’s history. Previous research has suggested that rapid climate change resulting from volcanic activity likely triggered the widespread collapse of biodiversity. However, how each of the resulting environmental impacts contributed to the extinction remains unclear. Justin Penn and colleagues investigated the roles of rapid greenhouse warming and the accompanying loss of oxygen in the ocean, the two best-supported aspects of end-Permian environmental change, according to the authors. Penn et al. investigated the dynamics of the ancient extinction using an Earth system model coupled with data representing a diverse collection of living species to simulate the effects of end-Permian ocean warming and deoxidation on habitat loss and animal survival. The results revealed distinct patterns of extinction — animals that lived in higher latitudes were more prone to extinction. As waters warmed and oxygen became scarce, their low tolerance for hypoxic environments meant they had nowhere to run. Tropical marine animals, pre-adapted to low-oxygen and high temperatures, were better-equipped to survive the environmental changes, according to the authors.
Press Package Team<|endoftext|>
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# Similar Shapes
Here we will learn about similar shapes in maths, including what they are and how to identify similar shapes. We will also solve problems involving similar shapes where the scale factor is known or can be found.
There are similar shapes worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
## What are similar shapes?
Similar shapes are enlargements of each other using a scale factor.
All the corresponding angles in the similar shapes are equal and the corresponding lengths are in the same ratio.
E.g.
These two rectangles are similar shapes.
The scale factor of enlargement from shape A to shape B is 2 .
The angles are all 90^o
The ratio of the bases is 2:4 which simplifies to 1:2
The ratio of the heights is also 1:2
E.g.
These two parallelograms are similar shapes.
The scale factor of enlargement from shape A to shape B is 3.
The corresponding angles are all equal, 45^o and 135^o .
The ratio of the bases are 3:9 which simplifies to 1:3
The ratio of the perpendicular heights is also 1:3
### Scale factor for length, area and volume
The scale factors for length, area and volume are not the same.
In Higher GCSE Maths similar shapes are extended to look at area scale factor and volume scale factors
To work out the length scale factor we divide the length of the enlarged shape by the length of the original shape.
To work the area scale factor we square the length scale factor.
To work the volume scale factor we cube the length scale factor.
E.g.
• Comparing length A and length B we can work out the scale factor to be 3 .
• Comparing area A and area B we can work out the scale factor to be 9 .
This is the same as 3^2 .
• Comparing volume A and volume B we can work out the scale factor to be 27 .
This is the same as 3^3 .
Step-by-step guide: Scale factor
## How to decide if shapes are similar
In order to decide if shapes are similar:
1. Decide which sides are pairs of corresponding sides.
2. Find the ratios of the sides.
3. Check if the ratios are the same.
### Related lessons on congruence and similarity
Similar shapes is part of our series of lessons to support revision on congruence and similarity. You may find it helpful to start with the main congruence and similarity lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
## Similar shapes examples
### Example 1: decide if shapes are similar
Are these shapes similar?
1. Decide which sides are pairs of corresponding sides.
The bases of the rectangles are a pair of corresponding sides.
The heights of the rectangles are a pair of corresponding sides.
2Find the ratios of the sides.
When writing the ratios the order is very important.
Here the ratio is length A : length B
The ratio of the bases is 1:2
The ratio of the heights is 2:4 which simplifies to 1:2
3Check if the ratios are the same.
The rectangles are similar shapes. The ratios for the corresponding lengths are the same 1:2.
The scale factor of enlargement for shape A to shape B is 2 .
### Example 2: decide if shapes are similar
Are these shapes similar?
Decide which sides are pairs of corresponding sides.
Find the ratios of the sides.
Check if the ratios are the same.
## How to find a missing length
In order to find a missing side in a pair of similar shapes:
1. Decide which sides are pairs of corresponding sides.
2. Find the scale factor.
3. Use the scale factor to find the missing length.
## Missing length examples
### Example 3: finding a missing length
Here are two similar shapes. Find the length QR.
Decide which sides are pairs of corresponding sides.
Find the scale factor.
Use the scale factor to find the missing length.
### Example 4: finding a missing length
Here are two similar triangles. Find the length BC.
Decide which sides are pairs of corresponding sides.
Find the scale factor.
Use the scale factor to find the missing length.
## How to find a missing length in a triangle
In order to find a missing side in a pair of triangles when you are not told that the triangles are similar:
1. Use angle facts to determine which angles are equal.
2. Redraw the triangles side by side.
3. Decide which sides are pairs of corresponding sides.
4. Find the scale factor.
5. Use the scale factor to find the missing length.
## Missing length in a triangle examples
### Example 5: finding a missing length in a triangle
Work out the value of x.
Use angle facts to determine which angles are equal.
Redraw the triangles side by side.
Decide which sides are pairs of corresponding sides.
Find the scale factor.
Use the scale factor to find the missing length.
### Example 6: finding a missing length in a triangle
Work out the value of x.
Use angle facts to determine which angles are equal.
Redraw the triangles side by side.
Decide which sides are pairs of corresponding sides.
Find the scale factor.
Use the scale factor to find the missing length.
## How to find an area or volume using similar shapes
In order to find an area or volume using similar shapes:
1. Find the scale factor.
2. Use the scale factor to find the missing value.
## Area or volume using similar shapes examples
### Example 7: finding an area or volume
These two figures are similar.
The area of shape A is 60 \; cm^2
Find the area of shape B:
Find the scale factor.
Use the scale factor to find the missing value.
### Example 8: finding an area or volume
These two shapes are similar.
The volume of shape A is 400 \; cm^3
Find the volume of shape B:
Find the scale factor.
Use the scale factor to find the missing value.
### Common misconceptions
• Take care with the order of ratios
Make sure that you are consistent with your ratios.
E.g.
In this example, always write the A value first, and then the B value.
1 : 3 and 2 : 6.
The ratios are equal, so these shapes are similar shapes.
• In most diagrams the diagrams are NOT drawn to scale
Often diagrams for questions involving similar shapes are NOT drawn to scale. So, use the measurements given, rather than measuring for yourself.
• Shapes can be similar but in different orientations
The second shape may be in a different orientation to the first shape. The shapes can still be similar.
E.g.
Here shape A and Shape B are similar.
Shape B is an enlargement of shape A by scale factor 2.
Here shape B has been rotated to make the similarity easier to see.
• Scaling up or down
If you are finding a missing length in the larger shape you can multiply by the scale factor. The scale factor will be a number greater than 1 .
If you are finding a missing length in the smaller shape you can multiply by the scale factor, but the scale factor will be a number between 0 and 1.
### Practice similar shapes questions
1. Consider if these shapes are similar:
Yes – sides in ratio 1:4
No – sides in ratio 1:3 and 1:4
No – sides in ratio 1:3 and 1:2
Yes – sides in ratio 1:3
The shapes are similar as the ratio of the corresponding sides are the same.
The ratio of the bases is \;\; 3:9
the ratio of the heights is \; 1:3
2. Consider if these shapes are similar:
Yes – sides in ratio 2:1
Yes – sides in ratio 3:1
No – sides in ratio 2:1 and 1:2
No – sides in ratio 3:1 and 2:1
The shapes are similar as the ratio of the corresponding sides are the same.
The ratio of the short sides is \;\; 4:2
the ratio of the long sides is \quad 8:4
3. These shapes are similar. Find the value of x.
x=11
x=10
x=9
x=8
The ratio of the bases is \;\; 6:12
The scale factor of enlargement is 2
x=5 \times 2 =10
4. These shapes are similar. Find the value of x.
x=11
x=10
x=12
x=12.5
The ratio of the bases is \;\; 6:9
The scale factor of enlargement is 1.5
x=8 \times 1.5=12
5. Find the value of x.
x=11
x=13
x=16
x=9
Use the parallel lines to identify equal angles.
Then we can find pairs of corresponding sides.
The ratio of the corresponding sides is \;\; 4:12
The scale factor of enlargement is \; 3
x=3 \times 3= 9
6. Find the value of x.
x=8
x=9
x=10
x=11
Use the parallel lines to identify equal angles.
Then we can find pairs of corresponding sides.
The ratio of the corresponding sides is \;\; 9:6
The scale factor of enlargement is \; \frac{2}{3}
x=12 \times \frac{2}{3}= 8
### Similar shapes GCSE questions
1. Which shape is similar to shape X?
(1 mark)
Shape D
(1)
2. Triangles ABC and DEF are similar.
(a) Write down the size of angle y
(b) Work out the value of x
(3 marks)
(a)
y=56
(1)
(b)
x:11=4:8 the scale factor is \frac{1}{2}
(1)
x=11\times \frac{1}{2}=5.5
(1)
3.
ABC and AED are straight lines.
BE is parallel to CD.
AE = 7.5 \; cm
BE = 6.8 \; cm
Work out the length CD
(2 marks)
10.5\div 7.5=1.4
(1)
CD=6.8\times 1.4 = 9.52
(1)
## Learning checklist
You have now learned how to:
• Compare lengths using ratio notation and/or scale factors
• Solve problems with similar shapes using ratio notation and/or scale factors
• Solve problem with areas and volumes using ratio notation and/or scale factors (HIGHER)
## Still stuck?
Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.
Find out more about our GCSE maths tuition programme.<|endoftext|>
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The hallmark of a safe school is whole-school wellbeing. This means that all members feel a sense of belonging, are valued, respected and free from discrimination and harassment.
The active participation of parents and caregivers in their child’s education is crucial for student wellbeing that leads to outcomes such as:
Bullying can take many forms and can have lasting impacts if no action is taken. Find advice on bullying and how to talk to your child about building a safe and positive learning space.
Respect for diversity is related to a young person’s sense of belonging and connection to others. Help your child to develop respect and acceptance of individual differences among people and groups.
Positive relationships are the key to students feeling safe, valued and accepted by all members of the school community. Find ways you can help create and maintain a school culture that is respectful and inclusive and values the participation of all.
Developing a young person's skills to make informed choices and feel positive about their decisions is an important responsibility for parents and caregivers. Find information to help them build these lifelong skills.
The online world has great benefits for learning but young people need to learn how to keep safe from bullying or unwanted contact. Find resources to help build their online safety skills.
Funded by the Australian Government Department of Education and Training.
© 2018 Commonwealth of Australia or Education Services Australia Ltd, unless otherwise indicated. Creative Commons BY 4.0, unless otherwise indicated.<|endoftext|>
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# Question b9b03
Sep 13, 2017
Your $\frac{3}{10}$ should be $\frac{3}{10}$ of the number of students who cast votes
#### Explanation:
Let the number of students who cast votes be represented by the variable $\textcolor{b l u e}{s}$
We are told that
color(white)("XXX")30%" of " s <= 57
$\textcolor{w h i t e}{\text{XXX}} \frac{3}{10} \times s \le 57$
$\textcolor{w h i t e}{\text{XXX}} s \le 57 \times \frac{10}{3}$
$\textcolor{w h i t e}{\text{XXX}} s \le 190$
Sep 13, 2017
$x \le 190$ $\text{votes}$
No more than $190$ students cast votes.
#### Explanation:
We know that the candidate receives $57$ votes, which are at least 30% of the total number of votes.
This means that $57$ is greater than or equal to 30% of the total votes.
Let the total number of votes be $x$.
Rightarrow 30% "of"# $x \le 57$ $\text{votes}$
$R i g h t a r r o w 0.3 x \le 57$ $\text{votes}$
$\therefore x \le 190$ $\text{votes}$
Therefore, there were no more than $190$ students who cast votes.<|endoftext|>
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With the target only 1 km away, there's no mirage, because ray curvature is so small that no ray crossing can occur in this short distance. (The smallness of the ray curvature is a direct result of the small curvature of the temperature profile.)
Notice that the transfer curve (in the lower left of the figure) is almost a straight line.
Consequently, the target hardly appears distorted.
Notice that it's much closer than the sea horizon, which can be seen
behind it in the simulation at the right.
If the lapse rate were constant, we'd expect the target to have half the angular subtense it had at 1 km, which was 10′. Half of 10 is 5; but the actual subtense of the target here is larger: about 6′. That's because differential refraction has stretched the lower part of the target (note the curve at the bottom of the transfer curve, and the corresponding curvature in the lowest parts of the target.) This vertical stretching, or magnification, constitutes towering.
Because the curvature of the
is marked only near the sea surface, the top of the target still looks
fairly normal. Notice that its apex is still very nearly a right angle.
The folding occurs at the minimum in the transfer curve, where the vertical magnification of the image is infinite. The region of large vertical magnification covers more of the target here, so the towering is more pronounced.
The apparent size of the mirage can be measured on the transfer curve,
from the vertical axis at the apparent horizon to the minimum in the
curve: it's a little less than a minute of arc in extent.
Although the target is now four times as far away as it was at 1 km, where it subtended about 10′ of arc, it remains much larger than the 2.5′ one would expect from simple distance scaling if the whole target were still visible. Actually, you can see from the disappearance of another stripe that only about half of the whole target is still in sight; so the expected height should be only 1.25′. But the apparent height of the target is really almost 4′. The magnification near the fold line makes the image considerably larger than one would expect.
At larger distances, the target rapidly shrinks to a thin vertical feature at the fold line, where it disappears completely at a range between 5.3 and 5.4 km.
While the inferior-mirage atmosphere hides half of the target below the fold line at 4 km range, the Standard Atmosphere allows the target to be more than twice as far away — 9.5 km — before half of it is hidden.
Copyright © 2008, 2012, 2013 Andrew T. Young
main mirage page
GF home page
or the website overview page<|endoftext|>
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# In isosceles ABC, B is the vertex. The measure of angle B can be represented as (8x+2). The measure of angle A can be represented as (9x-2). Find the measure of all three angles of the triangle. Be sure to name the angle with it’s measurement (ex: angle A= ).
## two angles are the same (given it is isoslese)
the angles add up to 180 degrees
use this to answer the question
## find x
and then plug x in to find respective angles
## To find the measure of all three angles of the isosceles triangle ABC, we can use the fact that the sum of the angles in any triangle is always 180 degrees.
Let's start by labeling the angles:
Angle B = 8x + 2 (Given)
Angle A = 9x - 2 (Given)
Angle C = Other angle of the triangle
Since ABC is an isosceles triangle, angle A and angle C are congruent. This means they have the same measure. So we can set up an equation:
Angle A + Angle C + Angle B = 180
Replacing the values of angle A and angle B:
(9x - 2) + (9x - 2) + (8x + 2) = 180
Combine like terms:
18x - 4 + 8x + 2 = 180
Combine the x terms:
26x - 2 = 180
Add 2 to both sides of the equation:
26x = 182
Divide both sides by 26:
x = 7
Now we can find the measure of angle A:
Angle A = 9x - 2
Angle A = 9(7) - 2
Angle A = 63 - 2
Angle A = 61
Similarly, we can find the measure of angle B:
Angle B = 8x + 2
Angle B = 8(7) + 2
Angle B = 56 + 2
Angle B = 58
Since ABC is an isosceles triangle, angles A and C have the same measure. Therefore, angle C will also measure 61 degrees.
So, the measure of the three angles in isosceles triangle ABC are:
Angle A = 61 degrees
Angle B = 58 degrees
Angle C = 61 degrees<|endoftext|>
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Pupils who have access to physical educational resources, such as books and computers at home, tend to perform better in science than those who don’t. This has been proved by a great deal of international research.
It’s true in South Africa, too, but our research has found that less tangible factors also play a role in pupils’ science achievement. These factors include parental education levels, parental involvement in homework – and, crucially, home language.
Language is a factor that makes the South African context unique and has a noteworthy role in pupils’ science achievements. The language of teaching and learning is often different from the language spoken in a pupil’s home. Only 26% of pupils who participated in the 2011 Trends in International Mathematics and Science Study (TIMSS) spoke the language of the test at home. For our research, we studied data from 11 969 Grade 9 pupils – who were, on average, 16 years old – who participated in TIMSS in 2011.
Successive apartheid governments used language policy as a tool to create socio-economic and educational division. This history means that language as a home resource can’t be overlooked when it comes to understanding pupils’ performance in science at school. Our results proved just how important language is: the language most often spoken in a pupil’s home was the single most important predictor.
In developing countries such as South Africa, science, technology and innovation have become forces that drive economic growth and competitiveness and have the potential for improving the quality of life. The number of skilled people (such as scientists, engineers and other technically skilled personnel) in a country is associated with its economic growth and ability to compete in the global economy.
The development of these skilled people begins at the school level. So it’s cause for concern that the 2011 TIMSS found the average science achievement of Grade 9 South African students to be well below the international centre point of 500 points. Tackling language policy can, we believe, improve pupils’ performance in this important subject.
Historically, the state provided educational resources in an unbalanced way. Schools designated for white pupils were well resourced, while those for black learners were under-resourced. Today, these imbalances persist. There are vast differences in physical resources at poor and affluent schools.
The school resources we included in our study were the condition of the school building; the use of workbooks or worksheets as the basis of instruction and class size. We also explored the capacity of the school to provide instruction based on the availability of resources such as textbooks, science equipment and computer software.
For home resources, we asked the pupils to report on how often the language of the test was spoken in the home, the number of books at home, the number of home assets, parental education levels, and parental involvement in school homework.
Language emerged strongly as a success factor. Pupils who used the language most frequently spoken at home in the TIMSS test scored 62 points higher, on average, than those who seldom spoke the language of the test.
The number of home assets present in a pupil’s home had the second strongest positive association with science achievement. It was found that for each additional asset (such as a fridge, television, computer etc.) in a pupil’s home, they scored an average of 11 points higher in science than their peers.
The third most important predictor of science achievement was the condition of the school building. Pupils who attended schools with minor problems with the building performed 24 points higher, on average, than those who attended schools that reported moderate to serious problems with the buildings.
So what does this all mean?
Language development is recognised as crucial for all other learning to take place. Our findings suggest that the language of instruction (and of testing) has not been mastered by the time pupils are in Grade 9. This is unsurprising. Most of the learners who were tested using TIMSS were, in essence, learning science through a foreign language.
This means that pupils are likely to be at a disadvantage because their knowledge of the language of instruction is below the expected level for their age and grade. The implication is that education policies must seek both to improve the manner in which the language of instruction is taught to students who don’t speak that language at home, and concurrently, the policies that promote instruction in the home language must be strengthened.
It’s important that we understand the determinants of science achievement for South African pupils. This has far reaching implications for the country’s broader growth and development. This is because successful interventions at school level may contribute to increasing the pool of matriculants who are eligible to study science-related subjects at a tertiary level and who will later join the skilled workforce.
Disregarding these environmental factors may hinder the success of policies designed to improve achievement and further economic growth.<|endoftext|>
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RAID, which is an acronym of Redundant Array of Independent Disks, is a software or hardware storage virtualization technology which allows a system to use several hard drives as one single logical unit. Simply put, all drives are used as one and the data on all of them is the same. Such a configuration has two major advantages over using a single drive to store data - the first one is redundancy, so in case one drive stops working, the data will be accessible from the others, and the second one is improved performance because the input/output, or reading/writing operations will be spread among several drives. There are different RAID types based on what amount of drives are employed, if reading and writing are both done from all the drives simultaneously, whether data is written in blocks on one drive after another or is mirrored between drives in the same time, and so on. Determined by the exact setup, the error tolerance and the performance may differ.<|endoftext|>
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# Manually Calculating the Amount Hidden by Curvature
In order to calculate the amount that should be hidden we may use the Theory of Pythagoras.
Diagram & Transformations Credit: dizzib | Theory Credit: Pythagoras
## Important Equations
To find the distance from the observer's eye to the horizon (d1) we first perform the following:
1. d1 = sqrt(h0^2 + 2 * R * h0)
Next, we place d1 in the following equation to find the amount hidden below the horizon of the earth (h1):
2. h1 = sqrt((d0 - d1)^2 + R^2) - R
### Example 1: Sea Level
To calculate the amount hidden at sea level, over 6.23 miles, with an observer height of 32 inches, we convert to a like unit (ie. km) and perform the following:
R = 6371 km
h0 = 0.0008128 km (32 inches)
d0 = 10.02621 km (6.23 miles)
1. d1 = sqrt(0.0008128^2 + 2×6371×0.0008128) = 3.21818
2. h1 = sqrt((10.02621 - 3.21818)^2 + 6371^2) - 6371 = 0.00363752
0.00363752 km converted to feet = 11.93412073491 feet hidden
### Example 2: Lake Above Sea Level
To calculate the amount hidden across a lake with an altitude of 1368 meters above sea level, we make a slight adjustment to R and perform the following:
R = 6372.368 km (6371 km + 1368 m)
h0 = 0.0008128 km (32 inches)
d0 = 10.02621 km (6.23 miles)
1. d1 = sqrt(0.0008128^2 + 2×6372.368×0.0008128) = 3.21853
2. h1 = sqrt((10.02621 - 3.21853)^2 + 6372.368^2) - 6372.368 = 0.00363636
0.00363636 km converted to feet = 11.93031496063 feet hidden
# Online Calculator
Based on the above, dizzib has provided an online calculator for computing the drop distance and the amount which should be hidden per observer height according to the curvature of the earth.<|endoftext|>
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- Staff Selection Commission - Number system (Type I and Type II)
# Staff Selection Commission Mathematics - Number system (Type I and Type II)
### TYPE-II
Ans .
(3) 8
1. Explanation :
$$a^2-b^2 = (a^2+b^2)(a+b)(a-b)$$
Let a = 3, b = 1
Required number
= (3 + 1) (3 – 1) = 8
Ans .
(1) 4
1. Explanation :
82. Let m = n = p and m – n = 2p m + n = 2p
(m – n) (m + n) = 4p2
Ans .
(3) 6
1. Explanation :
83. A number is divisible by 9, if sum of its digits is divisible by 9.
Let the number be x.
5 + 4 + 3 + 2 + x + 7 = 21 + x
x = 6
Ans .
(1) 5
1. Explanation :
84.A number is divisible by 9 if the sum of its digits is divisible by 9.
Here, 6 + 7 + 0 + 9 = 22
Now, 22 + 5 = 27, which is divis- ible by 9. Hence 5 must be add- ed to 6709.
Ans .
(2) 6
1. Explanation :
A number is divisible by 9 and 6 both, if it is divisible by LCM of 9 and 6 i.e., 18. Hence, the numbers are 108, 126, 144, 162, 180, 198.
Ans .
(2) 150
1. Explanation :
86.First 3–digit number divisible by 6 = 102
Last such 3-digit number =996
996 = 102 + (n –1) 6
(n – 1)6 = 996 – 102 = 894
n – 1 = $$\frac{894}{6}=149$$
n = 150
Ans .
(2) 6
1. Explanation :
$$n^3 – n = n (n^2 – 1)$$
= n (n + 1) (n – 1)
For n = 2, n3 – n = 6
Ans .
(3) 6
1. Explanation :
$$n^3 – n = n (n + 1) (n – 1)$$
$$n = 4, n^3 – n = 4 × 5 × 3 = 60$$
60 ÷ 6 = 10
Ans .
(3)divisible by 9
1. Explanation :
Number = 100x + 10y + z
Sum of digits = x + y + z
Difference = 100x + 10y + z – x – y – z
= 99x + 9y = 9 (11x + y)
Ans .
divisible by (11 × 13)
1. Explanation :
divisible by (11 × 13)
Ans .
(3) 5
1. Explanation :
91.Any number is divisible by 11 when the differences of alterna- tive digits is 0 or multiple of 0, 11 etc. Here,
5 + 2 + * = 7 + *
8 + 4 = 12
* = 12 – 7 = 5
Ans .
(4) 5
1. Explanation :
92.A number is divisible by 11, if the difference of the sum of its dig- its at odd places and the sum of its digits of even places, is either 0 or a number divisible by 11.
(5 + 9 + * + 7) – (4 + 3 + 8) = 0
or multiple of 11
21 + * – 15
* + 6 = a multiple of 11
* = 5
Ans .
(4) 1
1. Explanation :
93. A number is divisible by 11, if the difference of sum of its dig- its at odd places and the sum of its digits at even places is either 0 or a number divisible by 11.
Difference = (4 + 3 + 7 + 8) – (2 + 8 + ) = 22 – 10 – * = 12 – *
Clearly, * = 1
Ans .
(4) 4
1. Explanation :
94. A number is divisible by 11 if the difference of the sum of digits at odd and even places be either zero or multiple of 11.
If the middle digit be 4, then 24442 or 244442 etc are divisi- ble by 11.
Ans .
(2) 12
1. Explanation :
$$n^2(n^2–1) = n^2 (n + 1) (n – 1)$$
Now, we put values n = 2, 3..... When n = 2
$$n^2(n^2 –1)$$ = 4 × 3 × 1 = 12, which is a multiple of 12
When n = 3. n$$n^2(n^2 –1)$$ = 9 × 4 × 2 = 72,
which is also a multiple of 12. etc
Ans .
(4)Smallest 3-digit prime num- ber
1. Explanation :
Let the unit digit be x and ten’s digit be y.
Number = 1000y + 100x + 10y + x = 1010y + 101x = 101(10y + x)
Clearly, this number is divisible by 101, which is the smallest three-digit prime number.
Ans .
(2) 10004
1. Explanation :
The least number of 5 digits = 10000
Required number = 10000 + (41–37)
= 10004
Ans .
(1) $$2^{96} + 1$$<
1. Explanation :
$$2^{96} + 1 = (2^{32})^3 + 1^3$$
Clearly, $$2^{32}+1$$ is a factor of $$2^{96} +1$$
Ans .
(4) 48
1. Explanation :
For n = 2
$$n^4 + 6n^3 + 11n^2 + 6n + 24$$= 16 + 48 + 44 + 12 + 24 = 144
which is divisible by 48. Clearly, 48 is the required num- ber. = 144
which is divisible by 48. Clearly, 48 is the required num- ber.
Ans .
(2)18
1. Explanation :
When we divide 1000 by 225, quotient = 4
When we divide 5000 by 225, quotient = 22
Required answer = 22 – 4 = 18
Ans .
(3) 24
1. Explanation :
$$(n^3 – n) (n – 2) = n (n – 1) (n + 1) (n – 2)$$
When n = 3, Number = 3 × 2 × 4 = 24
Ans .
(1) 1440
1. Explanation :
LCM of 16 and 18 = 144 Multiple of 144 that is less than 1500 = 1440
Ans .
(2)6
1. Explanation :
The largest 4-digit number = 9999
Required number = 345 – 339 = 6
Ans .
(2) 10
1. Explanation :
$$4^{61} + 4^{62} + 4^{63} + 4^{64} = 4^{61}*(1 + 4 + 16 + 64) = 4^{61} * 85$$
Which is a multiple of 10.
Ans .
(2) 9
1. Explanation :
Let the number be 10x + y
where y < x. Number obtained by interchang- ing the digits = 10y + x
Difference = 10x + y – 10y – x
= 9x – 9y = 9 (x – y)
Hence, the difference is always exactly divisible by 9.
Ans .
(3) 303375
1. Explanation :
Check through option
$$\frac{303375}{25} = \frac{303375*4}{25*4} = 12135$$
A number is divisible by 25 if the last two digits are divisible by 25 or zero.
Ans .
(1) 132
1. Explanation :
307 × 32 = 9824
307 × 33 = 10131
Required number = 10131 – 9999 = 132
Ans .
(1) 15999879
1. Explanation :
a = 4011, b = 3989
ab = 4011 × 3989
= (4000 + 11) (4000 – 11) = (4000)2 – (11)2
= 16000000 – 121 = 15999879
Ans .
(2) 2
1. Explanation :
Expression =$$3^{2n} + 9^n + 5$$ = $$3^{2n} + 9^n + 3$$ + 2
Clearly, remainder = 2
Ans .
(3) 7
1. Explanation :
Resulting number = 3957 + 5349 – 7062 = 2244 which is divisible by 4, 3 and 11.
2244 ÷ 4 = 561
2244 ÷ 3 = 748
2244 ÷ 11 = 204
Ans .
(3) 7387
1. Explanation :
Prime numbers between 80 and 90.
= 83 and 89
Required product = 83 × 89 = 7387
Ans .
(2) 35
1. Explanation :
When n = 2, $$6^n – 1 = 6^2 – 1 = 36 – 1 = 35$$
When, n = an even number,
$$a^n–b^n$$ is always divisible by $$(a^2–b^2)$$.
Ans .
(2) 5
1. Explanation :
Total number of marbles = x
+ x + 3 + x – 3 = 3x
3x = 15 x = 5
Ans .
(3) 10 kg
1. Explanation :
Bucket + full water = 17 kg.
Bucket + 1/2 water = 13.5 kg.
Water = 2 × 3.5 = 7 kg.
Weight of empty bucket 122.
= 17 – 7 = 10 kg.
Ans .
(4) 30
1. Explanation :
A cow and a hen each has a head.
If the total number of cows be x,
then Number of hens = 180 – x
A cow has four legs and a hen has two legs.
(180 – x) × 2 + 4x = 420
360 – 2x + 4x = 420
2x = 420 – 360 = 60
Ans .
(4) 6
1. Explanation :
On putting n = 1
n(n +1) (n + 2) = 1 × 2 × 3 = 6
Ans .
(2) 3
1. Explanation :
2736 ÷ 24 = 114
Hence, first divisor (2736) is a multiple of second divisor (24).
Required remainder = Remainder obtained on dividing 75 by 24 = 3
Ans .
(2) 9
1. Explanation :
5 E9 + 2 F8 + 3 G7 = 1114
Value of ‘F’ will be maximum if the values of E and G are mini- mum.
509 + 2 F8 + 307 = 1114
2 F8 = 1114 – 509 – 307= 298
F = 9
Ans .
(4)6, 10, 14, 18
1. Explanation :
Let four numbers be a, b, c and d respectively.
a + b + c + d = 48 (i) and,
a + 5 = b + 1 = c – 3 = d – 7 = x (let)
a = x – 5; b = x – 1, c = x + 3, d = x + 7 From equation (i),
x – 5 + x – 1 + x + 3 + x + 7 = 48 4x + 4 = 48
4x = 48 – 4 = 44
x = 44 = 11
a = x – 5 = 11 – 5 = 6
b = x – 1 = 11 – 1 = 10
c = x + 3 = 11 + 3 = 14
d = x + 7 = 11 + 7 = 18
Ans .
(2) 24
1. Explanation :
$$\frac{2055}{27}$$
Remainder is 3
Required number = 27 – 3 = 24
Ans .
$$\frac{n+1}{2}$$
1. Explanation :
122.Sum of first n bers = $$\frac{n*(n+1)}{2}$$
Required avg. = $$\frac{n*(n+1)}{2*n}$$ = $$\frac{(n+1)}{2}$$
Ans .
(3) 9
1. Explanation :
123.Here, the first divisor (361) is a multiple of second divisor (19).
Required remainder = Re- mainder obtained on dividing 47 by 19 = 9
Ans .
(3) 990
1. Explanation :
Largest number = 3995
Smallest number = 3005
Difference = 3995 – 3005 = 990
Ans .
(2) 1250
1. Explanation :
Let the numbers be x and y. According to the question,
x + y = 75
x – y = 25
Q (x + y)2 – (x – y)2 = 4xy
752 – 252 = 4xy
4xy = (75 + 25) (75 – 25)
4xy = 100 × 50
xy = 1250
Ans .
(4) 95
1. Explanation :
Required difference = 97 – 2 = 95
Ans .
(4) 10
1. Explanation :
) xy = 24
(x, y)
= (1 × 24), (2 ×12), (3 × 8), (4 × 6)
Minimum value of (x + y) = 4 + 6 = 10.
Ans .
(3)Both 3 and 9
1. Explanation :
128. Let the 3–digit number be 100x + 10y + z.
Sum of the digits = x + y + z According to the question, Difference
= 100x + 10y + z – (x + y + z)
= 99x + 9y = 9 (11x + y)
Clearly, it is a multiple of 3 and 9.
Ans .
(1) 60
1. Explanation :
129. Let the numbers be x and y
where x > y.
According to the question, (x + y) – (x – y) = 30
x + y – x + y = 30
2y = 30
y = 30/2 = 15
xy = 900
15x = 900
x = 60
Ans .
(3) 5336
1. Explanation :
130. According to the question, Divisor (d) = 5r = 5 × 46 = 230 Again, Divisor (d) = 10 × Quo- tient (q)
230 = q × 10
q = 230/10 = 23
Dividend = Divisor × Quotient + Remainder
= 230 × 23 + 46
= 5290 + 46 = 5336
Ans .
(3) 5
1. Explanation :
Divided = 44 × 432 = 19008
19008/31 Remainder = 5
Ans .
(2) 2
1. Explanation :
Here, first divisor (729) is a multiple of second divisor (27).
Required remainder = Remainder got on dividing 56 by 27 = 2.
Ans .
(4) 100008
1. Explanation :
Smallest number of six dig- its = 100000
108/100000 remainder = 100
Required number = 100000 + (108 – 100) = 100008
Ans .
(2) 14
1. Explanation :
Let the number be x. According to the question, x + 25 = 3x – 3
3x – x = 25 + 3
2x = 28
x = 14
Ans .
(1) 2
1. Explanation :
334 × 545 × 7p is divisible by 3340.
334 × 5 × 109 × 7 × p, is
divisible by 334 × 2 × 5
Clearly, p = 2
Ans .
(2) 1
1. Explanation :
Let the number be a. According to the question,
a + 1/a = 2
a2 + 1 = 2a a2 – 2a + 1 = 0
(a – 1)2 = 0 a – 1 = 0
a = 1
Ans .
(3) 5
1. Explanation :
First divisor (56) is a mul- tiple of second divisor (8).
Required remainder = Remainder obtained after divid- ing 29 by 8 = 5
Ans .
(2) 7
1. Explanation :
Let the number be x. According to the question,
x – 4 = 21/x
x2 – 4x = 21
x2 – 4x – 21 = 0
(x + 3) (x – 7) = 0
x = 7 because x is not equal to – 3.
Ans .
(2) 2
1. Explanation :
Let quotient be 1.
n = 4 × 1 + 3 = 7
2n = 2 × 7 = 14,
On dividing 14 by 4, remainder = 2
Ans .
paste_right_option
1. Explanation :
Divisor = 555 + 445 = 1000
Quotient = (555 – 445) × 2 = 110 × 2 = 220
Remainder = 30
Dividend = Divisor × Quotient + Remainder
= 1000 × 220 + 30 = 220030
Ans .
(1) 6480
1. Explanation :
According to the question, Divisor = 2 × remainder = 2 × 80 = 160
Again, 4 × quotient = 160
Quotient = 160/4 = 40
x = Divisor × Quotient + re- mainder = 160 × 40 + 80 = 6480
Ans .
(2) 11
1. Explanation :
Here, first divisor (342) is a multiple of second divisor (18). i.e. 342 ÷ 18 = 19
Required remainder = Remainder on dividing 47 by 18 = 11
Ans .
(3) 42
1. Explanation :
Let second number = x. First number = 3x
Third number = 2/3 × 3x = 2x
According to the question, 3x + x + 2x = 252
6x = 252
x = 252/6 = 42
Ans .
(3) 65952
1. Explanation :
Five-digit numbers formed by 2, 5, 0, 6 and 8 :
Largest number = 86520
Smallest number = 20568
Required difference = 86520 – 20568 = 65952
Ans .
(1) 21
1. Explanation :
Let the number of cows be x. Q A hen or a cow has only one head.
Number of hens = 50 – x
A hen has two feet. A cow has four feet.
According to the question, 4x + 2 (50 – x) = 142
4x + 100 – 2x = 142
2x = 142 – 100 = 42
x = 21
Ans .
(2) 1683
1. Explanation :
Firstly, we find LCM of 5, 6, 7 and 8.
LCM = 2 × 5 × 4 × 3 × 7= 840
Required number = 840x + 3 which is exactly divis- ible by 9.
Now, 840x + 3 = 93x × 9 + 3x + 3
When x = 2 then 840x + 3, is di- visible by 9.
Required number = 840 × 2 + 3 = 1683
Ans .
(4) 9
1. Explanation :
A 3–digit number = 100x + 10y + z
Sum of digits = x + y + z
Difference = 100x + 10y + z – x – y – z
= 99x + 9y = 9 (11x + y)
i.e., multiple of 9.
Ans .
(1) 27
1. Explanation :
8961/84 Remainder is 57
Required number = 84 – 57 = 27
Ans .
(1) 27
1. Explanation :
Number of numbers lying be- tween 67 and 101
101 – 67 – 1 = 33
Prime numbers 71, 73, 79, 83, 89 and 97 = 6
Composite numbers = 33 – 6 = 27
Ans .
(3) 1
1. Explanation :
LCM of 9, 11 and 13 = 9 × 11 × 13 = 1287
Required lowest number that leaves 6 as remainder = 1287 + 6 = 1293
Required answer = 1294 – 1293 = 1
Ans .
paste_right_option
1. Explanation :
A number is divisible by 8 if number formed by the last three
If * is replaced by 3, then 632
Ans .
(4) 69
1. Explanation :
13851/87 Remainder = 18
Required no. = 87 - 18 = 69
Ans .
(2) 8
1. Explanation :
If the sum of the digits of a number be divisible by 9, the number is divisible by 9.
Sum of the digits of 451 * 603
= 4 + 5 + 1 + * + 6 + 0 + 3 = 19 + * If * = 8, then 19 + 8 = 27 which is divisible by 9.
Ans .
(2) 9944
1. Explanation :
The largest 4-digit number = 9999
9999/88 Remainder is 55 Required number = 9999 – 55 = 9944
Ans .
(1) 57717
1. Explanation :
= 5 + 7 + 7 + 1 + 7 = 27 which is divisible by 9
Required number = 57717
Ans .
(3) 187
1. Explanation :
Prime numbers between 58 and 68 59, 61 and 67
Required sum = 59 + 61 + 67 = 187
Ans .
(3) 38
1. Explanation :
Let the two digit number be 10x + y.
According to the question,
xy = 24 (i)
and, 10x + y + 45 = 10y + x
y - x = 5 ....(ii)
x + y = 11 ....(iii)
On adding equations (ii) and (iii),
y – x + x + y = 5 + 11
2y = 16 y = 8
xy = 24 8x = 24
x = 3
Required number = 10x + y = 10 × 3 + 8 = 38
Ans .
(3) 3
1. Explanation :
A number is divisible by 11 if the difference between the sum of digits at odd places and that at even places is either zero or a multiple of 11.
Sum of the digits at odd places = 6 + 8 + 5 = 19
Sum of the digits at even places = 9 + 6 + 7 = 22
Required number=22–19 = 3
Ans .
(3) 12
1. Explanation :
According to the question
$$\frac{2+2*5}{3} = 4$$
Second No. $$\frac{48}{4} = 12$$<|endoftext|>
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Sales Toll Free No: 1-855-666-7446
# Foil Method
Top
Sub Topics In algebra, to multiply two binomials, a method is defined called as FOIL method. FOIL stands for "first, outer, inner and last term". Here,First means initial terms of each binomial are multiplied together. Outer means initial term of binomial is multiplied by second term of other binomial. Inner means inside terms or numbers are multiplied, which means first term of initial binomial and second term of other binomial are multiplied together. Last means final term of both binomials are multiplied. FOIL is an acronym to remember a set of rules for multiplying binomials and is considered as a special case of a more general method for multiplying algebraic expressions using the distributive law. It is used for a two term polynomial times another two term polynomial.
## Foil Method Definition
Foil method definition states that, it is a standard method to multiply two binomials. Binomials are expressions which contain two terms.
When we multiply two binomials, we will get four terms:
• First: Multiply the first terms of both binomials.
• Outside: Multiply first term of first binomial with last term of second binomial.
• Inner: Multiply last term of first binomial and first term of second binomial.
• Last: Multiply last terms of both the binomials.
Simplify the products and combine the like terms which may occur.
## Reverse Foil Method
Reverse foil method is a process of doing foil backward. It is used to write the information in a more organized way and helps us to understand how FOIL works, when multiplying two binomials.
Reverse foil method of factoring trinomials is in the form x$^{2}$ + bx + c. Through this method, we can convert the trinomial back into a double binomial expression.
Steps to be considered are given below :
1. Set two sets of parentheses for trinomial equation like (_ $\pm$ _) (_ $\pm$ _).
2. Factor the given polynomial.
3. Find the possible factors of the last term.
4. Plug in the obtained number groups in the given equation. Check whether the obtained factors hold true or not in the given equation.
5. Keep plugging the factors and check it with the FOIL. If the factors hold true in the given equation, then you've got the answer.
## Foil Method Polynomials
Polynomial is an expression constructed from variables and constants. Expressions with two or more terms are called polynomials. A polynomial can have constants, variables and exponents. But, division and square root is not allowed. They are written in decreasing order of terms. Highest exponent in the polynomial is usually written first.
Polynomial equation also known as algebraic equation is of the form $a_{n}x^{n}$ + $a_{n-1}x^{n-1}$ + .......+ $a_{1}$x + $a_{0}$ = 0
### Solved Example
Question: Solve (2x + 5) (x$^{3}$ + 10x$^{2}$ + 8x + 3).
Solution:
Given (2x + 5) [(x$^{3}$ + 10x$^{2}$) + (8x + 3)]
The second expression can be split as (x$^{3}$ + 10x$^{2}$) + (8x + 3), so that, we can apply FOIL.
It can be solved now as follows:
(2x + 5)( (x$^{3}$ + 10x$^{2}$) + (2x + 5) (8x + 3) (Use FOIL again)
= 2x(x$^{3}$) + 2x(10x$^{2}$) + 5x$^{3}$ + 50x$^{2}$ + 2x(8x) + 2x(3) + 5(8x) + 5(3)
= 2x$^{4}$ + 25x$^{3}$ + 66x$^{2}$ + 46x + 15
## Foil Method Trinomials
An expression with three terms is called a trinomial. Foil method cannot be applied to trinomials, as it holds only for binomials. Given below are some examples which can be solved in this way. 3p + q - 5xty, 2s - 9t + 8t, 16y - 5z + 1 are all trinomials.
### Solved Example
Question: Solve (5x - 2) (5x$^{2}$ + 7x - 5)
Solution:
5x (5x$^{2}$) + (5x) (7x) + 5x (-5) - 2 (5x$^{2}$) - 2 (7x) - 2 (-5)
= 25x$^{3}$ + 35x$^{2}$ - 25x - 10x$^{2}$ - 14x + 10
= 25x$^{3}$ + 25x$^{2}$ - 39x + 10
## Foil Method Examples
Given below are some of the examples on FOIL method.
### Solved Examples
Question 1: Multiply (5x - 4) (x - 4) using FOIL method.
Solution:
First: $5x \times x = 5x^{2}$
Outside: $5x \times (-4) = -20x$
Inside: $- 4 \times x = -4x$
Last: $(-4) \times (-4) = 16$
Summing it all up, we get the trinomial equation as $5x^{2} - 24x + 16$
Question 2: Solve x$^{2}$ + 13x + 36 using reverse FOIL method.
Solution:
Step 1:
Set two sets of parentheses for trinomial equation like: (_ $\pm$ _) (_ $\pm$ _).
Coefficient of first term is 1. So, the binomials of the first terms are x and x.
So, we have, (x $\pm$ _) (x $\pm$ _).
Step 2:
Factor the given polynomial.
As the last sign is positive, find the factors of 36 that add to give 13.
Step 3:
Possible factors of the last term '36' are (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)
Step 4:
As 4 and 9 only add to give 13. These factors are the last numbers in the binomial factors as the other factors doesn't hold true for the given equation (x + 4) (x + 9).<|endoftext|>
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# How do you solve 6(y^2-2)+y<0 using a sign chart?
May 20, 2017
Solution : $- \frac{3}{2} < y < \frac{4}{3}$. In interval notation $\left(- \frac{3}{2} , \frac{4}{3}\right)$
#### Explanation:
$6 {y}^{2} - 12 + y < 0 \mathmr{and} 6 {y}^{2} + y - 12 < 0 \mathmr{and} \left(3 y - 4\right) \left(2 y + 3\right) < 0$
Critical points are $3 y - 4 = 0 \mathmr{and} y = \frac{4}{3} , 2 y + 3 = 0 \mathmr{and} y = - \frac{3}{2}$
When $y < - \frac{3}{2}$ sign of $\left(3 y - 4\right) \left(2 y + 3\right) i s \left(-\right) \cdot \left(-\right) = \left(+\right)$ i.e $> 0$
When $- \frac{3}{2} < y < \frac{4}{3}$ sign of $\left(3 y - 4\right) \left(2 y + 3\right) i s \left(-\right) \cdot \left(+\right) = \left(-\right)$ i.e $< 0$
When $y > \frac{4}{3}$ sign of $\left(3 y - 4\right) \left(2 y + 3\right) i s \left(+\right) \cdot \left(+\right) = \left(+\right)$ i.e $> 0$
So Solution : $- \frac{3}{2} < y < \frac{4}{3}$. In interval notation $\left(- \frac{3}{2} , \frac{4}{3}\right)$
graph{6x^2+x-12 [-40, 40, -20, 20]}
The graph also confirms above result #[Ans]<|endoftext|>
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# Scalar and Vector Quantities
Physics
## definition
### Scalar Quantities
A scalar quantity is a quantity which is defined by only magnitude. Some examples of scalar quantities are Mass, Charge, Pressure, etc.
## definition
### Vector quantities
Vector quantities are those which have both magnitude and direction and obey vector laws of addition. Some examples of vectors are displacement, velocity, force, etc.
A quantity is called a vector only if it follows all the above three conditions. For example, current is not a vector despite having both magnitude and direction because it does not follow vector laws of addition.
## result
### Scalars and Vectors
Following are some differences listed between scalars and vectors.
S.No. Scalars Vectors 1. Have only magnitude Have both magnitude and direction 2. Algebra: Same as real numbers Algebra: Follow vector laws of addition 3. Examples: Mass, charge, etc Examples: velocity, force, electric field. etc.
## definition
### Scalar
Scalar Quantities:
1. A physical quantity which is having magnitude only but not direction those are scalar quantities.
2. These are one-dimensional quantities.
3. It follows ordinary rules of algebra.
4. The scalar can be divided by any other scalar quantity.
5. It changes due to change in their magnitude only.
6. e.g. mass, work etc
## definition
### vector
Vector Quantity:
1. A vector quantity is one, that has both magnitude and direction.
2. Are multi-dimensional quantities.
3. It changes with the change in their direction or magnitude or both.
4.
5.
Two vectors can never divide.
## diagram
### Direction of vector
Vector shown in diagram can be represented as
Magnitude of vector a
a =
- component of vector along x-axis
- component of vector along y-axis
- component of vector along z- axis
- unit vectors along x,y and z direction.<|endoftext|>
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As seen in the first chapter, a string is a group of characters created by surrounding text in single or double quotes.
<?php $firstname = 'Joey'; $lastname = "Johnson";
A double quoted string can interpret special characters starting
with a back slash to create formatting. The
\n creates a newline
between the names and after them.
Double quoted strings can also embed variables in the text. This code outputs "Cindy Smith".
$firstname = 'Cindy'; echo "$firstname Smith\n";
Another feature of strings is the ability to combine them together. To combine two strings, use the period character in between them.
$firstname = 'Jenny'; $lastname = 'Madison'; $fullname = $firstname . $lastname; echo $fullname;<|endoftext|>
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The Space project day started with watching video presentations made prior the project day by partner schools in UK, Belgium and Norway as well as sharing our video.
Each of the videos contained different information crucial for a successful travel in space or colonisation of a new planet. Topics introduced where photosynthesis, resource management (recycling), energy and food and water. Each of these topics were further discussed further and how they can be a hindrance in space with the students.
In the next part of the project day LEGO Mindstrorm robots are introduced. These robots will scout for resources and information about the new planet. The students build the robots to have a sensor for obstacles and finding water. Later on, they code the robots to perform tasks like travel a certain distance, reverse and turn when close to an obstacle and sound when finding water (blue paper acted as water).
The students test the robots’ accuracy by coding the robot to travel exactly 100 cm, as accuracy in space or foreign planet may separate the expedition from life and death. This wasn’t as easy as it seemed and students needed to do this several times to get it right. Finally, a round of competition was held to find out whose robot was coded the best.
After the robots are able to find water, an electrolysis experiment is made showing how humans could create oxygen to survive in the space by using electric current to break the water atoms to oxygen and nitrogen. Solar cells are used to power the test to demonstrate how the energy could be produced.
Once the coding process is done and the equipment is working, they are placed in a green screen corner made of green cardboard and blue cardboard watery areas. Each robot is seen interacting reacting to the “water” and reversing and turning to a different ankle when getting close to the wall. This is filmed and using Google Earth to find pictures from other planets, such as Mars, and the films and pictures are combined to show how the robots might look on a new planet.
As a conclusion for the day the distances between the planets and the size of our solar system and space in general is discussed. The robot videos are shared with the partner school.<|endoftext|>
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Three sections of the Berlin Wall made of reinforced concrete from the area of the Staaken checkpoint.
These three sections of the wall are one of the largest sections of the wall outside of Berlin. They were brought back to Chatham, Kent by 38 Field Squadron November to be put on display in the Museum. The museum displays objects representing the breadth and wide ranging activities of the Kent-based regiment.
The wall symbolised the difference between the western democrats and the eastern communists and the way they thought Germany should be led. It also symbolised the inner conflict of Germany and the division between ‘free’ or democratic.
Dismantled by 38 (Berlin) Field Squadron November 1990. The wall completely cut off (by land) West Berlin from surrounding East Germany and from East Berlin until it was opened in November 1989. Its demolition officially began on 13 June 1990 and was completed in 1992. The barrier included guard towers placed along large concrete walls, which circumscribed a wide area (later known as the “death strip”) that contained anti-vehicle trenches, “fakir beds” and other defences. The Eastern Block claimed that the wall was erected to protect its population from fascist elements conspiring to prevent the “will of the people” in building a socialist state in East Germany. In practice, the Wall served to prevent the massive emigration and defection that had marked East Germany and the communist Eastern Bloc during the post-World War II period.
The Berlin Wall was officially referred to as the “Anti-Fascist Protective Wall” (German: Antifaschistischer Schutzwall) by GDR authorities, implying that the NATO countries and West Germany in particular were “fascists” by GDR propaganda. Along with the separate and much longer Inner German border (IGB), which demarcated the border between East and West Germany, it came to symbolize the “Iron Curtain” that separated Western Europe and the Eastern Bloc during the Cold War.
Before the Wall’s erection, 3.5 million East Germans circumvented Eastern Bloc emigration restrictions and defected from the GDR, many by crossing over the border from East Berlin into West Berlin; from which they could then travel to West Germany and other Western European countries. Between 1961 and 1989, the wall prevented almost all such emigration. During this period, around 5,000 people attempted to escape over the wall, with an estimated death toll ranging from 136 to more than 200 in and around Berlin. The fall of the Berlin Wall paved the way for German reunification, which was formally concluded on 3 October 1990. The part of the Berlin Wall on display in the museum is one of the largest sections of the wall outside of Berlin.
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» All Research News
February 28, 2019
» Vocal information about food availability more prevalent in the mornings
Wintering songbirds have to find food while avoiding predators. Previous research has demonstrated that birds benefit by forming groups: they use information from others to find food sources while per-capita predation risk decreases through dilution. However, much less is known about in what way birds produce information about food availability, e.g. calls which attract others. Attracting others to food decreases per-capita risk of predation, but increases competition. However, these costs and benefits do not covary linearly with group size, and the effect of recruiting an additional group member is not constant.
Friederike Hillemann, lead author on the paper, said: ‘Using a combined observational and experimental approach, we show that wintering songbirds make economic decisions about when to produce information about food availability: As the day progresses and foraging group sizes increase, the costs of producing calls that attract others outweigh the benefits, causing a decrease in vocal activity into the afternoon.’
Read the paper, published in the Proceedings of the Royal Society B, here: ‘Diurnal variation in the production of vocal information about food supports a model of social adjustment in wild songbirds‘.
May 9, 2018
» Birds migrate to save energy
Birds migrate in order to optimise the balance between their energy intake and expenditure, finds a paper published online this week in Nature Ecology & Evolution. EGI postdoctoral researcher Marius Somveille and his collaborators from Cambridge and Montpellier, found that this rule also applies to non-migratory species, and provides a general explanation for the global distribution of all birds.
Around 15% of the world’s bird species migrate between breeding and non-breeding habitats, allowing them, for example, to escape food shortages and unfavourable weather during winter months. However, identifying driving factors that are common to the movement of all migratory and non-migratory species has not been possible until now. Marius Somveille and colleagues designed a model that simulates a ‘virtual world’ in which birds distribute in an optimal fashion with regards to energy. The outputs of this model match very well the true (i.e. empirical) seasonal distribution patterns of birds, in contrast to the outputs of other ‘virtual worlds’ in which species do not optimise their annual energy balance. These results provide strong support for the important role of energy efficiency in determining the way birds distribute on the planet. The authors also suggest that the model is general enough to be applicable to other highly mobile animals such as fish and whales. Link to the paper here.
March 22, 2018
» Frontiers for Young Minds – How do birds cope with losing members of their group?
‘Frontiers for Young Minds’ is a new, free-to-publish, open-access journal which aims to make the latest scientific discoveries accessible to children. Through collaboration with a primary school teacher, the EGI has made its first contribution to this new initiative in a new article explaining how great tits in Wytham Woods cope with losing members of their group. You can read the article here. The paper explains the original manuscript (published in Proceedings of the Royal Society B) which examined how wild birds adjust their social network positions in response to experimental removal of their flockmates and the consequences of this for understanding the resilience of animal societies. You can read the original article here. For more information regarding this how to contribute to making the latest scientific research accessible to children using this new sci-comm method, please visit the Frontiers for Young Minds information page here or contact Josh Firth.
August 18, 2017
» Birds choose their neighbours based on personality
Birds of a feather nest together, according to a new study which has found that male great tits (Parus major) choose neighbours with similar personalities to their own. Oxford University researchers investigated whether the personality of birds influences their social lives – in particular who they choose to nest near. The study involved analysing social network structure in a population of wild great tits at Wytham Woods over six consecutive breeding seasons. Lead author and doctoral student Katerina Johnson explained: “We found that males, but not females, were picky about personalities, with males opting for like-minded neighbours. Our results emphasise that social interactions may play a key role in animal decisions.”
This tendency for males to associate with other males of similar personality may be particularly important during the breeding season when aggression peaks. Males fiercely defend their territories and compete for opportunities to mate with females and so shyer males may avoid setting up home near bolder, more aggressive individuals. Females, however, likely choose where to nest based on the attractive qualities of males.
The results also showed that this personality assortment amongst males was not affected by local environmental conditions. “Just like students choosing their flatmate”, Katerina commented, “birds may pay more attention to who they share their living space with than simply location.” She added: “Animal personalities can influence their social organisation and humans are likewise known to form social networks based on shared attributes including personality.”
Just like us, animals display individual behavioural differences that are consistent over time and stable across different situations and so may be thought of as personality traits. The researchers test the personality of great tits by introducing them to a novel environment and measuring how they respond. Whilst bold birds are keen to actively explore their new surroundings, shy birds tend to be more hesitant and cautious.
Katerina said: “This novel research finding may also help explain the evolution of personality and why individuals in a population differ in their behaviour. Rather than one particular personality type being favoured by natural selection as ‘the best’, different behavioural strategies may be equally good depending on who you choose to be your friends and neighbours.” Perhaps by nesting closer to others of similar character, this may improve a bird’s chances of survival and passing on their genes to the next generation. For example, although having bold neighbours may result in more skirmishes between males, they might also gain a shared benefit by more effectively repelling intruders. Link to paper here.
Here are some media links:<|endoftext|>
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Caption: The proof of the law of sines illustrated.
Proof:
1. Consider the triangle defined by the vertices/angles A, B, C.
We replace B and C by B' and C' as needed.
Note that h is a perpendicular dropped from vertex B.
2. From the sine function of trigonometry, we see that:
c*sin(A) = h = a*sin(180°-C) = a*sin(C) ,
where we have used the trig identity sin(180°-C) = sin(C).
We now immediately see that sin(A)/a = sin(C)/c.
3. The same result follows if the perpendicular is inside the triangle, mutatis mutandis, except that trig identity sin(180°-C) = sin(C) is NOT needed. The diagram illustrates this case with the triangle defined by the vertices/angles A, B', C'.
4. The result sin(A)/a = sin(C)/c is now proven in general.
5. Now note that vertices/angles A and C are general. It follows that the result extends to vertex/angle B, mutatis mutandis.
Thus, we arrive at the final result
``` sin(A) sin(B) sin(C)
------- = -------- = ------- which is the law of sines, QED.
a b c ```
The law of sines plus the law of cosines allow you to solve a triangle given three knowns (from the sides and angles) in most cases:
1. If for a triangle you are given a side and 2 angles, the triangle can be completely solved for. The third angle follows from the triangle-angles-sum-to-180° rule. The other sides follow from the law of sines.
2. If for a triangle you are given 2 sides adjacent to a given angle, then the law of cosines allows to find the third side. Using the law of cosines again, you can find a second angle and the third angle then follows from the triangle-angles-sum-to-180° rule.
3. If for a triangle you are given 2 sides and one angle which is NOT adjacent to one of the sides, the triangle may or may not be completely solved for.
There are 2 possible solutions. This arises from the fact the law of sines gives the sine of an angle, not the angle itself.
1. Say you were given A, a, and c as in the diagram. The law of sines would then give you sin(C) or sin(C'=180°-C) which are equal according to the trig identity sin(180°-C) = sin(C). Which angle C or C' is the solution that applies to the triangle?
Well both are allowed solutions if they satisfy the triangle-angles-sum-to-180° rule: i.e., A + C < 180° and A + C' = A + (180°-C) < 180°.
If both these inequalities hold, then both solutions lead to a triangle. Then B = 180°-A-C < 180° and B' = 180°-A-C' < 180° and then sin(B or B') is valid and then side b or b' can determined from the law of sines.
If the problem is specified has having only one solution, then more information is needed to determine which of the possible solutions is that one solution.
2. If one of A + C < 180° and A + C' = A + (180°-C) < 180° does NOT hold, then the other must and leads to a unique solution for the triangle.
3. Note that if you add A + C < 180° and A + (180°-C) < 180°, you get A < 90° which is a weaker condition than the 2 original ones.
If A < 90° holds, there may be 2 solutions, but there may still be a unique solution. For example, say that A = 80°, C = 5°, and C' = (180° - 5°) = 175°. In this case, B = 180° - 80° - 5° = 95° for a solution, but B' = 180° - 80° - 175° = -75° does NOT give a solution.
However, if A < 90° is violated (i.e., A ≥ 90°), then one or both of A + C < 180° and A + (180°-C) < 180° is violated and there can only be a unique solution or NO solution (e.g., with A = 179° and C = 5°).
Credit/Permission: © User:Ant.ton.t, 2013 / CC BY-SA 3.0.
Image link: Wikimedia Commons.
File: Trigonometry file: law_of_sines.html.<|endoftext|>
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Here’s a common source of confusion for computer users: the difference between memory and hard drive storage space. A reader was confused because she was getting a warning that she didn’t have enough memory to complete an action, even though she had several gigs of hard drive space free.
Memory and hard drive space are two different things.
The hard drive is where you store files and programs. When you install a program, create a document, download a picture… it is all stored on your hard drive. It’s kind of like a giant file cabinet.
Memory, on the other hand, means Random Access Memory or RAM.
Your PC uses RAM to load programs and processes. When you’re actively using a program, it’s loaded into RAM so you can speedily access it. When you’re finished with the program, it’s cleared to make room for other processes.
If your computer tells you that you don’t have enough memory, deleting files from your hard drive won’t help. If it says you don’t have enough hard drive space, removing files is the way to go.
If your computer says you don’t have enough memory, you may need more RAM or you may need to check into what’s hogging your memory. You might have programs running the background.
Try opening the Task Manager and looking under the Processes tab to see what programs are using the most memory.
Hard drive space is the amount of storage you have available to keep things long term. Memory is more like amount of space you have on your table to actually take the item out and actively use it.<|endoftext|>
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In computing, cross-platform software (also multi-platform software or platform-independent software) is computer software that is implemented on multiple computing platforms. Cross-platform software may be divided into two types; one requires individual building or compilation for each platform that it supports, and the other one can be directly run on any platform without special preparation, e.g., software written in an interpreted language or pre-compiled portable bytecode for which the interpreters or run-time packages are common or standard components of all platforms.
For example, a cross-platform application may run on Microsoft Windows on the x86 architecture, Linux on the x86 architecture and macOS on either the PowerPC or x86-based Apple Macintosh systems. Cross-platform programs may run on as many as all existing platforms, or on as few as two platforms. Cross-platform frameworks (such as Qt, Xamarin, Phonegap, or Ionic, React Native) exist to aid cross-platform development.
Platform can refer to the type of processor (CPU) or other hardware on which a given operating system or application runs, the type of operating system on a computer or the combination of the type of hardware and the type of operating system running on it. An example of a common platform is Microsoft Windows running on the x86 architecture. Other well-known desktop computer platforms include Linux/Unix and macOS - both of which are themselves cross-platform. There are, however, many devices such as smartphones that are also effectively computer platforms but less commonly thought about in that way. Application software can be written to depend on the features of a particular platform—either the hardware, operating system, or virtual machine it runs on. The Java platform is a virtual machine platform which runs on many operating systems and hardware types, and is a common platform for software to be written for.
A hardware platform can refer to an instruction set architecture. For example: x86 architecture and its variants such as IA-32 and x86-64. These machines often run one version of Microsoft Windows, though they can run other operating systems as well, including Linux, OpenBSD, NetBSD, macOS and FreeBSD.
Software platforms can either be an operating system or programming environment, though more commonly it is a combination of both. A notable exception to this is Java, which uses an operating system independent virtual machine for its compiled code, known in the world of Java as bytecode. Examples of software platforms are:
- Android for smartphones and tablet computers (x86, ARM)
- iOS (ARM)
- Microsoft Windows (x86, ARM)
- Linux (x86, PowerPC, ARM, and other architectures)
- macOS (x86, PowerPC (on 10.5 and below))
- Solaris (SPARC, x86)
- PlayStation 4 (x86), PlayStation 3 (PowerPC based) and PlayStation Vita (ARM)
- AmigaOS (m68k), AmigaOS 4 (PowerPC), AROS (x86, PowerPC, m68k), MorphOS (PowerPC)
- Atari TOS, MiNT
- BSD (many platforms; see NetBSDnet, for example)
- DOS-type systems on the x86: MS-DOS, IMB PC DOS, DR-DOS, S
- OS/2, eComStation
As previously noted, the Java platform is an exception to the general rule that an operating system is a software platform. The Java language typically compiles to a virtual machine: a virtual CPU which runs all of the code that is written for the language. This enables the same executable binary to run on all systems that implement a Java Virtual Machine (JVM). Java programs can be executed natively using a Java processor. This isn't common and is mostly used for embedded systems.
Java code running in the JVM has access to OS-related services, like disk I/O and network access, if the appropriate privileges are granted. The JVM makes the system calls on behalf of the Java application. This setup allows users to decide the appropriate protection level, depending on an ACL. For example, disk and network access is usually enabled for desktop applications, but not for browser-based applets. JNI can also be used to enable access to operating system specific functions.
Currently, Java Standard Edition programs can run on Microsoft Windows, macOS, several Unix-like operating systems, and several more non-UNIX-like operating systems like embedded systems. For mobile applications, browser plugins are used for Windows and Mac based devices, and Android has built-in support for Java. There are also subsets of Java, such as Java Card or Java Platform, Micro Edition, designed for resource-constrained devices.
For a piece of software to be considered cross-platform, it must be able to function on more than one computer architecture or operating system. Developing such a program can be a time-consuming task because different operating systems have different application programming interfaces (API). For example, Linux uses a different API for application software than Windows does.
Software written for a particular operating system does not automatically work on all architectures that operating system supports. One example as of August 2006 was OpenOffice.org, which did not natively run on the AMD64 or Intel 64 lines of processors implementing the x86-64 standards for computers; this has since been changed, and the OpenOffice.org suite of software is “mostly” ported to these 64-bit systems. This also means that just because a program is written in a popular programming language such as C or C++, it does not mean it will run on all operating systems that support that programming language—or even on the same operating system on a different architecture.
Web applications are typically described as cross-platform because, ideally, they are accessible from any of various web browsers within different operating systems. Such applications generally employ a client–server system architecture, and vary widely in complexity and functionality. This wide variability significantly complicates the goal of cross-platform capability, which is routinely at odds with the goal of advanced functionality.
Basic web applications perform all or most processing from a stateless server, and pass the result to the client web browser. All user interaction with the application consists of simple exchanges of data requests and server responses. These types of applications were the norm in the early phases of World Wide Web application development. Such applications follow a simple transaction model, identical to that of serving static web pages. Today, they are still relatively common, especially where cross-platform compatibility and simplicity are deemed more critical than advanced functionalities.
Because of the competing interests of cross-platform compatibility and advanced functionality, numerous alternative web application design strategies have emerged. Such strategies include:
- Graceful degradation
Graceful degradation attempts to provide the same or similar functionality to all users and platforms, while diminishing that functionality to a least common denominator for more limited client browsers. For example, a user attempting to use a limited-feature browser to access Gmail may notice that Gmail switches to basic mode, with reduced functionality. This differs from other cross-platform techniques, which attempt to provide equivalent functionality, not just adequate functionality, across platforms.
- Multiple codebases
Multiple codebase applications maintain distinct codebases for different (hardware and OS) platforms, with equivalent functionality. This obviously requires a duplication of effort in maintaining the code, but can be worthwhile where the amount of platform-specific code is high.
- Single codebase
This strategy relies on having one codebase that may be compiled to multiple platform-specific formats. One technique is conditional compilation. With this technique, code that is common to all platforms is not repeated. Blocks of code that are only relevant to certain platforms are made conditional, so that they are only interpreted or compiled when needed. Another technique is separation of functionality, which disables functionality not supported by client browsers or operating systems, while still delivering a complete application to the user. (See also: Separation of concerns). This technique is used in web development where interpreted code (as in scripting languages) can query the platform it is running on to execute different blocks conditionally.
- Third-party libraries
Third-party libraries attempt to simplify cross-platform capability by hiding the complexities of client differentiation behind a single, unified API.
- Responsive Web design
Responsive web design (RWD) is a Web design approach aimed at crafting the visual layout of sites to provide an optimal viewing experience—easy reading and navigation with a minimum of resizing, panning, and scrolling—across a wide range of devices, from mobile phones to desktop computer monitors. Little or no platform-specific code is used with this technique.
One complicated aspect of cross-platform web application design is the need for software testing. In addition to the complications mentioned previously, there is the additional restriction that some web browsers prohibit installation of different versions of the same browser on the same operating system. Although, there are several development approaches that companies use to target multiple platforms, all of them result in software that requires substantial manual effort for testing and maintenance across the supported platforms. Techniques such as full virtualization are sometimes used as a workaround for this problem.
Using tools such as the Page Object Model, cross platform tests can be scripted in such a way that one test case is usable for multiple versions of an app. So long as the different versions have similar user interfaces, both versions can be tested at one time, with one test case.
Web applications are becoming increasingly popular but many computer users still use traditional application software which does not rely on a client/web-server architecture. The distinction between traditional and web applications is not always clear. Features, installation methods and architectures for web and traditional applications overlap and blur the distinction. Nevertheless, this simplifying distinction is a common and useful generalization.
Traditionally in modern computing, application software has been distributed to end-users as binary file, especially executable files. Executables only support the operating system and computer architecture that they were built for—which means that making a single cross-platform executable would be something of a massive task, and is generally replaced by offering a selection of executables for the platforms supported.
For software that is distributed as a binary executable, such as software written in C or C++, the programmer must build the software for each different operating system and computer architecture, i.e. must use a toolset that translates—transcompiles—a single codebase into multiple binary executables. For example, Firefox, an open-source web browser, is available on Windows, macOS (both PowerPC and x86 through what Apple Inc. calls a Universal binary), Linux, and *BSD on multiple computer architectures. The four platforms (in this case, Windows, macOS, Linux, and *BSD) are separate executable distributions, although they come from the same source code.
The use of different toolsets to perform different builds may not be sufficient to achieve a variety of working executables for different platforms. In this case, the software engineer must port it, i.e. amend the code to be suitable to a new computer architecture or operating system. For example, a program such as Firefox, which already runs on Windows on the x86 family, can be modified and re-built to run on Linux on the x86 (and potentially other architectures) as well. The multiple versions of the code may be stored as separate codebases, or merged into one codebase by conditional compilation (see above). Note that, while porting must be accompanied by cross-platform building, the reverse is not the case.
As an alternative to porting, cross-platform virtualization allows applications compiled for one CPU and operating system to run on a system with a different CPU and/or operating system, without modification to the source code or binaries. As an example, Apple's Rosetta, which is built into Intel-based Macintosh computers, runs applications compiled for the previous generation of Macs that used PowerPC CPUs. Another example is IBM PowerVM Lx86, which allows Linux/x86 applications to run unmodified on the Linux/Power operating system.
Scripts and interpreted languages
A script can be considered to be cross-platform if its interpreter is available on multiple platforms and the script only uses the facilities provided by the language. That is, a script written in Python for a Unix-like system will likely run with little or no modification on Windows, because Python also runs on Windows; there is also more than one implementation of Python that will run the same scripts (e.g., IronPython for .NET Framework). The same goes for many of the open-source programming languages that are available and are scripting languages.
Unlike binary executable files, the same script can be used on all computers that have software to interpret the script. This is because the script is generally stored in plain text in a text file. There may be some issues, however, such as the type of new line character that sits between the lines. Generally, however, little or no work has to be done to make a script written for one system, run on another.
Some quite popular cross-platform scripting or interpreted languages are:
- bash – A Unix shell commonly run on Linux and other modern Unix-like systems, as well as on Windows via the Cygwin POSIX compatibility layer.
- Perl – A scripting language first released in 1987. Used for CGI WWW programming, small system administration tasks, and more.
- PHP – A scripting language most popular in use for web applications.
- Python – A modern scripting language where the focus is on rapid application development and ease-of-writing, instead of program run-time efficiency.
- Ruby – A scripting language whose purpose is to be object-oriented and easy to read. Can also be used on the web through Ruby on Rails.
- Tcl – A dynamic programming language, suitable for a wide range of uses, including web and desktop applications, networking, administration, testing and many more.
Cross-platform or multi-platform is a term that can also apply to video games released on a range of video game consoles, specialized computers dedicated to the task of playing games. Examples of cross-platform video games include:
The characteristics of a particular system may lengthen the time taken to implement a video game across multiple platforms. So, a video game may initially be released on a few platforms and then later released on remaining platforms. Typically, this situation occurs when a new gaming system is released, because video game developers need to acquaint themselves with the hardware and software associated with the new console.
Some games may not become cross-platform because of licensing agreements between developers and video game console manufacturers that limit development of a game to one particular console. As an example, Disney could create a game with the intention of release on the latest Nintendo and Sony game consoles. Should Disney license the game with Sony first, Disney may in exchange be required to release the game solely on Sony’s console for a short time or indefinitely—effectively prohibiting a cross-platform release for the duration.
Several developers have implemented means to play games online while using different platforms. Psyonix, Epic Games, Microsoft, and Valve Corporation all possess technology that allows Xbox 360 and PlayStation 3 gamers to play with PC gamers, leaving the decision of which platform to use to consumers. The first game to allow this level of interactivity between PC and console games was Quake 3.
Games that feature cross-platform online play include Rocket League, Final Fantasy XIV, Street Fighter V, Killer Instinct, Paragon and Fable Fortune, and Minecraft with its Better Together update on Windows 10, VR editions, Pocket Edition and Xbox One.
Cross-platform programming is the practice of actively writing software that will work on more than one platform.
Approaches to cross-platform programming
There are different ways of approaching the problem of writing a cross-platform application program. One such approach is simply to create multiple versions of the same program in different source trees—in other words, the Windows version of a program might have one set of source code files and the Macintosh version might have another, while a FOSS *nix system might have another. While this is a straightforward approach to the problem, it has the potential to be considerably more expensive in development cost, development time, or both, especially for corporate entities. The idea behind this is to create more than two different programs that have the ability to behave similarly to each other. It is also possible that this means of developing a cross-platform application will result in more problems with bug tracking and fixing, because the two different source trees would have different programmers, and thus different defects in each version.
Another approach that is used is to depend on pre-existing software that hides the differences between the platforms—called abstraction of the platform—such that the program itself is unaware of the platform it is running on. It could be said that such programs are platform agnostic. Programs that run on the Java Virtual Machine (JVM) are built in this fashion.
Cross-platform programming toolkits and environments
- 8th: A cross-platform development language, which utilizes Juce as its GUI layer. The platforms it currently supports are: Android, iOS, Windows, macOS, Linux and Raspberry Pi.
- Anant Computing: A mobile application platform that works in all Indian languages, including their keyboards, which is also supports AppWallet and Native performance inside all operating systems.
- AppearIQ: A framework that supports the workflow of app development and deployment in an enterprise environment. Natively developed containers present hardware features of the mobile devices or tablets through an API to HTML5 code thus facilitating the development of mobile apps that run on different platforms.
- Cairo: A free software library used to provide a vector graphics-based, device-independent API. It is designed to provide primitives for 2-dimensional drawing across a number of different backends. Cairo is written in C and has bindings for many programming languages.
- Cocos2d: An open source toolkit and game engine for developing 2D and simple 3D cross-platform games and applications.
- Delphi: A cross platform IDE, which uses Pascal language for Development. Currently it supports Android, iOS, Windows, macOS.
- Ecere SDK: A cross platform GUI & 2D/3D graphics toolkit and IDE, written in eC and with support for additional languages such as C and Python. Currently it supports Linux, FreeBSD, Windows, Android, macOS and the Web through Emscripten or Binaryen (WebAssembly)
- Eclipse: An open source cross-platform development environment. Implemented in Java with a configurable architecture which supports many tools for software development. Add-ons are available for several languages, including Java and C++.
- FLTK: Another open source cross platform toolkit, but more lightweight because it restricts itself to the GUI.
- fpGUI: An open source widget toolkit that is completely implemented in Object Pascal. It currently supports Linux, Windows and a bit of Windows CE.
- GeneXus: A Windows rapid software development solution for cross-platform application creation and deployment based on knowledge representation and supporting C#, COBOL, Java including Android and BlackBerry smart devices, Objective-C for Apple mobile devices, RPG, Ruby, Visual Basic, and Visual FoxPro.
- GLBasic: A BASIC dialoect and compiler that generates C++ code. It includes cross compilers for many platforms and supports numerous platform (Windows, Mac, Linux, Android,iOS and some exotic handhelds).
- GTK+: An open source widget toolkit for Unix-like systems with X11 and Microsoft Windows.
- Haxe: An open source cross-platform language.
- Juce: An application framework written in C++, used to write native software on numerous systems (Microsoft Windows, POSIX, macOS), with no change to the code.
- Lazarus: A programming environment for the FreePascal Compiler. It supports the creation of self-standing graphical and console applications and runs on Linux, MacOSX, iOS, Android, WinCE, Windows and WEB.
- Max/MSP: A visual programming language that encapsulates platform-independent code with a platform-specific runtime environment into applications for macOS and Windows.
- MechDome: A cross-platform Android runtime. It allows unmodified Android apps to run natively on iOS and macOS
- Mendix: A cloud based low-code application development platform.
- MonoCross: An open-source model-view-controller design pattern where the model and controller are shared cross-platform but the view is platform-specific.
- Mono: An open-source cross-platform version of Microsoft .NET (a framework for applications and programming languages)
- MoSync: An open-source SDK for mobile platform app development in the C++ family
- Mozilla application framework: An open source platform for building macOS, Windows and Linux applications
- OpenGL: A cross-platform 3D graphics library.
- PureBasic: A proprietary cross-platform language and IDE for building macOS, Windows and Linux applications
- Qt: An application framework and widget toolkit for Unix-like systems with X11, Microsoft Windows, macOS, and other systems—available under both open source and proprietary licenses.
- Simple and Fast Multimedia Library: A multimedia C++ API that provides low and high level access to graphics, input, audio, etc.
- Simple DirectMedia Layer: An open-source cross-platform multimedia library written in C that creates an abstraction over various platforms’ graphics, sound, and input APIs. It runs on many operating systems including Linux, Windows and macOS and is aimed at games and multimedia applications.
- Ultimate++: A C++ cross-platform rapid application development framework focused on programmers productivity. It includes a set of libraries (GUI, SQL, etc..), and an integrated development environment. It supports Windows and Unix-like OS-s. The U++ competes with popular scripting languages while preserving C/C++ runtime characteristics. It has its own integrated development environment, TheIDE, which features BLITZ-build technology to speedup C++ rebuilds up to 4 times.
- Unity: Another cross-platform SDK which uses Unity Engine.
- Unreal: A cross-platform SDK which uses Unreal Engine.
- V-Play Engine: V-Play is a cross-platform development SDK based on the popular Qt framework. V-Play apps and games are created within Qt Creator.
- WaveMaker: A Cross-platform low-code development tool to create responsive web and hybrid mobile (Android & iOS) applications.
- WinDev: Integrated Development Environment for Windows, Linux, .Net and Java (also with support for Internet and Intranet)
- wxWidgets: An open source widget toolkit that is also an application framework. It runs on Unix-like systems with X11, Microsoft Windows and macOS. It permits applications written to use it to run on all of the systems that it supports, if the application does not use any operating system-specific programming in addition to it.
- Xojo: A RAD IDE developed by Xojo, Inc. that uses an object-oriented programming language to create desktop, web and iOS apps. Xojo makes native, compiled desktop apps for macOS, Windows, Linux and Raspberry Pi. It creates compiled web apps that can be run as standalone servers or through CGI. And it recently added the ability to create native iOS apps.
Challenges to cross-platform development
There are certain issues associated with cross-platform development. Some of these include:
- Testing cross-platform applications may be considerably more complicated, since different platforms can exhibit slightly different behaviors or subtle bugs. This problem has led some developers to deride cross-platform development as "write once, debug everywhere", a take on Sun Microsystems' "write once, run anywhere" marketing slogan.
- Developers are often restricted to using the lowest common denominator subset of features which are available on all platforms. This may hinder the application's performance or prohibit developers from using the most advanced features of each platform.
- Different platforms often have different user interface conventions, which cross-platform applications do not always accommodate. For example, applications developed for macOS and GNOME are supposed to place the most important button on the right-hand side of a window or dialog, whereas Microsoft Windows and KDE have the opposite convention. Though many of these differences are subtle, a cross-platform application which does not conform appropriately to these conventions may feel clunky or alien to the user. When working quickly, such opposing conventions may even result in data loss, such as in a dialog box confirming whether the user wants to save or discard changes to a file.
- Scripting languages and virtual machines must be translated into native executable code each time the application is executed, imposing a performance penalty. This penalty can be alleviated using advanced techniques like just-in-time compilation; but even using such techniques, some computational overhead may be unavoidable.
- Different platforms require the use of native package formats such as RPM and MSI. Multi-platform installers such as InstallAnywhere address this need.
- Cross-platform execution environments may suffer cross-platform security flaws, creating a fertile environment for cross-platform malware.
- "Design Guidelines: Glossary". java.sun.com. Retrieved 2011-10-19.
- "Encyclopedia > cross platform". PC Magazine Encyclopedia. Retrieved 2011-10-19.
- Lee P Richardson (2016-02-16). "Xamarin vs Ionic: A Mobile, Cross Platform, Shootout".
- "Platform Definition". The Linux Information Project. Retrieved 2014-03-27.
- "About Mono". mono-project.com. Retrieved 2015-12-17.
- Porting to x86-64 (AMD64, EM64T) – Apache OpenOffice Wiki. Wiki.services.openoffice.org (2012-06-22). Retrieved on 2013-07-17.
- Corti, Sascha P. (October 2011). "Browser and Feature Detection". MSDN Magazine. Retrieved 28 January 2014.
- Choudhary, S.R. (2014). "Cross-platform testing and maintenance of web and mobile applications". Companion Proceedings of the 36th International Conference on Software Engineering - ICSE Companion 2014.
- Cross Platform Mobile Testing with the Page Object Model
- Cribba. Quake III Arena, Giant Bombcast, February 15, 2013.
- The GUI Toolkit, Framework Page
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In social psychology, a stereotype is an over-generalized belief about a particular category of people. Stereotypes are generalized because one assumes that the stereotype is true for each individual person in the category. While such generalizations may be useful when making quick decisions, they may be erroneous when applied to particular individuals. Stereotypes encourage prejudice and may arise for a number of reasons.
Explicit stereotypes are those people who are willing to verbalize and admit to other individuals. It also refers to stereotypes that one is aware that one holds, and is aware that one is using to judge people. People can attempt to consciously control the use of explicit stereotypes, even though their attempt to control may not be fully effective.
Only males play video games is a common stereotype. In fact, almost half of all gamers are female, when including mobile phone gaming. Women are more likely to play mobile phone games than traditional video games.
Implicit stereotypes are those that lay on individuals' subconsciousness, that they have no control or awareness of.
In social psychology, a stereotype is any thought widely adopted about specific types of individuals or certain ways of behaving intended to represent the entire group of those individuals or behaviors as a whole. These thoughts or beliefs may or may not accurately reflect reality. Within psychology and across other disciplines, different conceptualizations and theories of stereotyping exist, at times sharing commonalities, as well as containing contradictory elements.
The term stereotype comes from the French adjective stéréotype and derives from the Greek words στερεός (stereos), "firm, solid" and τύπος (typos), impression, hence "solid impression on one or more idea/theory."
The term comes from the printing trade and was first adopted in 1798 by Firmin Didot to describe a printing plate that duplicated any typography. The duplicate printing plate, or the stereotype, is used for printing instead of the original.
Outside of printing, the first reference to "stereotype" was in 1850, as a noun that meant image perpetuated without change. However, it was not until 1922 that "stereotype" was first used in the modern psychological sense by American journalist Walter Lippmann in his work Public Opinion.
Stereotypes, prejudice, and discrimination are understood as related but different concepts. Stereotypes are regarded as the most cognitive component and often occurs without conscious awareness, whereas prejudice is the affective component of stereotyping and discrimination is one of the behavioral components of prejudicial reactions. In this tripartite view of intergroup attitudes, stereotypes reflect expectations and beliefs about the characteristics of members of groups perceived as different from one's own, prejudice represents the emotional response, and discrimination refers to actions.
Although related, the three concepts can exist independently of each other. According to Daniel Katz and Kenneth Braly, stereotyping leads to racial prejudice when people emotionally react to the name of a group, ascribe characteristics to members of that group, and then evaluate those characteristics.
Possible prejudicial effects of stereotypes are:
Stereotype content refers to the attributes that people think characterize a group. Studies of stereotype content examine what people think of others, rather than the reasons and mechanisms involved in stereotyping.
Early theories of stereotype content proposed by social psychologists such as Gordon Allport assumed that stereotypes of outgroups reflected uniform antipathy. For instance, Katz and Braly argued in their classic 1933 study that ethnic stereotypes were uniformly negative.
By contrast, a newer model of stereotype content theorizes that stereotypes are frequently ambivalent and vary along two dimensions: warmth and competence. Warmth and competence are respectively predicted by lack of competition and status. Groups that do not compete with the in-group for the same resources (e.g., college space) are perceived as warm, whereas high-status (e.g., economically or educationally successful) groups are considered competent. The groups within each of the four combinations of high and low levels of warmth and competence elicit distinct emotions. The model explains the phenomenon that some out-groups are admired but disliked, whereas others are liked but disrespected. This model was empirically tested on a variety of national and international samples and was found to reliably predict stereotype content.
Early studies suggested that stereotypes were only used by rigid, repressed, and authoritarian people. This idea has been refuted by contemporary studies that suggest the ubiquity of stereotypes and it was suggested to regard stereotypes as collective group beliefs, meaning that people who belong to the same social group share the same set of stereotypes. Modern research asserts that full understanding of stereotypes requires considering them from two complementary perspectives: as shared within a particular culture/subculture and as formed in the mind of an individual person.
Stereotyping can serve cognitive functions on an interpersonal level, and social functions on an intergroup level. For stereotyping to function on an intergroup level (see social identity approaches: social identity theory and self-categorization theory), an individual must see themselves as part of a group and being part of that group must also be salient for the individual.
Craig McGarty, Russell Spears, and Vincent Y. Yzerbyt (2002) argued that the cognitive functions of stereotyping are best understood in relation to its social functions, and vice versa.
Stereotypes can help make sense of the world. They are a form of categorization that helps to simplify and systematize information. Thus, information is more easily identified, recalled, predicted, and reacted to. Stereotypes are categories of objects or people. Between stereotypes, objects or people are as different from each other as possible. Within stereotypes, objects or people are as similar to each other as possible.
Gordon Allport has suggested possible answers to why people find it easier to understand categorized information. First, people can consult a category to identify response patterns. Second, categorized information is more specific than non-categorized information, as categorization accentuates properties that are shared by all members of a group. Third, people can readily describe objects in a category because objects in the same category have distinct characteristics. Finally, people can take for granted the characteristics of a particular category because the category itself may be an arbitrary grouping.
A complementary perspective theorizes how stereotypes function as time- and energy-savers that allow people to act more efficiently. Yet another perspective suggests that stereotypes are people's biased perceptions of their social contexts. In this view, people use stereotypes as shortcuts to make sense of their social contexts, and this makes a person's task of understanding his or her world less cognitively demanding.
In the following situations, the overarching purpose of stereotyping is for people to put their collective self (their in-group membership) in a positive light:
As mentioned previously, stereotypes can be used to explain social events. Henri Tajfel described his observations of how some people found that the anti-Semitic contents of The Protocols of the Elders of Zion only made sense if Jews have certain characteristics. Therefore, according to Tajfel, Jews were stereotyped as being evil and yearning for world domination to match the anti-Semitic ‘facts’ as presented in The Protocols of the Elders of Zion.
People create stereotypes of an outgroup to justify the actions that their in-group has committed (or plans to commit) towards that outgroup. For example, according to Tajfel, Europeans stereotyped Turkish, Indian, and Chinese people as being incapable of achieving financial advances without European help. This stereotype was used to justify European colonialism in Turkey, India, and China.
An assumption is that people want their ingroup to have a positive image relative to outgroups, and so people want to differentiate their ingroup from relevant outgroups in a desirable way. If an outgroup does not affect the ingroup’s image, then from an image preservation point of view, there is no point for the ingroup to be positively distinct from that outgroup.
People can actively create certain images for relevant outgroups by stereotyping. People do so when they see that their ingroup is no longer as clearly and/or as positively differentiated from relevant outgroups, and they want to restore the intergroup differentiation to a state that favours the ingroup.
Stereotypes can emphasize a person’s group membership in two steps: Stereotypes emphasize the person’s similarities with ingroup members on relevant dimensions, and also the person’s differences from outgroup members on relevant dimensions. People change the stereotype of their ingroups and outgroups to suit context. Once an outgroup treats an ingroup member badly, they are more drawn to the members of their own group. This can be seen as members with in a group are able to relate to each other though a stereotype because of identical situations. A person can embrace a stereotype to avoid humiliation such as failing a task and blaming it on a stereotype.
Stereotypes are an indicator of ingroup consensus. When there are intragroup disagreements over stereotypes of the ingroup and/or outgroups, ingroup members take collective action to prevent other ingroup members from diverging from each other.
John C. Turner proposed in 1987 that if ingroup members disagree on an outgroup stereotype, then one of three possible collective actions follow: First, ingroup members may negotiate with each other and conclude that they have different outgroup stereotypes because they are stereotyping different subgroups of an outgroup (e.g., Russian gymnasts versus Russian boxers). Second, ingroup members may negotiate with each other, but conclude that they are disagreeing because of categorical differences amongst themselves. Accordingly, in this context, it is better to categorise ingroup members under different categories (e.g., Democrats versus Republican) than under a shared category (e.g., American). Finally, ingroup members may influence each other to arrive at a common outgroup stereotype.
Different disciplines give different accounts of how stereotypes develop: Psychologists may focus on an individual's experience with groups, patterns of communication about those groups, and intergroup conflict. As for sociologists, they may focus on the relations among different groups in a social structure. They suggest that stereotypes are the result of conflict, poor parenting, and inadequate mental and emotional development. Once stereotypes have formed, there are two main factors that explain their persistence. First, the cognitive effects of schematic processing (see schema) make it so that when a member of a group behaves as we expect, the behavior confirms and even strengthens existing stereotypes. Second, the affective or emotional aspects of prejudice render logical arguments against stereotypes ineffective in countering the power of emotional responses.
Correspondence bias refers to the tendency to ascribe a person's behavior to disposition or personality, and to underestimate the extent to which situational factors elicited the behavior. Correspondence bias can play an important role in stereotype formation.
For example, in a study by Roguer and Yzerbyt (1999) participants watched a video showing students who were randomly instructed to find arguments either for or against euthanasia. The students that argued in favor of euthanasia came from the same law department or from different departments. Results showed that participants attributed the students' responses to their attitudes although it had been made clear in the video that students had no choice about their position. Participants reported that group membership, i.e., the department that the students belonged to, affected the students' opinions about euthanasia. Law students were perceived to be more in favor of euthanasia than students from different departments despite the fact that a pretest had revealed that subjects had no preexisting expectations about attitudes toward euthanasia and the department that students belong to. The attribution error created the new stereotype that law students are more likely to support euthanasia.
Nier et al. (2012) found that people who tend to draw dispositional inferences from behavior and ignore situational constraints are more likely to stereotype low-status groups as incompetent and high-status groups as competent. Participants listened to descriptions of two fictitious groups of Pacific Islanders, one of which was described as being higher in status than the other. In a second study, subjects rated actual groups – the poor and wealthy, women and men – in the United States in terms of their competence. Subjects who scored high on the measure of correspondence bias stereotyped the poor, women, and the fictitious lower-status Pacific Islanders as incompetent whereas they stereotyped the wealthy, men, and the high-status Pacific Islanders as competent. The correspondence bias was a significant predictor of stereotyping even after controlling for other measures that have been linked to beliefs about low status groups, the just-world hypothesis and social dominance orientation.
Research has shown that stereotypes can develop based on a cognitive mechanism known as illusory correlation – an erroneous inference about the relationship between two events. If two statistically infrequent events co-occur, observers overestimate the frequency of co-occurrence of these events. The underlying reason is that rare, infrequent events are distinctive and salient and, when paired, become even more so. The heightened salience results in more attention and more effective encoding, which strengthens the belief that the events are correlated.
In the intergroup context, illusory correlations lead people to misattribute rare behaviors or traits at higher rates to minority group members than to majority groups, even when both display the same proportion of the behaviors or traits. Black people, for instance, are a minority group in the United States and interaction with blacks is a relatively infrequent event for an average white American. Similarly, undesirable behavior (e.g. crime) is statistically less frequent than desirable behavior. Since both events "blackness" and "undesirable behavior" are distinctive in the sense that they are infrequent, the combination of the two leads observers to overestimate the rate of co-occurrence. Similarly, in workplaces where women are underrepresented and negative behaviors such as errors occur less frequently than positive behaviors, women become more strongly associated with mistakes than men.
In a landmark study, David Hamilton and Richard Gifford (1976) examined the role of illusory correlation in stereotype formation. Subjects were instructed to read descriptions of behaviors performed by members of groups A and B. Negative behaviors outnumbered positive actions and group B was smaller than group A, making negative behaviors and membership in group B relatively infrequent and distinctive. Participants were then asked who had performed a set of actions: a person of group A or group B. Results showed that subjects overestimated the frequency with which both distinctive events, membership in group B and negative behavior, co-occurred, and evaluated group B more negatively. This despite the fact the proportion of positive to negative behaviors was equivalent for both groups and that there was no actual correlation between group membership and behaviors. Although Hamilton and Gifford found a similar effect for positive behaviors as the infrequent events, a meta-analytic review of studies showed that illusory correlation effects are stronger when the infrequent, distinctive information is negative.
Hamilton and Gifford's distinctiveness-based explanation of stereotype formation was subsequently extended. A 1994 study by McConnell, Sherman, and Hamilton found that people formed stereotypes based on information that was not distinctive at the time of presentation, but was considered distinctive at the time of judgement. Once a person judges non-distinctive information in memory to be distinctive, that information is re-encoded and re-represented as if it had been distinctive when it was first processed.
One explanation for why stereotypes are shared is that they are the result of a common environment that stimulates people to react in the same way.
The problem with the ‘common environment’ is that explanation in general is that it does not explain how shared stereotypes can occur without direct stimuli. Research since the 1930s suggested that people are highly similar with each other in how they describe different racial and national groups, although those people have no personal experience with the groups they are describing.
Another explanation says that people are socialised to adopt the same stereotypes. Some psychologists believe that although stereotypes can be absorbed at any age, stereotypes are usually acquired in early childhood under the influence of parents, teachers, peers, and the media.
If stereotypes are defined by social values, then stereotypes only change as per changes in social values. The suggestion that stereotype content depend on social values reflects Walter Lippman's argument in his 1922 publication that stereotypes are rigid because they cannot be changed at will.
Studies emerging since the 1940s refuted the suggestion that stereotype contents cannot be changed at will. Those studies suggested that one group’s stereotype of another group would become more or less positive depending on whether their intergroup relationship had improved or degraded. Intergroup events (e.g., World War Two, Persian Gulf conflict) often changed intergroup relationships. For example, after WWII, Black American students held a more negative stereotype of people from countries that were the United States’s WWII enemies. If there are no changes to an intergroup relationship, then relevant stereotypes do not change.
According to a third explanation, shared stereotypes are neither caused by the coincidence of common stimuli, nor by socialisation. This explanation posits that stereotypes are shared because group members are motivated to behave in certain ways, and stereotypes reflect those behaviours. It is important to note from this explanation that stereotypes are the consequence, not the cause, of intergroup relations. This explanation assumes that when it is important for people to acknowledge both their ingroup and outgroup, they will emphasise their difference from outgroup members, and their similarity to ingroup members.
The dual-process model of cognitive processing of stereotypes asserts that automatic activation of stereotypes is followed by a controlled processing stage, during which an individual may choose to disregard or ignore the stereotyped information that has been brought to mind.
A number of studies have found that stereotypes are activated automatically. Patricia Devine (1989), for example, suggested that stereotypes are automatically activated in the presence of a member (or some symbolic equivalent) of a stereotyped group and that the unintentional activation of the stereotype is equally strong for high- and low-prejudice persons. Words related to the cultural stereotype of blacks were presented subliminally. During an ostensibly unrelated impression-formation task, subjects read a paragraph describing a race-unspecified target person's behaviors and rated the target person on several trait scales. Results showed that participants who received a high proportion of racial words rated the target person in the story as significantly more hostile than participants who were presented with a lower proportion of words related to the stereotype. This effect held true for both high- and low-prejudice subjects (as measured by the Modern Racism Scale). Thus, the racial stereotype was activated even for low-prejudice individuals who did not personally endorse it. Studies using alternative priming methods have shown that the activation of gender and age stereotypes can also be automatic.
Subsequent research suggested that the relation between category activation and stereotype activation was more complex. Lepore and Brown (1997), for instance, noted that the words used in Devine's study were both neutral category labels (e.g., "Blacks") and stereotypic attributes (e.g., "lazy"). They argued that if only the neutral category labels were presented, people high and low in prejudice would respond differently. In a design similar to Devine's, Lepore and Brown primed the category of African-Americans using labels such as "blacks" and "West Indians" and then assessed the differential activation of the associated stereotype in the subsequent impression-formation task. They found that high-prejudice participants increased their ratings of the target person on the negative stereotypic dimensions and decreased them on the positive dimension whereas low-prejudice subjects tended in the opposite direction. The results suggest that the level of prejudice and stereotype endorsement affects people's judgements when the category – and not the stereotype per se – is primed.
Research has shown that people can be trained to activate counterstereotypic information and thereby reduce the automatic activation of negative stereotypes. In a study by Kawakami et al. (2000), for example, participants were presented with a category label and taught to respond "No" to stereotypic traits and "Yes" to nonstereotypic traits. After this training period, subjects showed reduced stereotype activation. This effect is based on the learning of new and more positive stereotypes rather than the negation of already existing ones.
Empirical evidence suggests that stereotype activation can automatically influence social behavior. For example, Bargh, Chen, and Burrows (1996) activated the stereotype of the elderly among half of their participants by administering a scrambled-sentence test where participants saw words related to age stereotypes. Subjects primed with the stereotype walked significantly more slowly than the control group (although the test did not include any words specifically referring to slowness), thus acting in a way that the stereotype suggests that elderly people will act. In another experiment, Bargh, Chen, and Burrows also found that because the stereotype about blacks includes the notion of aggression, subliminal exposure to black faces increased the likelihood that randomly selected white college students reacted with more aggression and hostility than participants who subconsciously viewed a white face. Similarly, Correll et al. (2002) showed that activated stereotypes about blacks can influence people's behavior. In a series of experiments, black and white participants played a video game, in which a black or white person was shown holding a gun or a harmless object (e.g., a mobile phone). Participants had to decide as quickly as possible whether to shoot the target. When the target person was armed, both black and white participants were faster in deciding to shoot the target when he was black than when he was white. When the target was unarmed, the participants avoided shooting him more quickly when he was white. Time pressure made the shooter bias even more pronounced.
Stereotypes can be efficient shortcuts and sense-making tools. They can, however, keep people from processing new or unexpected information about each individual, thus biasing the impression formation process. Early researchers believed that stereotypes were inaccurate representations of reality. A series of pioneering studies in the 1930s found no empirical support for widely held racial stereotypes. By the mid-1950s, Gordon Allport wrote that, "It is possible for a stereotype to grow in defiance of all evidence."
Research on the role of illusory correlations in the formation of stereotypes suggests that stereotypes can develop because of incorrect inferences about the relationship between two events (e.g., membership in a social group and bad or good attributes). This means that at least some stereotypes are inaccurate.
Empirical social science research shows that stereotypes are often accurate. Jussim et al. reviewed four studies concerning racial and seven studies that examined gender stereotypes about demographic characteristics, academic achievement, personality and behavior. Based on that, the authors argued that some aspects of ethnic and gender stereotypes are accurate while stereotypes concerning political affiliation and nationality are much less accurate. A study by Terracciano et al. also found that stereotypic beliefs about nationality do not reflect the actual personality traits of people from different cultures.
Attributive ambiguity refers to the uncertainty that members of stereotyped groups experience in interpreting the causes of others' behavior toward them. Stereotyped individuals who receive negative feedback can attribute it either to personal shortcomings, such as lack of ability or poor effort, or the evaluator's stereotypes and prejudice toward their social group. Alternatively, positive feedback can either be attributed to personal merit or discounted as a form of sympathy or pity.
Crocker et al. (1991) showed that when black participants were evaluated by a white person who was aware of their race, black subjects mistrusted the feedback, attributing negative feedback to the evaluator's stereotypes and positive feedback to the evaluator's desire to appear unbiased. When the black participants' race was unknown to the evaluator, they were more accepting of the feedback.
Attributional ambiguity has been shown to affect a person's self-esteem. When they receive positive evaluations, stereotyped individuals are uncertain of whether they really deserved their success and, consequently, they find it difficult to take credit for their achievements. In the case of negative feedback, ambiguity has been shown to have a protective effect on self-esteem as it allows people to assign blame to external causes. Some studies, however, have found that this effect only holds when stereotyped individuals can be absolutely certain that their negative outcomes are due to the evaluators's prejudice. If any room for uncertainty remains, stereotyped individuals tend to blame themselves.
Attributional ambiguity can also make it difficult to assess one's skills because performance-related evaluations are mistrusted or discounted. Moreover, it can lead to the belief that one's efforts are not directly linked to the outcomes, thereby depressing one's motivation to succeed.
Stereotype threat occurs when people are aware of a negative stereotype about their social group and experience anxiety or concern that they might confirm the stereotype. Stereotype threat has been shown to undermine performance in a variety of domains.
Claude M. Steele and Joshua Aronson conducted the first experiments showing that stereotype threat can depress intellectual performance on standardized tests. In one study, they found that black college students performed worse than white students on a verbal test when the task was framed as a measure of intelligence. When it was not presented in that manner, the performance gap narrowed. Subsequent experiments showed that framing the test as diagnostic of intellectual ability made black students more aware of negative stereotypes about their group, which in turn impaired their performance. Stereotype threat effects have been demonstrated for an array of social groups in many different arenas, including not only academics but also sports, chess and business.
Not only has stereotype threat been widely criticized by on a theoretical basis, but has failed several attempts to replicate its experimental evidence. The findings in support of the concept have been suggested by multiple methodological reviews to be the product of publication bias.
Stereotypes lead people to expect certain actions from members of social groups. These stereotype-based expectations may lead to self-fulfilling prophecies, in which one's inaccurate expectations about a person's behavior, through social interaction, prompt that person to act in stereotype-consistent ways, thus confirming one's erroneous expectations and validating the stereotype.
Word, Zanna, and Cooper (1974) demonstrated the effects of stereotypes in the context of a job interview. White participants interviewed black and white subjects who, prior to the experiments, had been trained to act in a standardized manner. Analysis of the videotaped interviews showed that black job applicants were treated differently: They received shorter amounts of interview time and less eye contact; interviewers made more speech errors (e.g., stutters, sentence incompletions, incoherent sounds) and physically distanced themselves from black applicants. In a second experiment, trained interviewers were instructed to treat applicants, all of whom were white, like the whites or blacks had been treated in the first experiment. As a result, applicants treated like the blacks of the first experiment behaved in a more nervous manner and received more negative performance ratings than interviewees receiving the treatment previously afforded to whites.
A 1977 study by Snyder, Tanke, and Berscheid found a similar pattern in social interactions between men and women. Male undergraduate students were asked to talk to female undergraduates, whom they believed to be physically attractive or unattractive, on the phone. The conversations were taped and analysis showed that men who thought that they were talking to an attractive woman communicated in a more positive and friendlier manner than men who believed that they were talking to unattractive women. This altered the women's behavior: Female subjects who, unknowingly to them, were perceived to be physically attractive behaved in a friendly, likeable, and sociable manner in comparison with subjects who were regarded as unattractive.
A 2005 study by J. Thomas Kellow and Brett D. Jones looked at the effects of self-fulfilling prophecy on African American and Caucasian high school freshman students. Both white and black students were informed that their test performance would be predictive of their performance on a statewide, high stakes standardized test. They were also told that historically, white students had outperformed black students on the test. This knowledge created a self-fulfilling prophecy in both the white and black students, where the white students scored statistically significantly higher than the African American students on the test. The stereotype threat of underperforming on standardized tests effected the African American students in this study.
Because stereotypes simplify and justify social reality, they have potentially powerful effects on how people perceive and treat one another. As a result, stereotypes can lead to discrimination in labor markets and other domains. For example, Tilcsik (2011) has found that employers who seek job applicants with stereotypically male heterosexual traits are particularly likely to engage in discrimination against gay men, suggesting that discrimination on the basis of sexual orientation is partly rooted in specific stereotypes and that these stereotypes loom large in many labor markets. Agerström and Rooth (2011) showed that automatic obesity stereotypes captured by the Implicit Association Test can predict real hiring discrimination against the obese. Similarly, experiments suggest that gender stereotypes play an important role in judgments that affect hiring decisions.
Stereotypes can affect self-evaluations and lead to self-stereotyping. For instance, Correll (2001, 2004) found that specific stereotypes (e.g., the stereotype that women have lower mathematical ability) affect women's and men's evaluations of their abilities (e.g., in math and science), such that men assess their own task ability higher than women performing at the same level. Similarly, a study by Sinclair et al. (2006) has shown that Asian American women rated their math ability more favorably when their ethnicity and the relevant stereotype that Asian Americans excel in math was made salient. In contrast, they rated their math ability less favorably when their gender and the corresponding stereotype of women's inferior math skills was made salient. Sinclair et al. found, however, that the effect of stereotypes on self-evaluations is mediated by the degree to which close people in someone's life endorse these stereotypes. People's self-stereotyping can increase or decrease depending on whether close others view them in stereotype-consistent or inconsistent manner.
Stereotyping can also play a central role in depression, when people have negative self-stereotypes about themselves, according to Cox, Abramson, Devine, and Hollon (2012). This depression that is caused by prejudice (i.e., "deprejudice") can be related to group membership (e.g., Me–Gay–Bad) or not (e.g., Me–Bad). If someone holds prejudicial beliefs about a stigmatized group and then becomes a member of that group, they may internalize their prejudice and develop depression. People may also show prejudice internalization through self-stereotyping because of negative childhood experiences such as verbal and physical abuse.
Stereotypes are traditional and familiar symbol clusters, expressing a more or less complex idea in a convenient way. They are often simplistic pronouncements about gender, racial, ethnic, and cultural backgrounds and they can become a source of misinformation and delusion. For example, in a school when students are confronted with the task of writing a theme, they think in terms of literary associations, often using stereotypes picked up from books, films, and magazines that they have read or viewed.
The danger in stereotyping lies not in its existence, but in the fact that it can become a substitute for observation and a misinterpretation of a cultural identity. Promoting information literacy is a pedagogical approach that can effectively combat the entrenchment of stereotypes. The necessity for using information literacy to separate multicultural "fact from fiction" is well illustrated with examples from literature and media.
Stereotypes are common in various cultural media, where they take the form of dramatic stock characters. The instantly recognizable nature of stereotypes mean that they are effective in advertising and situation comedy. Alexander Fedorov (2015) proposed a concept of media stereotypes analysis. This concept refers to identification and analysis of stereotypical images of people, ideas, events, stories, themes and etc. in media context.
The characters that do appear in movies greatly effect how people worldwide perceive gender relations, race, and cultural communities. Because approximately 85% of worldwide ticket sales are directed toward Hollywood movies, the American movie industry has been greatly responsible for portraying characters of different cultures and diversity to fit into stereotypical categories. This has led to the spread and persistence of gender, racial, ethnic, and cultural stereotypes seen in the movies.
For example, Russians are usually portrayed as ruthless agents, brutal mobsters and villains in Hollywood movies. The portrayals of Latin Americans in film and print media are restricted to a narrow set of characters. Latin Americans are largely depicted as sexualized figures such as the Latino macho or the Latina vixen, gang members, (illegal) immigrants, or entertainers. By comparison, they are rarely portrayed as working professionals, business leaders or politicians.
In Hollywood films, there are several Latin American stereotypes that have historically been used. Some examples are El Bandido, the Halfbreed Harlot, The Male Buffoon, The Female Clown, The Latin Lover, The Dark Lady, The Wise Old Man, and The Poor Peon. Many hispanic characters in hollywood films consists of one or more of these basic stereotypes, but it has been rare to view Latin American actors representing characters outside of this stereotypical criteria.
Media stereotypes of women first emerged in the early 20th century. Various stereotypic depictions or "types" of women appeared in magazines, including Victorian ideals of femininity, the New Woman, the Gibson Girl, the Femme fatale, and the Flapper.
Stereotypes are also common in video games, with women being portrayed as stereotypes such as the "damsel in distress" or as sexual objects (see Gender representation in video games). Studies show that minorities are portrayed most often in stereotypical roles such as athletes and gangsters (see Racial representations in video games).
In literature and art, stereotypes are clichéd or predictable characters or situations. Throughout history, storytellers have drawn from stereotypical characters and situations to immediately connect the audience with new tales.
Examples of stereotypes
Beatnik was a media stereotype prevalent throughout the 1950s to mid-1960s that displayed the more superficial aspects of the Beat Generation literary movement of the 1950s. Elements of the beatnik trope included pseudo-intellectualism, drug use, and a cartoonish depiction of real-life people along with the spiritual quest of Jack Kerouac's autobiographical fiction.Blonde stereotype
Blonde stereotypes are stereotypes of blond-haired people, especially women. Sub-types include the "blonde bombshell" and the "dumb blonde". Blondes are differently stereotyped from brunettes as more desirable and less intelligent. There are many blonde jokes made on these premises. Although chiefly aimed at women, jokes of this style have also been aimed at similar stereotypes associated with men, such as the "dumb jock" and the "surfer dude".Chav
"Chav" ( CHAV) ("charver" in parts of Northern England) is a pejorative epithet used in the United Kingdom to describe a particular stereotype of anti-social youth dressed in sportswear. The word was popularised in the 2000s by the British mass media to refer to an anti-social youth subculture in the UK. The Oxford English Dictionary defines "chav" as an informal British derogatory, meaning "a young lower-class person who displays brash and loutish behaviour and wears real or imitation designer clothes". The derivative chavette has been used to refer to females, and the adjectives chavvy, chavvish and chavtastic have been used in relation to items designed for or suitable for use by chavs.Criminal stereotype of African Americans
The criminal stereotype of African Americans in the United States is an ethnic stereotype according to which African American males in particular are stereotyped to be dangerous criminals. The figure of the African-American man as criminal has appeared frequently in American popular culture and has been associated with consequences in the justice system such as racial profiling and harsher sentences for African American defendants in trials.Dragon Lady
A Dragon Lady is usually a stereotype of East Asian and occasionally South Asian and Southeast Asian women as strong, deceitful, domineering, or mysterious. The term's origin and usage is Western, not Chinese. Inspired by the characters played by actress Anna May Wong, the term comes from the female villain in the comic strip Terry and the Pirates. It has since been applied to powerful Asian women and to a number of racially Asian film actresses. The stereotype has generated a large quantity of sociological literature. "Dragon Lady" is sometimes applied to persons who lived before the term became part of American slang in the 1930s. It is also used to refer to any powerful but prickly woman, usually in a derogatory fashion.Ethnic and national stereotypes
An ethnic stereotype (national stereotype, or national character) is a system of beliefs about typical characteristics of members of a given ethnic group or nationality, their status, society and cultural norms.
National stereotypes may be either about ones' own ethnicity/nationality or about a foreign or differing nationality or ethnicity. Stereotypes about ones' own nation may aid in maintaining a national identity due to a collective relatability to a trait or characteristic.Gender role
A gender role, also known as a sex role, is a social role encompassing a range of behaviors and attitudes that are generally considered acceptable, appropriate, or desirable for people based on their actual or perceived sex. Gender roles are usually centered on conceptions of femininity and masculinity, although there are exceptions and variations. The specifics regarding these gendered expectations may vary substantially among cultures, while other characteristics may be common throughout a range of cultures. There is ongoing debate as to what extent gender roles and their variations are biologically determined, and to what extent they are socially constructed.
Various groups, most notably the feminist movement, have led efforts to change aspects of prevailing gender roles that they believe are oppressive or inaccurate.
The term gender role was first used by John Money and colleagues in 1954, during the course of his study of intersex individuals, to describe the manners in which these individuals expressed their status as a male or female in a situation where no clear biological assignment existed.Hillbilly
"Hillbilly" is a term (often derogatory) for people who dwell in rural, mountainous areas in the United States, primarily in southern Appalachia and the Ozarks.
The first known instances of "hillbilly" in print were in The Railroad Trainmen's Journal (vol. ix, July 1892), an 1899 photograph of men and women in West Virginia labeled "Camp Hillbilly", and a 1900 New York Journal article containing the definition: "a Hill-Billie is a free and untrammeled white citizen of Alabama, who lives in the hills, has no means to speak of, dresses as he can, talks as he pleases, drinks whiskey when he gets it, and fires off his revolver as the fancy takes him". The stereotype is twofold in that it incorporates both positive and negative traits: "Hillbillies" are often considered independent and self-reliant individuals who resist the modernization of society, but at the same time they are also defined as backward and violent. Scholars argue this duality is reflective of the split ethnic identities in white America.Jock (stereotype)
In the United States and Canada, a jock is a stereotype of an athlete, or someone who is primarily interested in sports and sports culture, and does not take much interest in intellectual culture. It is generally attributed mostly to high school and college athletics participants who form a distinct youth subculture. As a blanket term, jock can be considered synonymous with athlete. Jocks are usually presented as practitioners of team sports such as football, lacrosse, basketball, baseball, soccer and hockey.
Similar words that may mean the same as jock (in North America) include meathead, musclebrain, and musclehead. These terms are based on the stereotype that a jock is muscular but not very smart, and cannot carry a conversation on any topic other than one relating to sports and exercise.
"Jock" is also a derogatory word used by the English towards Scottish men.LGBT stereotypes
Lesbian, gay, bisexual and transgender (LGBT) stereotypes are conventional, formulaic generalizations, opinions, or images based on the sexual orientations or gender identities of LGBT people. Stereotypical perceptions may be acquired through interactions with parents, teachers, peers and mass media, or, more generally, through a lack of firsthand familiarity, resulting in an increased reliance on generalizations.Negative stereotypes are often associated with homophobia, lesbophobia, biphobia, or transphobia. Positive stereotypes, or counterstereotypes, also exist.Model minority
A model minority is a demographic group (whether based on ethnicity, race or religion) whose members are perceived to achieve a higher degree of socioeconomic success than the population average. This success is typically measured relatively by income, education, low criminality and high family/marital stability.The concept is controversial, as it has historically been used to suggest there is no need for government action to adjust for socioeconomic disparities between certain groups. This argument has most often been applied to contrast Asian Americans (both South & East Asians) against African Americans & Hispanic Americans in America, enforcing the idea that Asian Americans are good law-abiding, productive immigrants/citizens while promoting the stereotype that Hispanics and African Americans are criminally prone, welfare recipient immigrants/citizens.Generalized statistics are often cited to back up model minority status such as high educational achievement and a high representation in white-collar professions. A common misconception is that the affected communities usually hold pride in their labeling as the model minority. The model minority stereotype is considered detrimental to relevant minority communities because it is used to justify the exclusion of minorities in the distribution of assistance programs, both public and private, as well as to understate or slight the achievements of individuals within that minority. Furthermore, the idea of the model minority pits minority groups against each other by implying that non-model groups are at fault for falling short of the model minority level of achievement and assimilation. The concept has also been criticized by outlets such as NPR for potentially homogenizing the experiences of Asian Americans on one side and Hispanics & African Americans on the other, despite the different groups experiencing racism in different ways. The model minority stereotype, and the perpetuation of the belief that any minority has the capability to rise economically without assistance, also completely ignores the very different history of Asian Americans and African Americans, and sometimes Hispanics, in the U.S. Beginning with the legalized and widespread slavery of Africans that were kidnapped from Africa, then continuing with Black Codes, Jim Crow, and the prison–industrial complex.The concept of "model minority" is heavily associated with U.S. culture and is not extensively used outside the U.S., though many European countries have concepts of classism that stereotype ethnic groups in a similar manner to model minority.Nerd
A nerd is a person seen as overly intellectual, obsessive, introverted or lacking social skills. Such a person may spend inordinate amounts of time on unpopular, little known, or non-mainstream activities, which are generally either highly technical, abstract, or relating to topics of science fiction or fantasy, to the exclusion of more mainstream activities. Additionally, many so-called nerds are described as being shy, quirky, pedantic, and unattractive.Originally derogatory, the term "nerd" was a stereotype, but as with other pejoratives, it has been reclaimed and redefined by some as a term of pride and group identity.Redneck
Redneck is a derogatory term chiefly but not exclusively applied to white Americans perceived to be crass and unsophisticated, closely associated with rural whites of the Southern United States. Its usage is similar in meaning to cracker (especially regarding Texas, Georgia, and Florida), hillbilly (especially regarding Appalachia and the Ozarks), and white trash (but without the last term's suggestions of immorality).By the 1970s, the term had become offensive slang, its meaning expanded to include racism, loutishness, and opposition to modern ways.Patrick Huber, in his monograph A Short History of Redneck: The Fashioning of a Southern White Masculine Identity, emphasized the theme of masculinity in the 20th-century expansion of the term, noting, "The redneck has been stereotyped in the media and popular culture as a poor, dirty, uneducated, and racist Southern white man."Smith space
In functional analysis and related areas of mathematics, Smith space is a complete compactly generated locally convex space having a compact set which absorbs every other compact set (i.e. for some ).
Smith spaces are named after Marianne Ruth Freundlich Smith, who introduced them as duals to Banach spaces in some versions of duality theory for topological vector spaces. All Smith spaces are stereotype and are in the stereotype duality relations with Banach spaces:
In functional analysis and related areas of mathematics, stereotype spaces are topological vector spaces defined by a special variant of reflexivity condition. They form a class of spaces with a series of remarkable properties, in particular, this class is very wide (for instance, it contains all Fréchet spaces and thus, all Banach spaces), it consists of spaces satisfying a natural condition of completeness, and it forms a *-autonomous category with the standard analytical tools for constructing new spaces, like taking dual spaces, spaces of operators, tensor products, and in addition, immediate subspaces, immediate quotient spaces, products and coproducts, limits and colimits, etc.Stereotype threat
Stereotype threat is a situational predicament in which people are or feel themselves to be at risk of conforming to stereotypes about their social group. Stereotype threat is purportedly a contributing factor to long-standing racial and gender gaps in academic performance. It may occur whenever an individual's performance might confirm a negative stereotype because stereotype threat is thought to arise from a particular situation, rather than from an individual's personality traits or characteristics. Since most people have at least one social identity which is negatively stereotyped, most people are vulnerable to stereotype threat if they encounter a situation in which the stereotype is relevant. Situational factors that increase stereotype threat can include the difficulty of the task, the belief that the task measures their abilities, and the relevance of the stereotype to the task. Individuals show higher degrees of stereotype threat on tasks they wish to perform well on and when they identify strongly with the stereotyped group. These effects are also increased when they expect discrimination due to their identification with a negatively stereotyped group. Repeated experiences of stereotype threat can lead to a vicious circle of diminished confidence, poor performance, and loss of interest in the relevant area of achievement.Since its introduction into the academic literature, stereotype threat has become one of the most widely studied topics in the field of social psychology.
Stereotype threat has been argued to show a reduction in the performance of individuals who belong to negatively stereotyped groups. According to the theory, if negative stereotypes are present regarding a specific group, group members are likely to become anxious about their performance, which may hinder their ability to perform to their full potential. Importantly, the individual does not need to subscribe to the stereotype for it to be activated. It is hypothesized that the mechanism through which anxiety (induced by the activation of the stereotype) decreases performance is by depleting working memory (especially the phonological aspects of the working memory system).However, studies have cautioned that stereotype threat should not be interpreted as a factor in real-world performance gaps. Multiple reviews has raised concerns that the effect has been over-estimated for schoolgirls and that the field likely suffers from publication bias.The opposite of stereotype threat is stereotype boost, which is when people perform better than they otherwise would have, because of exposure to positive stereotypes about their social group. A variant of stereotype boost is stereotype lift, which is people achieving better performance because of exposure to negative stereotypes about other social groups.Stereotypes of African Americans
Generalizations and stereotypes of African Americans and their culture have evolved within American society dating back to the colonial years of settlement, particularly after slavery became a racial institution that was heritable.
A comprehensive examination of the restrictions imposed upon African-Americans in the United States of America through culture is examined by art historian Guy C. McElroy in the catalog to the exhibit "Facing History: The Black Image in American Art 1710-1940." According to McElroy, the artistic convention of representing African-Americans as less than fully realized humans began with Justus Engelhardt Kühn's colonial era painting Henry Darnall III as a child. Although Kühn's work existed "simultaneously with a radically different tradition in colonial America" as indicated by the work of portraitists such as Charles (or Carolus) Zechel, (see Portrait of a Negro Girl and Portrait of a Negro boy) the market demand for such work reflected the attitudes and economic status of their audience.
From the colonial era through the American Revolution ideas about African-Americans were variously used in propaganda either for or against the issue of slavery. Paintings like John Singleton Copley's Watson and the Shark (1778) and Samuel Jennings' Liberty Displaying the Arts and Sciences (1792) are early examples of the debate underway at that time as to the role of Black people in America. Watson represents an historical event, while Liberty is indicative of abolitionist sentiments expressed in Philadelphia's post revolutionary intellectual community. Nevertheless, Jennings' painting represents African-Americans as passive, submissive beneficiaries of not only slavery's abolition, but knowledge, which liberty has graciously bestowed upon them.
As a stereotypical caricature "performed by white men disguised in facial paint, minstrelsy relegated black people to sharply defined dehumanizing roles." With the success of T. D. Rice and Daniel Emmet the label of "blacks as buffoons" was created. One of the earliest versions of the "black as buffoon" can be seen in John Lewis Krimmel's Quilting Frolic. The violinist in the 1813 painting, with his tattered and patched clothing, along with a bottle protruding from his coat pocket, appears to be an early model for Rice's Jim Crow character. Krimmel's representation of a "[s]habbily dressed" fiddler and serving girl with "toothy smile" and "oversized red lips" marks him as "...one of the first American artists to utilize physiognomical distortions as a basic element in the depiction of African-Americans."Stereotypes of Jews
Stereotypes of Jews are generalized representations of Jews, often caricatured and of a prejudiced and antisemitic nature. The Jewish diaspora have been stereotyped for over 2,000 years as scapegoats for a multitude of societal problems such as: Jews always acting with unforgiving hostility towards the Christians, Jews' religious rituals thought to have specifically undermined the church and state, and Jews' habitual assassinations of Christians as their most extreme deeds. Antisemitism continued throughout the centuries and reached a climax in the Third Reich during World War II. Modern-day Jews are still stereotyped as greedy, nit-picky, stingy misers and are often depicted in caricatures, comics, and propaganda posters counting money or collecting diamonds. Early films such as Cohen's Advertising Scheme (1904, silent) stereotyped Jews as "scheming merchants".Common objects, phrases and traditions used to emphasize or ridicule Jewishness include bagels, playing violin, klezmer, undergoing circumcision, kvetching, haggling and uttering various Yiddish phrases like mazel tov, shalom, and oy vey. Other Jewish stereotypes are the rabbi, the complaining and guilt-inflicting Jewish mother, often along with a meek and nerdy nice Jewish boy, and the spoiled and materialistic Jewish-American princess.Watermelon stereotype
The watermelon stereotype is a stereotype of African Americans that states that African Americans have an unusually great appetite for watermelons. This stereotype has remained prevalent into the 21st century.
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# Complex integration
### Basic concepts and principles
One of the most important ways to get involved in complex variable analysis is through complex integration. When we talk about complex integration we refer to the line integral.
Line integral definition begins with γ a differentiable curve such that
$$\begin{matrix}\gamma : [a,b] \mapsto \mathbb{C}\\ \;\;\;\;\; \;\;\;\;\;\;\; x \mapsto \gamma(x) \end{matrix}$$
Now we split the interval [a, b] in n parts zi such that z0a, and zn=b
For each subinterval we take $E_{i}=f(\zeta_{i})(z_{i}-z_{i-1}), \; i=1,..,n$.
Then we take partial sums $\sum_{i=1}^{n}E_{i} = \sum_{i=1}^{n}f(\zeta_{i})(z_{i}-z_{i-1})$. Making the limit when n tends to infinity we get the line integral as $$\int_{a}^{b}f(z)dz \;\;, \;\; \int_{C}f(z)dz$$
Both two formulas are analogous
The complex integral over a C curve is defined as
$\int_{C}f(z)dz = \int_{C}(u+iv)(dx+idy)$ $= \int_{C}udx -vdy + i\int_{C}vdx -udy$.
A very interesting property of the integral and that is used in most of proofs and arguments is the follwing
$$\left | \int_{a}^{b }f(z)dz \right | \le \int_{a}^{b }\left |f(z) \right |dz$$ Click here to see a proof of this fact.
### Line integral definitionº
Given f, a complex variable function and γ a piecewise-differentiable curve. We define the line integral of f over γ as: $$\int_{\gamma}f(z)dz = \int_{a}^{b}f(\gamma(t))\gamma'(t)dt$$
### Extended theory
The most important therorem called Cauchy's Theorem which states that integral over a closed and simple curve is zero on simply connected domains. Cauchy gave a first demonstration by supposing that function f has a continuous first derivative, later Eduard Gousart discovered that this hypothesis was actually redundant, for this reason Cauchy's theorem is sometimes called Cauchy-Gousart's Theorem. This will be the version that we will see here.
In the following theorems, C is a closed and simple curve and contained in a simply connected open R region (this is a domain).
### Theorem of Cauchy - Gousart
Given f a, holomorphic function over R, then $$\int_{C}f(z)dz = 0$$
Click here to see a proof of Cauchy's theorem.
### Green's Theorem in the plane
Let P and Q be continuous functions and with continuous partial derivatives in R and on their boundary C. Then
$\int_{C} P dx+ Q dy$ $= \int\int_{R}[\frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}]dx dy$
It is relatively simple to put Green's theorem in complex form:
### Green's theorem in complex form
Given F, with continuous partial derivatives in R and on their boundary C. Then
$\int_{C}F(z, \bar{z})dz$ $= 2i\int\int_{R}\frac{\partial F}{\partial \bar{z}}dA$
Following Theorem is sometimes called reciprocal of the Cauchy's theorem:
### Morera's Theorem
Given f, a complex variable function, lets suposse that it verifies $$\int_{C}f(z)dz = 0$$ then, f is Holomorphic over R.
### Consequences of the Cauchy's theorem
The following theorems are consequences of Cauchy's theorem
### Theorem 1
If a and b are two points of R then the integral $$\int_{a}^{b} f(z) dz$$ It is independent of the path followed between a and b.
The proof of this theorem is simple, it is enough to observe if C is any path between a and b and C' is another different path, then for Cauchy's theorem, the whole integral between C and C' is zero, as the path C 'does not matter, both line integrals are in fact equals.
### Theorem 2
Lets a and b two points of R and F'(z) = f(z) then $$\int_{a}^{b} f(z) dz = F(b) - F(a)$$
Reciprocally, if a and z are points of R and it is fulfilled
$$F(z) = \int_{a}^{z} f(z) dz$$
then, F is Holomorphic in R and F'(z) = f(z)
The following theorem very important, it talk about that the value of an integral over a closed and simple curve that surrounds a singularity does not depend on the curve:
### Theorem 3
Given f a function Holomorphic in a region bounded by two closed and simple curves C and C' $$\int_{C} f(z) dz = \int_{C'} f(z) dz$$ Where C and C' are traversed positively oriented, therefore counterclockwise.
The following theorem is an extension of the previous one to regions with n singularities instead two.
Figura 1: Region closed between curves C and C'
### Teorema 4
Given f a function Holomorphic in a region bounded by nclosed and simple curves $C_{1}, C_{2}, ..., C_{n}$. Which are enclosed by another larger curve C. Then.
$\int_{C} f(z) dz = \int_{C_{1}} f(z) dz + \int_{C_{2}} f(z) dz +$ $... + \int_{C_{n}} f(z) dz$
Where the curves curves $C_{1}, C_{2}, ..., C_{n}$ are traversed positively oriented, that is, counterclockwise.
Figure 2: Region enclosed between C and $C_{1}, C_{2}, ..., C_{n}$ curves.
## Was useful? want add anything?
Post here
### kamaljitsingh gill:
2019-10-21 14:33:23
the theorem together make things clear for understanding .The content is explained nicely<|endoftext|>
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Humans are limited. We can’t fly, breathe underwater, or survive in high temperatures. So we’ve left much of our scientific endeavors to instruments that extend and collect information beyond our senses, like seismometers that detect waves moving through Earth, lidar instruments that can measure elevation, and sensors that can determine the salinity of the oceans.
Increasingly, a new technology is furthering scientists’ ability to capture Earth’s vast, unreachable spaces: drones. These remote-controlled vehicles—sometimes called unmanned aerial vehicles (UAV) or autonomous underwater vehicles (AUV)—can travel to hazardous places humans try to avoid, like up close to a volcanic eruption or underneath meters of sea ice. Now more than ever, robots are scattered across the planet, engaged in some heavy science.
This week marks the 20th annual Earth Science Week, founded by the American Geosciences Institute. This year’s theme, “Earth and Human Activity,” emphasizes the ways in which humans affect Earth and the ways Earth affects humans. In celebration of Earth Science Week, we want to highlight some of the many ways drones help us understand our planet and the changes we make to it.
Here are some of the ways scientist use drones.
1. To Find Sources of Pollution
Scientists have invented an instrument that when affixed to a drone, can help them detect and find the source of air pollution at ultrafine scales. The instrument uses spectroscopy to pinpoint the composition of a gas in a given area and precisely measure its concentration. Applications range from understanding dynamics at the boundaries of atmospheric layers to detecting chemical weapons.
2. To Study Why Climate Change Affects the Arctic So Severely
Scientists working on Alaska’s North Slope are using drones to monitor gas concentrations in three dimensions over the Arctic. Some drones have even dropped buoys into the surface ocean. Combined, these instruments help researchers study the physics of the transition region between frozen sea and open water. This transition is a crucial element of climate models in the Arctic.
3. To Peer into Volcanoes
Researchers used drones to peer down the throat of one of the world’s most active volcanoes: Stromboli, in Italy. The drone flew through clouds of ash plumes above the erupting volcano and snapped high-resolution pictures and videos of Stromboli’s active and nonactive vents.
4. To Show that Typical Ground Truths Overestimate Earth’s Albedo
Researchers use satellites to study how Earth reflects and absorbs light. They rely on on-the-ground weather stations to calibrate their measurements. However, some drone-operating scientists recently found that these weather stations may be overestimating how much light gets reflected because they don’t take into account certain surrounding topography. The drones helped to show that in some cases, albedo in certain places can be overestimated by up to 10%.
5. To Explore the Expanse of Sea Ice Beyond the Decks of Their Icebreaker Ships
Researchers working in the Arctic and Antarctic have piloted drones to altitudes of 500 meters over the ice. Images collected helped scientists study the distribution of ice floes, mechanisms of pancake ice formation, and ocean wave–sea ice interactions. Drones in these environments can provide crucial information on how sea ice recovers in autumn, a time when harsh conditions hinder many research expeditions.
6. To Study Propagating Cracks in Glaciers
In 2015, researchers used a drone to monitor a crack that had sprouted in Greenland’s Bowdoin Glacier. Over several days, the drone captured thousands of images, allowing researchers to model the crack’s propagation. Someday, these kinds of data may allow scientists to predict when an iceberg might calve from a glacier.
7. To See an Entire Floodplain from the Sky
Scientists are employing drones to help them study the geologic remnants of the Lake Missoula floods, in which more than 2000 cubic kilometers of water burst from a glacial dam and flowed across the Pacific Northwest as many as 100 times between 18,000 and 13,000 years ago. The drones survey wide swaths of land, searching for large-scale ripple marks and flow scars—features too vast to see from the ground.
8. To Help Detect Gas Leaks Underwater
Researchers have tested a method that can help AUVs pinpoint dangerous gas emissions from underwater sources, such as from oil wells or stored natural gas. The idea goes like this: When broad surveys detect some form of underwater leak, researchers deploy a drone to find it. Then, each location where the drone fails to detect the origin of the leak will update a map so that it shows regions where the drone has the next highest probability of finding the leak.
9. To Map the Bottom of Highly Acidic Lakes
Across Earth, there are 35 volcanic lakes that can build up gas to the point where waters sometimes explode. In 2016, scientists traveled to Laguna Caliente in Costa Rica to test out an autonomous underwater vehicle—basically, a drone that can swim.
The rugged AUV survived in 55°C water that was 3 times more acidic than battery acid. Unfortunately, on its second venture into the lake, the AUV succumbed to an explosion that sent acidic waves over its hull. The team has since built a new drone.
10. To Visit Under-Ice Ecosystems
In 2014, some scientists tested an AUV in the ocean near Greenland, sending it to examine the under-ice ecosystem. The drone revealed a world that surprised the researchers—high amounts of algae and other biology that likely supports a diverse food web under the ice. Researchers hope that a better understanding of the existing ecosystems will help them understand how climate change may alter those ecosystems.
11. To Help Track Underwater Avalanches
In 2015, researchers deployed AUVs to help them place beach ball–shaped benthic event detectors on the seafloor in Monterey, Calif. The balls are engineered to roll with material that slides down continental slopes. Through these sensors, scientists hope to understand how quickly and how far sediment flows during underwater avalanches.
12. To Find Shipwrecks
In 2014, National Oceanic and Atmospheric Administration scientists sent an AUV equipped with sonar capabilities to investigate an ancient shipwreck about 5 kilometers off the coast of Southeast Farallon Island. Five years prior, a routine sonar survey of that patch of ocean floor revealed a previously overlooked shipwreck. The drone, along with another remotely operated vehicle that took pictures and videos, discovered that the shipwreck was the long-missing USS Conestoga, which researchers had been searching for since it vanished 95 years ago.
13. To Detect Ever-So-Slight Changes in Earth’s Elevation
Drones in the Jet Propulsion Laboratory’s Uninhabited Aerial Vehicle Synthetic Aperture Radar program help researchers study Earth’s surface. The autopiloted crafts are programmed to fly the same path again and again, using radar signals to gather information about the surface. When these signals are compared to one other, scientists can see how the land has changed between flights. These craft can detect land subsidence or even tiny changes in elevation after an earthquake.
—JoAnna Wendel (@JoAnnaScience), Staff Writer<|endoftext|>
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Black Hawk, born Ma-ka-tai-me-she-kia-kiak in 1767 in the village of Saukenuk (present-day Rock Island, Ill.) on the Rock River. Black Hawk's father Pyesa was the tribal medicine man of the Sauk people. At age 15, Black Hawk accompanied his father on a raid against the Osage. He won approval by killing and scalping his first enemy. The young Black Hawk then tried to establish himself as a war captain by leading other raids. He had limited success until, at age 19, he led 200 men in a battle against the Osage, in which he personally killed five men and one woman. Soon after, he joined his father in a raid against the Cherokee along the Meramec River in Missouri. After Pyesa died from wounds received in the battle, Black Hawk inherited the Sauk medicine bundle which his father had carried, giving him an important role in the tribe. He did not belong to a clan that provided the Sauk with hereditary chiefs. He achieved status through his exploits as a warrior and by leading successful raiding parties.
During the War of 1812, Black Hawk had fought on the side of the British against the U.S., hoping to push white American settlers away from Sauk territory. Later he led a band of Sauk and Fox warriors, known as the “British Band,” against white settlers in Illinois and present-day Wisconsin in the 1832 Black Hawk War. The war stretched from April to August 1832, with a number of battles, skirmishes and massacres on both sides. Black Hawk led his men in another conflict, the Battle of Wisconsin Heights. Afterward, the Illinois and Michigan Territory militias caught up with Black Hawk's "British Band" for the final confrontation at Bad Axe. At the mouth of the Bad Axe River, pursuing soldiers, their Indian allies, and a U.S. gunboat killed hundreds of Sauk and Potawatomi men, women and children.
On 27 August 1832, Black Hawk asked to surrender to the Indian agent Joseph Street but was instead taken to Gen. Zachary Taylor. He surrendered to Lt. Jefferson Davis, future president of the Confederacy. After the war, Black Hawk lived with the Sauk along the Iowa River and later the Des Moines River near Iowaville. Black Hawk died on 3 October 1838 after two weeks of illness. He was buried on the farm of his friend James Jordan on the north bank of the Des Moines River in Davis County, Iowa.
(Id. No. 2140: displacement 13,500; length 404'6"; beam 53'9"; draft 28'5"; speed 13 knots; complement 442; armament 4 5-inch, 1 3-inch)
The steamship Santa Catalina – launched in 1913 by William Cramp & Sons Ship & Engine Building Co., Philadelphia, Pa. -- was purchased by the Navy on 3 December 1917 and, given the identification number (Id. No.) 2140, was renamed Black Hawk on 26 December, the day after Christmas of 1917. Converted into a tender and repair ship at Fletcher’s Shipyard, Hoboken, N.J., Black Hawk was commissioned there on 15 May 1918, Cmdr. Roscoe C. Bulmer in command.
Black Hawk was assigned as tender and flagship to Mine Squadron One on 22 April 1918, also assigned to the squadron that day were the minelayers Aroostook (Id. No. 1256) and Shawmut (Id. No. 1255). The ship departed New York on 12 June and arrived at President Roads, Boston, Mass., on 14 June. Black Hawk departed Boston with Shawmut, Aroostook, and Saranac (Id. No. 1702) to take station for mining duties in Scotland on 16 June 1918, during the German submarines' activity on the New England coast. Uncompleted work had not delayed them like the other ships of the force, but the trial runs of Shawmut and Aroostook had shown their fuel consumption to be much larger than had been estimated, no data having been available when their conversion was planned. This made their fuel capacity insufficient for the passage to Europe. Indefinite delay, until a tanker could accompany them, was averted by the ships’ captains. By expeditious management the three mine planters, together with Black Hawk, were able to sail in company on 16 June. The only oil hose obtainable quickly was of 4-inch diameter, nearly twice as heavy as that ordinarily used for fueling at sea. The first fueling was done in a gale of wind, and it was a novel undertaking for all concerned. Yet it was successfully accomplished. The second time fueling was done it was easier; and without further noteworthy incident the detachment arrived in Scotland during the evening on 29 June, just before departure on the second excursion to lay the North Sea Mine Barrage.
The Navy Department designated the base at Inverness as Base No. 18, and the one at Invergordon as Base No. 17. The two mine bases were so organized that there were two executive officers, representatives of the commanding officer, in complete charge of all administrative and industrial activities at then: respective bases. Each base was organized with military, industrial, supply, medical, and transportation departments. The work of preparing and outfitting the mine bases was done by contract through the Admiralty. The construction work was done through the controller's department of the Admiralty. Rear Adm. Lewis Clinton-Baker, RN, was the Admiralty's representative in general charge of the work. The intent for all U.S. Naval forces in European Waters was to be self-supporting. In the case of the U.S. Mine Squadron, as a result of the initial supply of mines and regular replenishment by the mine carriers, the force had to draw on the British stocks for very little. With Black Hawk’s arrival this became even less. As of 10 July, Black Hawk served as the flagship for Rear Adm. Joseph Strauss, Commander, Mine Force, Atlantic Fleet. She remained moored off Inverness, and did not take part in minelaying, but her equipment of machine tools and repair material made the Mine Force largely independent in regard to upkeep. Except for docking, U.S. minelayers asked very little of the British in the way of repairs.
After receiving a re-supply of Mk. VI mines, the U.S. squadron got underway on 14 July 1918, and laid 5,395 mines the following day in 4 hours and 22 minutes, the largest number so far laid in a single operation. At 4.20 a.m. on 16 July, while just north of Cromarty Firth, one of the escorting destroyers sheered close in to the squadron’s flagship, San Francisco (Cruiser No. 5), and reported that they were too close inshore. The squadron turned out, stopped and backed, but before headway had been checked Roanoke (Id. No. 1695) and Canonicus (Id. No. 1696) had grounded. Canonicus was able to back off, but attempts to clear Roanoke proved unavailing until she was lightened as much as possible. She came off easily on the following high tide. In light of the fact that neither vessel sustained any damage, the Commander, Mine Force, recommended no further proceedings and the matter was disposed of by Vice Adm. William S. Sims, Commander, U.S. Naval Forces in European Waters in a letter.
The fifth British operation was carried out on 21 July 1918, in Area C. Several days delay was encountered before the fourth U.S. operation on account of again having to wait for mining materiel. The squadron was reported ready to sail on 25 July, but it was necessary to wait four days more for the escorting and supporting forces from the Grand Fleet. The British and U.S. operations had recently been overlapping each other in such a manner that one squadron was out at sea while the other was loading in port. This required keeping a large part of the Grand Fleet at sea almost constantly, the Commander-in-Chief, therefore, desired that the U.S. squadron should wait until the British squadron had again loaded, so that it would only be necessary to send one force to support both squadrons.
The U.S. squadron sailed on 29 July 1918, laying 5,399 mines the following day. The premature explosions, much more numerous than on any of the previous excursions -- approximately 14% of the mines going off – proved most disconcerting. Instead of the explosions decreasing as experience was gained in the assembly and laying of the mines, the percentage had been gradually increasing and then had suddenly jumped to 14% on this excursion. Losses of 3-4% could be tolerated, but this latter figure was prohibitive, and the causes of the explosions had to be determined and eliminated. Due to the large number of premature explosions which occurred in the fourth operation, the Force Commander ordered the suspension of further minelaying operations until the cause of the explosions had been ascertained and corrected. The next excursion, a joint effort by the British and U.S. squadrons began on 8 August. The efforts to cure the premature explosions on this excursion were found even less successful than before; approximately 19% of the mines had exploded. After laying 1,596 mines, the operation was discontinued and the squadron returned to the bases. It was found that the rubber insulation between the copper plates on the firing device caused a slight current in the firing circuit in the direction necessary to operate it. Although the current was in most cases small, there was a possibility that if it were eliminated the mines would then have sufficient stability so as not to explode after they had been planted. In order to carry out the practical part of the experiments after the theoretical tests had been completed at the bases, San Francisco proceeded to the mine field on 12 August. The improvement obtained in this test was sufficient to enable minelaying to be resumed. The squadron sailed on the sixth excursion on 18 August, and the minelaying was completed on the 19th. The squadron got underway for the seventh excursion on 26 August, and stood out toward the minefield. Saranac (Id. No. 1702) broke down shortly after leaving the base and had to return to Inverness with her full cargo of mines. San Francisco and the remaining eight ships, however, continued and carried out the operation. Unfortunately, dense fog was encountered practically throughout the operation; so thick at times that it was impossible for the vessels to see the next ship abeam, distant only 500 yards.
The eighth excursion was intended as a surprise. Neutral nations had not been notified that Area B was dangerous to shipping, and with this knowledge, enemy U-boats were constantly passing through it on their way to the Atlantic. It was accordingly decided not to notify neutrals about the area, but to secretly route all shipping so as to avoid it, with the hope that U-boats might still attempt to use it after it had been mined. In order to prevent the enemy observing the mining while it was in progress, an elaborate patrol was arranged, beginning the day before the operation and continuing until after its completion. British and U.S. mining squadrons rendezvoused off the Orkney Islands on 7 September and proceeded to carry out the operation. The U.S. laid six lines of surface mines across Area B, while the British laid one line of surface mines parallel. This was really the first joint operation carried out by the British and U.S. squadrons. On several previous occasions both squadrons had been at sea at the same time, but had not been working side by side, so as to necessitate appointing one officer to command the expedition. On this occasion, Rear Adm. Joseph Strauss, embarked on board San Francisco, was designated to take general charge of both squadrons while mining was in progress.
In the early morning of 20 September 1918, while the U.S. mining squadron was on its way to the minefield to carry out the ninth excursion, a submarine was sighted off Stronsay Firth. She was immediately attacked with depth charges by the escorting destroyers, and at the same time a smoke screen was laid by both the escort and the minelayers. Shortly afterwards, she was again sighted just ahead of San Francisco, and was again attacked. The squadron proceeded through Westray Firth and then to a position about 6 miles to the northward of the western end of the minefield which was laid on 7 September. In this excursion, 5,520 mines were laid in 3 hours and 50 minutes the record number laid by a minelaying force in a single operation. At the same time, the British squadron laid 1,300 mines in a single line parallel and to the northward of those laid by us. Rear Adm. Strauss, on board San Francisco, was in command of the American minelayers while Rear Adm. Lewis Clinton-Baker, CB, RN, commanded the combined forces. The firing devices had been adjusted and there was a reduction of premature explosions on this excursion, being between 5-6%, a marked improvement.
On 27 September 1918, 5,450 mines were laid, slightly over 4% of which exploded prematurely. Only nine of the mine layers, Baltimore (Cruiser No. 3) having returned to the U.S., took part in this operation. The eleventh operation was carried out on 4 October, again in Area A, and approximately 6% of the mines exploded prematurely. The U.S. mining squadron completed the twelfth excursion on 13 October, losing 4% by premature explosions. Roanoke and Canandaigua (Id. No. 1694) proceeded to Newcastle for docking upon the completion of the operation. Eight days' delay was encountered before the thirteenth and last operation could begin. On account of the sequence of the British and U.S. operations in Areas A and C, it had been impractical to extend the minefields so as to overlap each other. This left a gap between the two areas approximately 6 miles wide. In order to close this, the next excursion was planned to consist of six rows of surface mines to the southward of the gap, continuing with two rows into Area C, so as to complete the four rows which the U.S. squadron had agreed to lay in this area. The first of the winter weather was encountered in this operation, when it was necessary for the squadron to wait one day after having reached the mine field before the sea moderated sufficiently to enable the mines to be laid. Even then the ships were rolling as much as 20º to 30º on each side of the vertical. This provided an excellent test of the mining installations with the result that no difficulties were encountered by any of the ships, either in the stowing of their mines or in the actual planting under such severe conditions. The operation was completed 26 October, having laid 3,760 mines, of which slightly over 4% were lost by premature explosions.
Although the U.S. mining squadron was again ready for the next excursion by 30 October 1918, it was necessary to wait until the British squadron had completed the operation which they had planned before escort could be furnished us. Reliable information indicated that enemy submarines were crossing the eastern portion of Area A, and the British had decided to lay surface mines in this position to the southward of those laid on our first excursion so as to strengthen this part of the field which was the least effectively mined part of the area. Weather conditions, however, prevented them from going out for several days, and, in the meantime, the series of events during the latter part of October and 1 November brought the end of the war into view. Further mining would have been an unnecessary waste of time, effort, and material. The British squadron did not carry out their contemplated operation, nor did the U.S. squadron. With the armistice with Germany on 11 November, came the end of building the North Sea Mine Barrage.
Rear Adm. Strauss summed up the final status of the operation and the results obtained from it as follows: Had it been possible to carry out minelaying operations as fast as the necessary mining material was received and assembled, the U.S. portion of the North Sea barrage could have been completed by the latter part of September 1918. The frequent delays, especially during the latter part of the work, which were principally due to the necessity of awaiting for escort to be supplied by the Grand Fleet, or for the British mine squadron to complete its preparations so as to be able to go out at the same time, prevented the barrage from being completed prior to the signing of the armistice.
Throughout this entire time Black Hawk remained at Inverness. With the war’s end, the mission turned to sweeping the mines laid in the North Sea barrage and clearing those waters enabling safe navigation. On 1 December 1918, the minelayers sailed for the United States, leaving Black Hawk with Patapsco (Id. No. 3475) and Patuxent (Id. No. 2766), as the nucleus for the minesweeping force. Black Hawk remained the flagship and repair ship of this embryo organization, while Patapsco and Patuxent, two powerful tugs, were retained to carry out experiments to ascertain, if possible, some means of sweeping the barrage and to develop the gear that would be required. It had been decided to use Kirkwall, Scotland, in the Orkney Islands, as the primary base for operations given its proximity to the barrage. Arrangements had been completed to obtain from the Admiralty an oil ship, water boat and gasoline boat, also to obtain coal from the British coal barges which were maintained at Kirkwall. Since the transportation facilities from Great Britain to Kirkwall were so inadequate, Base No. 18 was to be used as a receiving base for the constant train of supplies which were required at Kirkwall. Vessels from the U.S. force would transport the supplies between the two bases. Also since the hospital facilities available on Black Hawk were inadequate for the entire force, it was necessary to retain those at Inverness to handle the more serious cases as they occurred.
On 20 April 1919, the first twelve sweepers anchored in Inverness Firth. A few hours later, Rear Adm. Strauss, arrived in Inverness and broke his flag on board Black Hawk. His instructions stated that operations were to begin at the earliest possible moment, and every effort must be made to complete the clearance of the North Sea barrage that year. Everything was in place to commence the first operation by 28 April, but sailing was delayed for 24 hours on account of a heavy snowstorm. The following morning the twelve sweepers which had arrived, accompanied by six submarine chasers, got underway for the barrage, while Black Hawk and the remaining submarine chasers sailed for their new base at Kirkwall. A few hours after Black Hawk had anchored on 29 April, she was joined by Heron (Minesweeper No. 10), Auk (Minesweeper No. 38), Sanderling (Minesweeper No. 37), and Oriole (Minesweeper No. 7), just arriving from the U.S.
The various procedures for communications had been carefully worked out and incorporated in the Minesweeping Orders which had been printed and issued to the force prior to the first operation. In general, all messages which could be transmitted without relaying were to be sent by flag hoist, searchlight, or semaphore. Messages which affected a division, squadron, or the entire force were sent by radiotelephone. Shape signals, consisting of combinations of balls, drums, diamonds, and flags, were also prescribed for the more common signals in connection with passing sweeps and maneuvering. On the whole, the system was highly satisfactory, and after the first few days no difficulties were encountered. The radiotelephone proved highly reliable as well as possessing a high degree of ruggedness, and in only the very severest of accidents was disabled by the explosion of mines. The range of audibility of the telephones for the minesweepers averaged approximately 30 miles. In some cases satisfactory communication was maintained at distances of 50 to 60 miles. On account of the short antenna of the submarine chasers their range of audibility was considerably less, in general, not being more than ten miles. Considerable difficulty was constantly encountered on the submarine chasers on account of the salt-water spray and leaks in their hulls, which caused a great amount of short circuiting in the apparatus. The spark sets on the sweepers, though only 1 kilowatt, proved entirely satisfactory, especially after all sweepers were equipped with audion panels and the flagships with two step amplifiers. Communication between Black Hawk and sweepers at anchor in a Norwegian fjord, 250 miles distant, was executed with ease.
On 2 May 1919, sweepers and submarine chasers completed the first operation and proceeded to Kirkwall. In all 221 American mines had been destroyed, which represented approximately 25% of the total mines which were laid in the areas over which they had swept. By 3 June 1919, the sweepers and submarine chasers were again ready to sail, but a storm delayed their departure until the afternoon of the 5th. That same day, four of the new sweepers, Chewink (Minesweeper No. 39), Flamingo (Minesweeper No. 32), Thrush (Minesweeper No. 32), and Penguin (Minesweeper No. 32), which had been requested by Rear Adm. Strauss upon completion of the first operation, arrived at Kirkwall. Black Hawk began at once the installation of the electrical protective devices which had been sent over on the first sweepers for spares. Group 9 was to be cleared on the third operation. This group, consisting of 5,520 mines, was the largest which had been laid on a single operation. The British had already cleared their single line of mines laid at a depth of 95 feet about 1,000 yards to the northward of our field. Out of the 1,300 British mines originally laid in this line 606 had survived until summer and were accounted for by the British sweepers.
The object in clearing this large group was twofold. On future operations it would shorten the distance that the vessels would have to steam in going back and forth from port and at the same time would reduce the number of mines which might break adrift and menace the ships in the western part of the barrage. Furthermore, this group of mines, although the largest laid, contained only two rows laid at the upper level. It was still desired to avoid as much as possible the chances of being damaged by countermining while experience was being gained. This danger was not, however, so great as had been originally anticipated, and Rear Adm. Strauss had decided to have one division of the minesweepers sweep their section of the field longitudinally in order to
compare this method with the transverse sweeping which had been used up until this time. If the division succeeded in demonstrating that the danger was no greater, our sweeping speed could probably be greatly increased and possibly enough to complete the clearance of the barrage within the year. In order to assess the relative merits of the two methods, Rear Adm. Strauss hoisted his flag on Eider (Minesweeper No. 17) and spent several days on the minefield observing the actual conditions and difficulties encountered. A few days after the operation had begun the admiral returned to Inverness. From what he had seen he was convinced that longitudinal sweeping could be used with equal safety.
On 5 August 1919, Capt. Roscoe C. Bulmer, Commander, Minesweeping Detachment, died on board Black Hawk from injuries received the day before when thrown from an automobile which had skidded. The death of Capt. Bulmer proved a severe loss. His unbounded enthusiasm and cheerfulness, coupled with resolute determination, at times in the face of overwhelming odds, had been invaluable in the early part of the minesweeping when the obstacles and accidents were so discouraging. His body, after being embalmed, was sent to Inverness and then transported to the U.S. for burial.
After seven sweeping operations the North Sea Mine Barrage was declared cleared on 30 September 1919. By the time that the sweeping operations had been completed, all the submarine chasers except two had been sent to Devonport, England, for docking prior to their trans-Atlantic voyage. Black Hawk, accompanied by Oriole, departed Kirkwall, on 1 October. While en route to Devonport, they stopped at Gravesend, England (4 October), to land Rear Adm. Strauss, in order that he might proceed to London. A general railroad strike was in progress at the time. As such, the admiral's automobile, which had been placed on board at Kirkwall, was landed to enable him to proceed to London, and later to Southampton, England, where he embarked upon the steamer Adriatic to return to the U.S. in order to arrange for the disbanding of the force upon its arrival home.
As the repairs at Devonport and Chatham, England, drew to an end, the minesweeping force was divided into two detachments for the trip home, as had been previously decided, with a view of reducing the congestion while the vessels were in harbors and thus enabling them to obtain fuel and water more easily. On 12 October 1919, the destroyer tender Panther and 12 of the submarine chasers sailed from Devonport to Brest, France. They were followed two days later by fourteen of the minesweepers. On 15 October, Black Hawk and the remaining ’sweepers and submarine chasers, except Swan (Minesweeper 34), Auk, S.C.164, S.C.178 and S.C.206, got underway for Brest. The repairs were completed on these latter vessels the day following Black Hawk’s departure, when they got underway. In the meantime three of the vessels which had been sent to Chatham were completed and were on their way to Brest to join the remainder of the detachment. The repairs on Penguin required more time than had been expected, and it was necessary to hold her at the dockyards for several additional days. Although there was plenty of fuel, water, and gasoline available at Brest, considerable difficulty was encountered in getting it on board the vessels. Four days were required before Panther and her detachment of 12 submarine chasers and 14 sweepers could get underway for Lisbon. Black Hawk and the remainder of the sweepers and chasers, however, managed to sail after two days in port (18 October). Panther and her convoy arrived in Lisbon on 20 October, followed by the Black Hawk detachment two days later (22 October). Stopping at Lisbon was intended to break the trip for the submarine chasers given the discomforts aboard those vessels while at sea. The weather, however, was very good and the force could have proceeded directly to the Azores from Brest. After two days spent in port the two detachments, separated by an interval of one day, got underway for Ponta Delgada, Azores.
The weather continued fine for the next leg of the cruise and the two detachments reached the Azores on 27 and 28 October 1919, respectively. The small harbor at Ponta Delgada, was taxed almost to its maximum capacity by the arrival of the 55 vessels in the two detachments. There was much difficulty in obtaining water and provisions and the facilities for supplying water were inadequate. In the meantime Penguin, which had completed her repairs at Chatham, had arrived at the Azores with the tug Concord (S.P. 773), which she was escorting. On the evening of 29 October, Panther and her detachment stood out for Bermuda. Two days later, 30 October, the Black Hawk detachment sailed, and they in turn were followed by Penguin and Concord on 31 October. The excellent weather to the Azores was replaced by gales for the next two weeks. At times the ships were barely able to make headway through the heavy seas. The third day out from the Azores, S.C. 256, which was then in tow of Falcon (Minesweeper No. 28), was destroyed by a fire following a gasoline explosion. Turkey (Minesweeper No. 13) was towed by Panther; Swallow (Minesweeper No. 4) and Auk were oiled at sea by Black Hawk; and several of the submarine chasers were given gasoline at sea. Seagull (Minesweeper No. 30) ran entirely out of oil, and could not operate her radio. The engineer officer, however, managed to connect the radio generator to enable Seagull to call for assistance. Black Hawk, which still had a reserve supply of oil, proceeded to her assistance.
The two detachments straggled in to Bermuda, arriving singly or in groups (9-13 November 1919). The weather made the entering the narrow channels to the inner harbor difficult. By the evening of 15 November, the Panther detachment was ready to sail. The following day the remaining vessels, except Black Hawk, weighed anchor and stood out of the harbor. The weather having been too severe for the dockyard to take Black Hawk alongside the oil dock, it was necessary for her to remain until the 17th in order to get refueled. The two detachments arrived off Tompkinsville [Staten Island], N.Y., on 19 and 20 November, respectively. Shortly after Black Hawk had anchored, Rear Adm. Strauss returned on board and re-hoisted his flag. On 21 November, the vessels shifted berth to the North River anchorage, the sweepers in two columns with the sub chasers tied up alongside.
At 10:00 a.m. 24 November 1919, Secretary of the Navy Josephus Daniels, reviewed the North Sea Mine Force from Meredith (Destroyer No. 165). After steaming up one side of the formation they returned on the other side, then the reviewing party went on board Heron to inspect her. The review was followed by a reception for the officers given by the Secretary on board Columbia (Cruiser No. 12). Simultaneously, a luncheon was held for the 2,000 enlisted men at the Astor Hotel. At midnight on 25 November, Rear Adm. Strauss, Commander, Mine Force, Atlantic Fleet, hauled down his flag, and the force disbanded at noon. At that time Black Hawk received orders assigning her to Destroyer Squadrons, Atlantic Fleet, until Panther returned from Europe.
Black Hawk remained at New York through the end of the year, then entered the New York Navy Yard, for the installation of a 12-foot rangefinder, on 8 January 1920. Clearing the yard, she steamed to Cuba for the fleet’s annual winter, arriving at Guacanayabo Bay on 25 January. She then shifted to Guantanamo Bay, on 29 January, before departing for the Canal Zone (C.Z.) on 8 February. She reached Cristóbal, on 11 February and remained there until 16 February, when she went to sea and then made her return to Guantanamo on 26 February. She remained there in port until 26 March, when she shifted to Guacanayabo Bay, arriving the next day. The tender got underway before month’s end and cruised Cuban waters making stops at Guantanamo Bay (2 April), Cienfuegos (3-6 April), then back in to Guacanayabo on 7 April. Moving to Guantanamo Bay on 17 April, she stood there until 25 April, when she cleared bound for New York. Steaming through the Ambrose Channel, she proceeded up the Hudson River, to the North River anchorage on 1 May. After five days, she got underway and steamed to Vera Cruz, Mexico. Arriving on 18 May, she remained there into June. During this time she sent a message to the Office of the Chief of Naval Operations (OPNAV) on 21 May, stating that the situation stemming from the revolution in Mexico had improved and were “more favorable than they have been for several years.” On 7 June, she received instructions to proceed to Newport when relieved by Des Moines (Cruiser No. 15) and to proceed to Newport to report by dispatch to Commandeer in Chief, Atlantic Fleet and Commander, Destroyer Squadrons, Atlantic Fleet. Getting underway on 15 June, she set a course for New York having received new orders, and reached on the 23rd. Six days later, on 29 June, she entered the New York Navy Yard for maintenance. She cleared the yard on 7 July, and steamed to Newport, R.I.; arriving the next day, she remained there in port until 7 September. During this time, she was re-designated (AD-9) as part of a Navy-wide administrative re-organization on 20 July. Between 7 September and 2 October, she shifted between Newport and Smithtown Bay, R.I. On 2 October, she got underway and proceeded to Hampton Roads, Va. Arriving on 4 October, Black Hawk operated in the waters around the Virginia capes until 27 October. Clearing the Southern Drill Grounds, she steamed to New York, and entered the Navy Yard on 28 October. She remained there through the end of the year undergoing overhaul and modification to include the installation of a torpedo workshop with stowage for torpedoes and air compressors, along with other equipment. During this yard period, she was also permanently assigned as tender for Destroyer Force, Atlantic Fleet, on 4 November.
After spending the holiday season finishing up her modifications, Black Hawk cleared the yard on 4 January 1921. Going to sea, she returned to the Southern Drill Grounds for trials (5 January), then continued on to the Caribbean for fleet winter training, arriving at Guantanamo on 9 January.
After a week at Guantanamo, the tender was dispatched to the Canal Zone. Departing on 16 January 1921, she arrived at Cristóbal, on 19 January, and transited the Panama Canal to Balboa, C.Z. On 22 January, she cleared Balboa, and steamed to Callao, Peru. Arriving on 31 January, she remained there until 5 February and returned to Balboa on the 14th. Passing back through the canal, Black Hawk returned to the Atlantic and continued on to Guantanamo, reaching on 24 February. She remained at Guantanamo until 13 March, then going to sea, she cruised Cuban waters, making several port visits into April. Standing out of Guantanamo on 24 April, she steamed to the New York Navy Yard, docking on 29 April. Undergoing overhaul until 15 June, she cleared the yard that day, and steamed south to undergo trials and training on the Southern Drill Grounds. Arriving on 16 June, she departed the drill grounds on 29 June, and steamed back northward. After passing through the Verrazano Narrows, she steamed up the Hudson River, and moored at North River on 30 June. She remained until 9 July, then proceeded to return to the Southern Drill Grounds (12-21 July), before returning back to the North River anchorage (22 July-1 August). Returning to sea, she steamed to Lynnhaven Roads, via the Southern Drill Grounds (4 August), and remained there until 25 August. Between 26 August and 10 October, she continued to shuttle between the lower Chesapeake Bay and New York with stays of varying length at each. She entered the Philadelphia Navy Yard on 11 October and docked until 8 November. Upon clearing the yard, she steamed directly to New York and entered the navy yard there and remained through the end of the year.
After undocking on 3 January 1922, Black Hawk went to sea and after steaming to the Southern Drill Grounds (4-6 January), she continued on to Guacanayabo Bay. Reaching on 12 January, she joined the fleet for the annual winter training and exercises. Shifting to Guantanamo Bay on 3 February, she remained there until 22 April. Getting underway, she steamed to New York and entered the Navy Yard on 28 April to prepare for distant service. The tender had received orders to serve as the Flagship, Destroyer Squadron, Asiatic Fleet. Clearing the yard on 5 June, she steamed to Newport (6-15 June), then crossed the Atlantic, raising Gibraltar on 26 June. Clearing the British possession on 6 July, she continued on to Marseilles, France (9 July); Malta (12-14 July); Ismailia, Egypt (17-22 July); Aden [Yemen] (27-30 July); Colombo, Ceylon [Sri Lanka] (6-11 August); Singapore (13-18 August), and arrived at her primary base in China at Chefoo [Yantai], on 27 August. She remained there until 30 September, when she got underway to cruise Chinese waters, visiting Shanghai (2-12 October), Amoy [Xiamen] (15-22 October), Hong Kong (23-28 October), before steaming across the South China Sea to Manila, Philippine Islands (P.I.), where she arrived on 30 October. Shifting to the Navy Yard at Cavite, she stood there for almost six months.
Black Hawk resumed operations underway when she cleared Cavite on 7 April 1923, bound for the International Settlement at Shanghai. Reaching on 12 April, she moored off the Bund there until 24 April. Going to sea, she steamed up the Chinese coast to Tsingtao [Qingdao] (8-10 June). After her visit to the port city on the Shantung [Shandong] Peninsula, she proceeded to Chefoo, where she arrived on 11 June. Standing out of Chefoo on 23 July, she moved to Chinwangtao [Qinhuangdao] (24 July-6 August), before returning to Chefoo on 7 August. Again underway on 27 August, she shifted to Darien [Dalian], China (28 August-3 September); before touching at Chefoo, en route to Tsingtao (4-6 September). In response to the Great Kantō earthquake that struck Japan on 1 September, Black Hawk cleared Tsingtao and steamed across the Sea of Japan with relief supplies. She stood at Yokohama, Japan, providing aid (10-27 September).
Departing the Home Islands on 27 September 1923, she steamed to Shanghai (1-12 October), before returning to the Philippines at Olongapo on 16 October. Remaining there until 13 November, she shifted to Manila, that same day and stood there into the spring.
Black Hawk cleared Manila on 12 April 1924 and steamed to Hong Kong, where she spent the next six weeks. Returning to Japan, the flagship visited Nagasaki (26-30 May) and Kagoshima (31 May-7 June), before making her return to China at Shanghai (7-17 June) and Tsingtao (18-27 June), before arriving back at Chefoo on 30 June. She went to sea again on 11 July for a visit to Chinwangtao (12-24 July) before returning to Chefoo (25 July-23 September).
Black Hawk relocated to Tsingtao (24-27 September 1924), then to a lengthy stay off the Bund at Shanghai (29 September-28 November), before returning to Manila on 2 December. As she did the previous year, Black Hawk stood at Manila into the following April.
Going to sea on 18 April 1925, Black Hawk set a course for Amoy. Arriving on 21 April, she weighed anchor on 5 May, and moved on to visit Tsingtao (9 May-3 June), before making her return to Chefoo on 3 June. Standing there until 8 September, Black Hawk made her return to Manila via (Tsingtao (9-15 September) and arriving on 20 September.
Between 9 February and 5 April 1926, Black Hawk departed Manila and moved between Olongapo (9-18 February) and Cavite (18 February-5 April). Clearing the Philippines on 5 April, she transited to Hong Kong (8-11 April) before visiting Shanghai (13 April-10 May) and Tsingtao (11-31 May), en route to Chefoo, where she arrived on 1 June. The flagship would remain here until 4 October. At this same time, China was in the midst of a period of great upheaval. The Kuomintang [Guomindang] Party which constituted the largest part of the Nationalist government, created the National Revolutionary Army (NRA) under the leadership of Chiang Kai-shek [Jiang Jieshi] to subjugate the various warlord cliques and their armies to unify the nation under Kuomintang leadership. They launched their Northern Expedition in July and proceeded to fight for the next two years until the warlords bowed to the Kuomintang’s authority in June 1928. In the meantime, Black Hawk cleared Chefoo on 4 October and went to sea to visit Tsingtao (5-19 October) and the International Settlement at Shanghai (21-28 October), en route to her regular seasonal return to the Philippines. She reached Manila on 1 November, and remained there until January.
Getting underway on 31 January 1927, Black Hawk shifted to Olongapo (31 January-3 February) before returning to Manila on the 3rd. She stood out of Manila to make her return to China on 18 April. Three days later, she touched briefly at Amoy before moving on to Shanghai. Reaching on 24 April, she remained there off the Bund until 6 June, when she got underway for Chefoo and arrived on 8 June. She would remain there in port over the next three months. Departing Chefoo on 14 September, she touched at Alacrity Bay (15-16 September) and then continued on to Southeast Asia en route to a return to the Philippines. Her first port visit was to Saigon, Cochin China, French Indochina [Ho Chi Minh City, Vietnam] (23-26 September). Continuing on, she moved to Singapore (28 September-1 October). After departing this British possession, the tender moved on to the Netherlands East Indies (N.E.I.) [Indonesia] visiting Batavia [Jakarta], Java (3-9 October), Soerabaja [Surabaya], Java (10-13 October), and Makassar [Ujung Pandang], Celebes (15-17 October). She then proceeded back to the Philippine archipelago touching at Tawi-Tawi (19 October), Zamboanga (20-22 October), and San Fernando (24-27 October), before returning to Manila on 28 October and remaining there through the end of the year.
Black Hawk got underway on 13 February for a brief visit to Olongapo (14-15 February) before returning to Manila. After a month in port, she went to sea and steamed to Hong Kong (23 March-3 April) before visiting other locations along the Chinese coast including Amoy (4-7 April), Magpie Bay (8 April), Samsa Inlet (8-10 April), and Alacrity Bay (12 April), before steaming eastward to the “Land of the Rising Sun” and calling at Kobe, Japan (16-26 April) and Ito Saki, Japan (26-27 April). Returning to Chinese waters, she made landfall and spent a fortnight at Chinwangtao (30 April-14 May), before arriving at Chefoo on 15 May. Remaining in port until 8 July, she got underway and visited Tsingtao (10-12 July) and Shanghai (14-18 July), before returning via Tsingtao (20 July) to Chefoo on the 21st. She continued to operate from there until 18 September, when she sortied to return to Manila, reaching on 25 September. Having returned to her primary base in the Philippines, the tender remained in port well into the next year.
While Black Hawk got underway to visit Olongapo (15-20 February 1929), she largely stood in port until 4 March, when she stood out of Manila Bay bound for Hong Kong. Arriving on 8 March, she remained until the 13th, when she moved on to Amoy (15-24 March). Departing on 24 March, she steamed to Japan for visits to Nagasaki (29 March-10 April) and Yokohama (13-23 April). Returning to China, she spent a month at Tsingtao (27 April-27 May), then arrived back at Chefoo on 28 May. Standing in port until 5 July, she got underway and conducted visits to Tsingtao (6-17 July) and the Bund at Shanghai (20 July-11 August), before returning to Chefoo via Tsingtao (13 August) on 14 August. Clearing Chefoo for the year on 16 September, she made a third visit to Tsingtao (17-23 September) and a second to Shanghai (26 September-2 October), before standing in to Manila on 6 October.
Black Hawk spent much of the first quarter of 1930, shifting to different locations in the Philippines with time at Manila, Olongapo, and Subic Bay. Steaming from Manila on 1 April, she reached Hong Kong on 3 April, remaining there until the 9th. She then proceeded to visit Amoy (9-22 April), Tsingtao (26 April-8 May), and Chinwangtao (11-24 May) before arriving at Chefoo on 24 May and settling in to a period in port. Sortieing from Chefoo on 30 September, she initiated her return to the Philippines via a roundabout route through Southeast Asia. Bypassing French Indochina, she steamed to Singapore (9-15 October), then moved on to the Netherlands East Indies, visiting Batavia (21-27 October), Soerabaja (27-29 October), Buleleng, Bali (2-3 November), and Makassar (4-7 November). Making her return to the Philippine Islands, she called at Tawi-Tawi (9-10 November) and Coron Bay (12-13 November) en route to her return to Manila on 14 November.
Black Hawk again made a visit to Olongapo early in the year (9-12 March 1931), in advance of her departure from Manila for China. Getting underway on 16 April, she reached Shanghai on 21 April. Remaining there until 3 May, she continued to move up the coast visiting Tsingtao (5-14 May), Chefoo (15-16 May), and Chinwangtao (17 May-1 June), before returning to Chefoo on 2 June. Operating from Chefoo, Black Hawk made visits to Tsingtao (9-13 September) and Shanghai (15-17 September and 23-24 September). Finally, standing out from Chefoo on 17 October, she made her return to Manila via Shanghai (20 October-2 November) and standing in to Manila Bay on 6 November.
On 4 February 1932, Black Hawk steamed in haste for Shanghai and arrived on 9 February. She would spend more than three months off the Bund. The flagship’s lengthy visit coincided with the Shanghai Incident (28 January-3 March) which saw fighting around the city between China and Japan. Though a cease-fire was eventually brokered by the League of Nations, tensions remained around the International Settlement. She finally steamed out of the Yangtze River on 23 May, bound for Chefoo she arrived two days later and began her period in port. Just over four months later on 3 October, she cleared Chefoo, for a return to Shanghai (5-25 October), before visits to Amoy (28-31 October) and Hong Kong (2-9 November). Departing the Chinese coast, she returned to Manila, where she remained through the end of the year with the exception of a 10-day period at Olongapo (5-15 December).
Still in port at Manila on 1 January 1933, Black Hawk went to sea on 11 April and proceeded to Hong Kong. Arriving on 13 April, she remained until the 17th, when she departed for Amoy (18-24 April) and Shanghai (27 April-16 May) before arriving at Chefoo for her period in port on 18 May. Remaining until 5 October, she cleared that same day for Shanghai (7-20 October) before proceeding to Hong Kong (24 October-1 November) en route to Manila, where she arrived on 4 November.
Black Hawk sortied on 9 April 1934 and steamed to Japan for visits to Yokohama (16-25 April) and Kobe (26 April-3 May). Making her annual return to Chinese waters, she arrived at Shanghai on 6 May, and after ten days, she cleared the Bund, and steamed to Chefoo. Arriving on 18 May, she began her in port period tending to the ships of DesDiv 5 until 5 October. After visits to Tsingtao (6-7 October); Shanghai (9-28 October); Amoy (31 October-4 November); and Hong Kong (6-14 November), she returned to the Philippines and entered the Navy Yard at Cavite on 16 November. With her maintenance period completed she shifted to Manila, where she remained into the next year.
Black Hawk got underway on 14 April 1935 and set a course for Shanghai. Arriving on 18 April, she remained until 30 April, when she departed for Kobe (3-17 May). After her visit to Japan, she made her return to China, and re-establishing her summer residence at Chefoo on 19 May. She remained until 21 September, when she cleared Chefoo en route to Hong Kong (26 September-14 October) and Louraine Bay (16-22 October) before returning to Saigon (24 October-2 November). Upon departing French Indochina, the tender, steamed on an easterly course and raised Manila on 6 November.
Going to sea from Manila on 13 April 1936, Black Hawk called at Olongapo (17-20 April) in advance of her return to China. She arrived at Shanghai on 24 April and remained until 13 May. She then proceeded to Chefoo, where she stood in port for two weeks (15-29 May), then moved to Chinwangtao (30 May-5 June), before returning to Chefoo and her summer time in port on 6 June. She stood there tending to her attached destroyer division until 14 October, when she departed for Shanghai. Lying off the Bund from 16 October until 2 November, she cleared the mouth of the Yangtze and steamed to Hong Kong (5-11 November). Departing this British Far Eastern possession, she steamed to the other, arriving at Singapore on 16 November. After a week’s visit, she got underway on 23 November and steaming directly to the Philippines, she stood in to Manila on 28 November.
Black Hawk cleared Manila on 5 April 1937 and arrived at Hong Kong on 8 April. Leaving the next day, she moved on to Shanghai (12 April-6 May). The tender then continued on to Chefoo where she arrived on the 8th to perform her tender duties. She got underway again on 2 July to visit Chinwangtao (2-11 July). During her time away from Chefoo, Japanese and Chinese forces clashed at the Marco Polo Bridge outside Peiping [Beijing] on 7 July. The incident sparked the outbreak of the Second Sino-Japanese War. The ship returned to Chefoo and remained there until 8 November, when she set a course for Tsingtao (9-10 November) en route to Shanghai. Arriving at the International Settlement on Armistice Day, she moored on “Man of War Row” amidst the fighting between the Chinese and Japanese forces in and around the city. After a week, she departed on 18 November, and steamed back in to Manila on 26 November for the winter.
Black Hawk departed Manila on 1 April 1938 and steamed to Hong Kong for a two-day visit (4-6 April). Bound for a return to Manila, she arrived on 10 April, and then departed the next day for two months cruising the Philippine archipelago. Clearing the island of Mindanao on 14 June, she steamed to Bangkok, Thailand. Arriving on 21 June, she remained a week before charting a course eastward, reaching Manila on 2 July. She stood at Manila until 15 July, then went to sea to return to China. After a time at Hong Kong (18-20 July), she shifted to Chefoo (24 July-15 September). Moving on to Shanghai (19-28 September), she returned to Manila (4-9 October), before again cruising the islands in the southern part of the archipelago (10-20 October). With the cruise completed, she returned to Manila on 22 October.
Black Hawk returned to cruising the Philippines when she departed Manila on 10 April 1939. Returning to the city on 22 May, she remained there until 3 June, when she sortied for China. Arriving at Hong Kong on the 5th, she stood there until 7 June, then continued on to Shanghai (11-16 June), before arriving at Chefoo on 18 June. After seven weeks in port, the ship got underway for a visit to Shanghai (13-19 August), before making a brief return to Chefoo on 20 August. Departing again that same day, she spent next five weeks cruising the Chinese coast making multiple port visits during that time. Steaming from Chefoo on 29 September, she made her return to the Philippines. Initially arriving at Manila on 5 October, she departed again on 18 October. Cruising the islands south of Manila, she made her return to the city on 2 December, and spent the Christmas and New Year’s holidays in port.
Black Hawk stood out of Manila on 24 April 1940, to make her return to war-ravaged China. Initially visiting the British possession at Hong Kong (29 April- 6 May), she returned to Chefoo (11 May-8 June), before getting underway to visit Tsingtao (9-10 June) and Chinwangtao (12-22 June) before returning to Chefoo on the 23rd. Black Hawk stood there until 20 July, when she departed for Shanghai (22-29 July) and then moved on to Tsingtao (29 July-9 September). Standing out of Tsingtao, she steamed for a return to the Philippines, reaching Olongapo on 15 September. She then spent the next three months cruising Philippine waters until standing in to Manila on 19 December. She remained in port until 15 April 1941 and then spent the succeeding months continuing to patrol the Philippine Islands
Black Hawk, now assigned to DesRon 29, Asiatic Fleet, cleared Manila, and arrived at Balikpapan, Borneo, N.E.I., on 30 November 1941. She got underway on 6 December in compliance with orders from Commander in Chief, Asiatic Fleet, directing her to Batavia, for “supplies and liberty.” She was in company with Whipple (DD-217), Alden (DD-211), Edsall (DD-219), and John D. Edwards (DD-216) as Destroyer Division (DesDiv) 57. With news of the Japanese attack on Pearl Harbor on 7 December [8 December west of the International Date Line], the tender received orders to change course and proceed to Soerabaja. Arriving on 9 December, she moored and performed repairs to various other ships through December. She received orders on 30 December from Commander in Chief, Asiatic Fleet, to move to Port Darwin [Darwin], Northern Territory, Australia. She departed that day in company with Boise (CL-47), Pope (DD-225), Barker (DD-213), and the tanker George C. Henry (later Victoria, AO-46).
Black Hawk arrived at Port Darwin, on 6 January 1942 and soon took Peary (DD-226) and Heron (AVP-2) alongside for repairs to damage suffered in the Japanese bombing of Cavite Navy Yard (8 December) and in the Straits of Molucca (31 December), respectively. On 20 January, Black Hawk was designated as flagship, Commander, Base Force, at Port Darwin. On 3 February, in compliance with order from Commander, Task Force (TF) 5, she departed on 3 February, bound for Tjilatjap [Chilachap], Java, N.E.I. Reaching her destination on 10 February, she would serve as a repair ship. Repairs of note include those to Marblehead (CL-12) after she suffered damage in the action at Makassar Strait on 4 February and those required by Whipple after her collision with the Dutch light cruiser HNLMS DeRuyter on 12 February. She was underway again on 20 February, at the direction of Commander, U.S. Naval Forces, Southwest Pacific. Bound for Exmouth, Western Australia, Australia, she arrived on 26 February, accompanied by the submarine tender Holland (AS-3), Barker (DD-213), Bulmer (DD-222), and the submarine Stingray (SS-186). While en route on 22 February, she rendezvoused with Pope and John D. Ford (DD-228) at Christmas Island for the purpose of transferring 23 Mk. 8 torpedoes, to the destroyers. Operating in Australian waters (26 February-29 May), Black Hawk departed Exmouth on 2 March and shifted to Fremantle, Western Australia, arriving on 6 March. She departed Fremantle on 10 May, and then moved to Melbourne, Victoria (18-21 May) and Sydney, New South Wales (24-29 May), before leaving Australia bound for the U.S. via Pago Pago, American Samoa (6 June). In company with Parrott (DD-218), she stood in to Pearl Harbor, on 15 June, and reported for duty to Commander Destroyers, Pacific. She was assigned to tender duty, and provided service to 13 destroyers at Pearl Harbor.
Black Hawk departed Pearl Harbor in company with the oiler Ramapo (AO-12), as Task Group (TG) 2.5 on 19 July 1942. Bound for Kodiak, Alaskan Territory, they rendezvoused en route with King (DD-242) on 27 July, and arrived at Kodiak on the 29th. Assigned to tender duty in the territory, she remained there until 4 November 1942, when she departed for San Francisco, Calif. Accompanied by Humphreys (DD-236), she reached San Francisco on 11 November, and that same day entered the United Engineering Co. Ship Yard for repairs and overhaul. Completing her overhaul on 16 March 1943, Black Hawk cleared the yard the next day and returned to Alaskan waters, arriving at Kodiak on 25 March. She departed three days later and shifted to Dutch Harbor, Unalaska, in the Aleutians (30 March-10 April), before proceeding to Adak Island in the Aleutians.
Black Hawk remained at Adak until 21 September 1943, when she accompanied vessels of Task Units (TU) 11, 15, and 16 to Pearl Harbor. Arriving on 30 September, she remained at Pearl Harbor, providing tender service into 1944. Clearing the naval base on 1 February 1944, Black Hawk returned to Adak on 10 February and remained there until 21 March 1945. Following overhaul at Alameda, Calif., she arrived at Pearl Harbor 30 May 1945; remained there until 11 September; and then proceeded to Okinawa to support Allied occupation forces in the western Pacific. Black Hawk served in the Far East tending vessels at Okinawa and in China until 20 May 1946, when she headed home for the last time.
Black Hawk decommissioned on 15 August 1946. Deemed surplus to Navy needs and made available for disposal on 5 September, she was stricken from the Navy list on 25 September. Transferred to the Maritime Commission on 4 September 1947, she was then sold to Kaiser Co. Inc. on 17 March 1948, and then resold that same day to Dulien Steel Products Co., for scrapping.
Black Hawk received one battle star for her service in World War II.
||Dates of Command
|Cmdr. Roscoe C. Bulmer
||15 March 1918 – 5 August 1919
|Cmdr. Ellis Lando
||5 August 1919 – 1 October 1919
|Cmdr. John Rodgers
||1 October 1919 – 28 November 1919
|Capt. Byron A. Long
||28 November 1919 – 22 December 1921
|Capt. John W. Timmons
||22 December 1921 – 1 January 1924
|Cmdr. Charles T. Hutchins Jr.
||1 January 1924 – 2 September 1925
|Cmdr. Adolphus Staton
||2 September 1925 – 21 March 1926
|Lt. Cmdr. Eugene M. Woodson
||21 March 1926 – 24 July 1926
|Cmdr. Guy E. Baker
||24 July 1926 – 23 July 1927
|Cmdr. Randall Jacobs
||23 July 1927 – 15 January 1929
|Lt. Cmdr. Carroll W. Hamill
||15 January 1929 – 27 February 1929
|Lt. Cmdr. George B. Wilson
||27 February 1929 – 11 March 1929
|Lt. Cmdr. Lee C. Carey
||11 March 1929 – 15 October 1929
|Capt. Aubrey K. Shoup
||15 October 1929 – 1 July 1930
|Cmdr. Aquilla G. Dibrell
||1 July 1930 – 8 September 1930
|Cmdr. Walter H. Lassing
||8 September 1930 – 5 July 1931
|Capt. William P. Gaddis
||5 July 1931 – 7 May 1932
|Cmdr. Rufus W. Mathewson
||7 May 1932 – 28 July 1934
|Cmdr. Kinchen L. Hill
||28 July 1934 – 12 November 1935
|Cmdr. John H. Everson
||12 November 1935 – 20 March 1936
|Cmdr. Jay K. Esler
||20 March 1936 – 9 February 1937
|Lt. Cmdr. Chester M. Holton
||9 February 1937 – 10 March 1937
|Cmdr. Jay K. Esler
||10 March 1937 – 13 April 1937
|Lt. Cmdr. Chester M. Holton
||13 April 1937 – 17 April 1937
|Cmdr. Howard D. Bode
||17 April 1937 – 19 March 1938
|Cmdr. Willard E. Cheadle
||19 March 1938 – 11 March 1939
|Cmdr. Marshall B. Arnold
||11 March 1939 – 5 June 1940
|Cmdr. George L. Harriss
||5 June 1940 – 13 January 1943
|Cmdr. Edward H. McMenemy
||13 January 1943 – 13 May 1944
|Cmdr. Charles J. Marshall
||13 May 1944 – 17 August 1945
|Cmdr. Francis W. Beard
||17 August 1945 – 15 August 1946
Christopher B. Havern Sr.
20 February 2018<|endoftext|>
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Trying to understand the overall effect of climate change on our food supply can be difficult. Increases in temperature and carbon dioxide (CO2) can be beneficial for some crops in some places, but overall changing climate patterns lead to frequent droughts and floods that put a severe strain on yields. It’s not all about production, however. Researchers at University of California, Davis found that rising CO2 levels are inhibits plants’ ability to assimilate nitrates into nutrients, altering their quality for the worse. Our whole food chain relies on the proteins found in plants, whose energy we assimilate directly or from animals that eat the same plants. Consequently, crop quality in the face of global warming is an aspect that needs to be thoughtfully addressed.
“Food quality is declining under the rising levels of atmospheric carbon dioxide that we are experiencing,” said the study’s lead author Arnold Bloom. “Several explanations for this decline have been put forward, but this is the first study to demonstrate that elevated carbon dioxide inhibits the conversion of nitrate into protein in a field-grown crop.”
Nitrogen and CO2
Food crops use nitrogen to produce the proteins that are vital for human nutrition. Nitrogen is the mineral element that plants and other living organisms require in the greatest quantity to survive and grow. Plants obtain most of their nitrogen from the soil and, in the moderate climates of the United States, absorb most of it through their roots in the form of nitrate. In plant tissues, those compounds are assimilated into organic nitrogen compounds, which have a major influence on the plant’s growth and productivity.
Previously, scientists proved that nitrates assimilation is inhibited by carbon dioxide in the leaves of grain and non-legume plants, however the present study marks the first time that this relation was investigated in field-grown crops.
Samples of wheat that had been grown in 1996 and 1997 at the Maricopa Agricultural Center in Arizona were analyzed by the UC Davis researchers. At that time, carbon dioxide-enriched air was released in the fields, creating an elevated level of atmospheric carbon at the test plots, similar to what is now expected to be present in the next few decades. Almost two decades later, the UC Davis researchers returned to these samples and subjected them to chemical analysis that was not available at the time of harvesting.
Three different measures of nitrate assimilation confirmed that the elevated level of atmospheric carbon dioxide had inhibited nitrate assimilation into protein in the field-grown wheat. It’s safe to assume that other food crops like barley, rice, and potatoes may be subjected to the same risks, though the same experiment needs to be repeated for each of these.
“These field results are consistent with findings from previous laboratory studies, which showed that there are several physiological mechanisms responsible for carbon dioxide’s inhibition of nitrate assimilation in leaves,” Bloom said.
Historical records have documented that the concentration of carbon dioxide in Earth’s atmosphere has increased by 39 percent since 1800. If current projections hold true, the concentration will increase by an additional 40 to 140 percent by the end of the century.
Extrapolating, the researchers found that, in all, global food proteins available for human consumption could drop by as much as 3% in the coming decades as a result of rising carbon dioxide levels. While heavy nitrogen fertilization could partially compensate for this decline in food quality, it would also have negative consequences including higher costs, more nitrate leaching into groundwater, and increased emissions of the greenhouse gas nitrous oxide.
The findings were reported in the journal Nature Climate Change.<|endoftext|>
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# Activity: Birthday Algorithm
This activity requires only pencil and paper (no need for a computer). In this lesson students follow a series of steps to complete a math trick.
Students are introduced to:
• Algorithms
• Program flow
• Debugging
## Algorithm
An Algorithm is series of steps, decisions and / or formulas. Some common examples include:
• Cooking recipes
• Map directions to an address
• Assembly instructions
## Birthday Algorithm
The Birthday Algorithm displays a birthday as `month.day` after following a set of calculations.
Result is in the format `month.day` example: March 1st would be `3.01`
Follow the Birthday Algorithm steps below working with a partner. Record the steps and the answer for each step so you can troubleshoot. Doing this with just paper and pencil adds to the challenge. If the trick doesn’t work, retrace the steps and check each calculation with partner.
Produce a total at the end of each step and use the total in the next step until producing the birthday `month.day`.
2. Multiply by month the person was born (number 1-12)
3. Subtract 1
4. Multiply by 13
6. Subtract 2
7. Add the day the person was born (number 1-31)
8. Multiply by 11
9. Subtract day the person was born (number 1-31)
10. Subtract month the person was born (number 1-12)
11. Divide by 5
13. Divide by 200
14. If not a match of the `month.day` of the persons birthday, carefully re-read the instructions and check answers at each step
## What did we learn?
1. Describe how well the algorithm worked. Explain what made the algorithm work or what was tried when it did not work.
2. Explain how the algorithm could be improved in terms of efficiency and usability for the participant. Be sure to describe how to reduce the number of steps while maintaining the trick.
3. Challenge: Write full equation at each step. Then try to simplify each answer.
Example: first 4 steps
1. `7`
2. `7m` (`m` = month)
3. `7m - 1`
4. `(7m - 1)13 = 91m - 13`<|endoftext|>
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Food Security, Environmental Justice and Protection
Food insecurity and hunger have increased considerably world-wide. Women & children, especially in low resource countries, continue to be overrepresented among people lacking access to adequate food to lead healthy and productive lives (Anderson, 1990; Nord, 2003; USDA, 2013).
Despite efforts directed at reducing food insecurity, progress in mitigating chronic hunger has been thwarted world-wide. The number of hungry people in the world is currently at an historic high. The Food and Agricultural Organization (FAO) estimates that about 1.02 billion people were chronically hungry and undernourished in 2009 (FAO, 2010). These figures represent 15.5 per cent of the world’s population.
Global food insecurity is linked to a number of factors including environmental justice, poor harvests, and subsequent increases in food price worldwide – especially the rising cost of dairy products and cereals (FAO, 2012). It is also connected to increases in unemployment and the recent economic downturn. Further, evidence suggest that hunger is linked to gender inequality. Indeed, data from the Global Hunger Index (GHI) (2009), suggests that countries reporting severe hunger also had high levels of gender inequality. Given the negative impact of food insecurity and hunger on welfare, the subject deserves attention.<|endoftext|>
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Although children ask questions from a very early age it does not mean that they automatically know how to pose questions that are useful for their learning. How can we help young children to ask productive questions—the type of questions that move their learning and thinking forward? How can we convey to children that their wonderings are welcomed in the classroom?
Any parent can confirm that most preschoolers ask what feels like innumerable questions each day, especially ‘how’ and ‘why’ questions. At the same time, studies of classroom talk suggest that the frequency and the quality of children’s questions drop as soon as they begin in an early childhood setting. The way educators treat children’s questions influences whether children will continue to pose questions in the classroom.
Supporting young children to ask productive questions explores how we can help children develop the habit and skill of asking questions that are useful for their learning. When children ask real questions, that is, questions that stem from their desire to understand the world around them, their mind is more open to connections and learning feels meaningful. When children are able to pose questions and explore the answers they feel motivated to exercise their sense of agency and build their independence skills.
Given young children’s natural capacity to ask questions, the challenge for educators is to keep children’s curiosity alive and help them to see questions as the powerful learning tools they are. This edition of the Research in Practice Series provides ideas and activities for creating environments that encourage children to notice, wonder, question and explore. Author Maria Birbili also discusses adult strategies for improving children’s questioning skills in meaningful ways.
Also available in e-version here.<|endoftext|>
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109 |
The learning and development requirements are introduced by the best available authentication on how children learn and improve the broad range of skills, knowledge and attitudes children need as a foundation for good future progress.
The areas of learning and development:
There are seven areas of learning and development. All areas of
learning and developments are important and inter-connected.
These three areas are the prime areas:
- Communication and language
- Physical development
- Social and emotional development
The specific areas are:
- Understand the surroundings
- Art & Design<|endoftext|>
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# In the xy coordinate plane the slope of line p is 1/2 and its x-intercept is -3. How do you find the equation of a perpendicular to p and intersects p at is x-intercept.?
Dec 29, 2016
$2 x + y + 6 = 0$
#### Explanation:
The equation of a line having a slope $m$ and passing through point $\left({x}_{1} , {y}_{1}\right)$ is $\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)$. As the slope of line is $\frac{1}{2}$ and its $x$-intercept is $- 3$ (i.e. it passes through $\left(- 3 , 0\right)$, its equation is
$\left(y - 0\right) = \frac{1}{2} \left(x - \left(- 3\right)\right)$ or $y = \frac{1}{2} x + \frac{3}{2}$ i.e. $x - 2 y + 3 = 0$.
As the slope of line $p$ is $\frac{1}{2}$, slope of line perpendicular to it is $\left(- 1\right) \div \frac{1}{2} = - 1 \times 2 = - 2$ and as it intersects $p$ at its $x$-intercept, it tpp passes through $\left(- 3 , 0\right)$ and its equation will be
$\left(y - 0\right) = - 2 \left(x - \left(- 3\right)\right)$ or $y = - 2 x - 6$ i.e. $2 x + y + 6 = 0$
graph{(x-2y+3)(2x+y+6)=0 [-13.96, 6.04, -4.16, 5.84]}<|endoftext|>
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Free and printable Math problems worksheets are available to use as your kids’ teaching resources! In these worksheets, word problems math become the most basic math skill that your kids have to explore. The exercises are created with interesting pictures to make the activity more fun. Check out the worksheets below!
These Math problems worksheets will help deepen a student’s understanding of mathematical concepts. Using these free and printable math worksheets, their skill in math will be reinforced. All the worksheets provided are the comprehensive collection of word math worksheets. More interesting word math worksheets are provided in the following images.
These worksheets will assist you to make your children get used to solve math problems provided in words. All pictures of the worksheets are provided in jpg with downloadable size. Choose the worksheets that will be best enhancing your kids’ skill. Just hit on the images and you are all set to get the worksheet. More worksheets are available below.
Print these worksheets and hand them to your children for their math activity’s worksheets. Print these fun math worksheets for practice in subtraction, addition, multiplication, division and more. Find more fun and interesting math worksheets in this site by browsing through our collection!<|endoftext|>
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The transport properties of a solid includes the Seebeck coefficient (“S” in Volts per degree), the electrical conductivity (“s” in 1/(ohm-meters)) and the thermal conductivity (“l” in Watts per meter-Kelvin). These three properties determine the thermoelectric figure of merit (Z) performance, Z º S2s/l, which has units of reciprocal temperature.
Thermoelectric Figure of Merit
From an engineering perspective, the Seebeck coefficient is defined as the ratio of electric potential difference to temperature difference in the absence of electrical current flow. The Seebeck coefficient of a material can be measured using two thermocouples plus a heater, which establishes a small temperature difference across the material sample. Accurate and precise measurement of a sample's temperature at two locations, typically a few millimeters apart, is the most challenging aspect of this measurement.
Electrical conductivity is defined as the ratio of electrical current density to electric field strength, in the absence of a temperature gradient. In practice, electrical conductivity is obtained by measuring the voltage drop (V) across a sample of length (L) and constant cross-section (A) when a uniform current (I) is flowing: s = LI/AV. For this measurement, it is important that the sample has a uniform current density and that the measured voltage does not contain any significant contact, or connector, resistances.
The thermal conductivity is defined as the ratio of heat flux to temperature gradient in the absence of electrical current flow. Thermal conductivity can be obtained by measuring the temperature difference (DT) across a sample of length (L) and constant cross-section (A) when a heat flow (Q) is established: l = LQ/ADT. Measurement errors in both heat flux and temperature make this a challenging property to quantify.
Many thermoelectric groups use a ‘4-probe’ method to collect S and s data concurrently on the same sample as a function of temperature. In this measurement method, current is injected through the end faces of bar-shaped samples and thermocouples are pressed against a side face of the sample. The resulting thermocouple-sample thermal contact may not necessarily be ideal and could possibly lead to an underestimation of temperature difference and consequent overestimation of S. Errors in 4-probe Seebeck measurements have been studied recently (J. Martin, W. Wong-Ng & M. Green, Journal of Electronic Materials 44 #6, pp 1998-2006.) and found to be significant, especially at higher temperatures.
To obtain thermal conductivity, many researchers use an indirect method rather than the direct method mentioned above. The indirect method requires measurement of the thermal diffusivity (cm2/sec, the ratio of the time derivative of temperature to the 2nd spatial derivative of temperature), the specific heat (Joules per gram-Kelvin) and the mass density (g/cm3). Thermal diffusivity often is measured using a laser flash apparatus (http://www.electronics-cooling.com/2002/05/flash-diffusivity-method-a-survey-of-capabilities/), while specific heat is measured using a differential scanning calorimeter or differential thermal analyzer.
For the measurement methods most used by materials research groups interested in thermoelectric power generation, a recent empirical assessment of errors in the thermoelectric transport properties, led by a group at Oak Ridge National Laboratory, found the following reproducibility results for measurements from 323 to 773 Kelvin: ~±7% for S, ~±8% for s, ~±8% for l, ~±15% for Z.
For temperatures from 180 to 400 Kelvin, covering essentially all thermoelectric cooling applications, II-VI Marlow uses a modified Harman method to measure the thermoelectric properties. This method uses the Peltier effect to create a temperature difference, so that no heater is required. A switched DC current is used, and the Seebeck and ohmic voltage contributions are separated transiently, taking advantage of the low thermal diffusivity of thermoelectric materials. Thermocouples are embedded in small copper blocks that are soldered to the material sample, which typically has a thin layer of nickel applied to the end faces. Fine current and thermocouple wires are used to minimize thermal losses, which are estimated and used to make corrections to the calculated thermal conductivity. II-VI Marlow believes the Z values obtained have an accuracy of ~±3%.
Whether you’re exploring outer space, working to develop the next cure for cancer or finding sustainable energy solutions, let us partner with you to help reach your goal. For specific information, contact II-VI Marlow and we will assist with the application and cooler selection
For more information about working with thermoelectric materials. Download our Guide to Working with Thermoelectric Materials!<|endoftext|>
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In this section we provide a brief introduction to the origin of money. Money is one of our most pervasive social institutions, yet it is also one of the most mysterious. Its history is the story of both a technology, an idea, power, and how we organise society. Over the centuries, human societies have developed various means for valuing and exchanging goods and for storing wealth.
As we will show the origin of money involves three different theoretical perspectives money as commodity, state/fiat and credit. To really understand all the aspects of money, one needs to understand the historical relevance and economic implications of each of these views.
The origins of money / What is money?
The commodity theory of money
The most common theory of the origins of money is the one found in “The Wealth of Nations” (1776) by Adam Smith. According to Smith’s story, money emerged with increasing productivity and the division of labour, as individuals found themselves without many of the necessities they required but at the same time an excess of their own products. Without a common means of payment individuals had to resort to barter in order to trade, which was problematic as both sides of the deal had to have something the other person wanted (the “double coincidence of wants”).
According to the theory certain types of commodities were introduced as means of exchange for goods and services to avoid this inconvenience. Particularly obvious as candidates for such a means were gold, silver and other kinds of metals, as the majority of people found them valuable and as they easily divisible into smaller units, most would accept them in exchange for their own goods or services.
This idea of gold, silver and other kinds of metal as money, is the so-called commodity theory of money, which is found in many text-books.
“Fiat money is the norm in most economies today, but most societies in the past have used a commodity with some intrinsic value for money.This type of money is called commodity money. The most widespread example is gold. When people use gold as money (or use paper money that is redeemable for gold), the economy is said to be on a gold standard. Gold is a form of commodity money because it can be used for various purposes—jewelry, dental fillings, and so on—as well as for transactions. The gold standard was common throughout the world during the late nineteenth century.”
– (Mankiw, 2012)
The theory has thus been widely perpetuated by mainstream or ‘orthodox’ economists and in economic models up until today. From this point of view, money, banks and debt are believed to have no substantial macroeconomic effect other than to “‘oil the wheels’ of trade” and thus can be largely ignored when considering the workings of the economy. This means that today banks, money or debt have almost no place in economic models.
The state-theory of money
The commodity theory is appealing and intuitively logical, as it coincides with the way we interact with money in our everyday economy, in which money functions as means of exchange when we trade goods and services, a store of value for later consumption, and a unit of account for measuring prices on goods, services and various kinds of assets.
Nonetheless, the theory is very problematic when it comes to explaining the actual historical origin and emergence of money, as well as the macroeconomic functionality of money. The commodity theory of money basically tells a story of why it is practically convenient to have a monetary system, but not really the story about what money is and how the monetary system have evolved over time. Why is something money and other things not? And if gold were really money, one should ask, why is it not money today?
Overall, the theory has two important shortcomings. The first problem with Smith’s theory about money is the lack of the factor Smith himself attempted to downplay: the political and physical power of monarchies and states. What historians and anthropologists point out is that if we look at the actual historical evidence, the extraction of gold and silver were under the monopoly of kings and states who then had the complete control of money production (Spufford, 2002).
In the UK, coinage began with the founding of the royal mint in 1886, which was under the full control of the sovereign, and in the US the first coins were issued by the Treasury in 1792 following the war of independence in 1775–1783.
The fact that money historically have been a means for concentrating power around the king, the state, the church and other forms of central institutions is underpinned by the so-called state theory of money (chartalism) (Knapp, 1905). According to this theory, the central authority issues coins (and other means of payment) in order to finance expenditures, for example soldiers’ wages, and then subsequently secures the circulation of money by collecting the same coins as taxes. Thus, the only way in which the butcher, brewer and baker in Smith’s story could get hold of gold and silver coins for the payment of taxes was if the central authority had spent them into circulation in the first place.
The theory thereby suggests the need for over-institutional legitimization of the value of money. It is not enough for there to be a simply a need for a mean of exchange, there must also be some political, legal, moral or religious institutions that support the value of money by making it eligible for payment of taxes and defining is as legal tender.
In his 1930s publication A Treatise on Money, John Maynard Keynes famously declared all money to be chartalist:
“To-day all civilised money is, beyond the possibility of dispute, chartalist.”
– Keynes (1930 p. 5)
This state theory of the historical origin of money is a very important basis for the MMT school of thought.
The credit-theory of money
The second problem with the Commodity theory of money is the idea that money emerged ‘spontaneously’ from barter into metal coins with intrinsic value. The problem with this idea is that, in the words of anthropologist David Graeber,
“… there’s no evidence that it ever happened, and an enormous amount of evidence suggesting that it did not.”
– Graeber, 2011, p. 29
This is not to say that barter does not exist or has not existed in the past, merely that money did not emerge from it. In reality, barter tends to happen only between people who are strangers (i.e. they have no ongoing relationship) or is reverted to amongst people who are used to cash transactions, but for whatever reason have no currency available, such as those in prisoner of war camps. (See Radford (1945) for a fascinating example.)
Historically the minting of coins was the privilege of sovereigns, but money as unit of account emerged long before coinage. The first money in the form of precious metals shaped into coins appeared in three separate places (northern China, northeast India, and around the Aegean Sea) between around 600 and 500 BC, but the emergence of money as a unit of account can be traced all the way back to the Mesopotamian temple and palace administrations in 3500 BC (Hudson, 2004).
Here, money evolved as a simple bookkeeping and accounting process to keep track of who owed what to whom. Money thus evolved not as metal and coins but as credit in form of debt relation between temples, states and merchants. As David Graeber explains:
“Our standard account of monetary history is precisely backwards. We did not begin with barter, discover money, and then eventually develop credit systems. It happened precisely the other way around. What we now call virtual money came first. Coins came much later, and their use spread only unevenly, never completely replacing credit systems. Barter, in turn, appears to be largely a kind of accidental by-product of the use of coinage or paper money: historically, it has mainly been what people who are used to cash transactions do when for one reason or another they have no access to currency.”
– Graeber, 2011, p. 40
Credit is a promise to pay in the future. The issuer promises to pay the bearer some specific species (coins, goods etc.) – either on demand or on some specific time in the future – in exchange for a deposit or a debt contract.
Over the course of history, credit systems have evolved in many different forms: sometimes driven by private banks, but more often controlled by state authorities in forms of temples, states and later central banks. What is special about credit is that it holds no intrinsic value in itself, but its value is almost solely based on the trust to the issuer.
One of the most well-known examples of the emergence of privately issued credit-money is the story about the London goldsmiths, who evolved into banks in the 16th century. First by issuing promissory notes in exchange for coins, bullion and other valuables, and later as people began to accept the promissory notes as money in place of coins, the goldsmiths exploited the opportunity of issuing and lending additional promissory notes (which would be seen by the borrower as a loan of money) in exchange for interest bearing debt, effectively giving them the status of money-issuing banks (He, Huang, & Wright, 2005).
The emergence of banking in the UK to some extent supports the idea of money evolving from the market’s need for a means of exchange, but only as a substitute for coins, and definitely not in accordance with the commodity theory and the idea of money as neutral medium of exchange. The goldsmiths made a lot of profit from issuing credit.
What banking crises later showed is that a private credit system eventually needs the support of a central authority and credit of a higher order in order to survive. Again, the existence of a central authority is absolutely crucial for these credit systems to work efficiently. As for credit to circulate as money it is of absolutely necessary that there is a general trust in the issuer. As Minsky puts it: “Everyone can create money; the problem is to get it accepted.”
– Minsky, 1986, p. 228.
Also, during history most credit systems have been either directly controlled by (or at least largely supported by) the public. In a sense, money issued by a state or a central bank can be seen as a special form of credit money. In the US, the first paper money was issued by the authorities at the Massachusetts Bay Colony in 1690, and later the first dollar notes came about during the civil war 1861-1865 as the North issued so-called Greenbacks (Demand Notes), which were not redeemable in gold and silver coins. Instead it derived its value from being legal tender, meaning it could be used to pay “all dues” to the federal government. This was the very first purely state-issued money. At the same time, private banks were allowed to issue their own bank notes according in accordance with the National Currency Act of 1863, but only if backed by government debt.
Today most money is credit issued by private banks.
The credit-theory of money thus best explains how money is created in the modern economy, but – just like the commodity theory – it does not explain the existence of denominations of currencies, such as. Pound Sterling. Why, for example, does Barclays not issue loans in their own “Barclay Pound”?
Overall, one can say that none of the three theories about money are unequivocally correct, but at the same time none of them are unambiguously wrong. The three theories of money live in symbiosis with changing domination to changing times.
In the UK/US, goods and services are exchanged for Pound/Dollars because the state legislates them to be legal tender and they are redeemable for paying taxes, but at the same time they are issued as credit by private banks denominated in Pound/Dollars, and used to settle the majority of transactions in the economy at a scale that would be absolutely impossible without an effective means of exchange.
Graber, D ., 2011. Debt: The First 5.000 years. Melville House
Hudson, M., 2004. The Development of Money-of-Account in Sumer’s Temples, in Hudson, M. and Wunsch, C. (eds.), Creating Economic Order. Bethesda: CDL.
Keynes, J.M., 1930. Treatise on money: Pure theory of money Vol. I.
Knapp, G.F., 1924. The state theory of money. McMaster University Archive for the History of Economic Thought.
Mankiw, G., 2012. Macroeconomics. 8th ed. New York: Worth Publishers
Minsky, H. P., 1986. Stabilizing An Unstable Economy. New Haven: Yale University Press.
Radford, R.A., 1945. The economic organisation of a POW camp. Economica, 12(48), pp.189-201.
Spufford, P., 2002. Power and profit: the merchant in medieval Europe (p. 27). London:: Thames & Hudson.<|endoftext|>
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## Want to keep learning?
This content is taken from the University of Padova's online course, Precalculus: the Mathematics of Numbers, Functions and Equations. Join the course to learn more.
1.13
## University of Padova
Skip to 0 minutes and 10 seconds Hello. Welcome back to a step in practice. We are dealing here with real numbers. In the first exercises, we are asked to describe the property that is used in each of the three formulas that are proposed. Well, in the first, 3 times x plus 4y equals 3 times x plus 12y. Well, this is the distributivity of the product over the sum. So distributivity of the product over the sum.
Skip to 0 minutes and 55 seconds The second, 4 plus, x plus y equals 4 plus x, plus y. Well, this is simply the associativity property of the sum.
Skip to 1 minute and 18 seconds So it’s the same to add 4 to x plus y or to add 4 plus x, to y. And the third property states that 3 plus x equals x plus 3. And this is the commutativity of the sum. So it’s the same to perform 3, and then to add x or to add 3 to x. So this is commutativity of the sum.
Skip to 2 minutes and 5 seconds In exercise two, we note that two integers, two real numbers are one strictly greater than the other. More precisely, x is strictly greater than y. Now, is it true that x is greater or equal than y? Yes, absolutely, because saying that x is greater or equal than y means that x is strictly greater than y or x equals y. So this proposition is true whenever at least one of the two propositions, x strictly greater than y or x equal to y, is true. So if x is strictly greater than y, the first proposition is true. So this is true.
Skip to 3 minutes and 0 seconds Now be careful that the opposite is not true, the converse is not true. If x is greater or equal than y, this does not imply that x is strictly greater than y. For instance, 3 is greater or equal than 3, but 3 is not strictly greater than 3.
Skip to 3 minutes and 23 seconds So see you in the next step.
# Real numbers in practice
The following exercises are solved in this step.
We invite you to try to solve them before watching the video.
In any case, you will find below a PDF file with the solutions.
### Exercise 1.
Which properties of the real numbers are being used in the following equalities? (Choose between: commutativity, associativity, distributivity over addition, here $$x,y$$ are real numbers):
1. $$3(x+4y)=3x+12y$$
2. $$4+(x+y)=(4+x)+y$$
3. $$3+x=x+3$$
### Exercise 2.
Let $$x,y\in\mathbb R$$ be such that $$x>y$$. Is it true or not that $$x\ge y$$?<|endoftext|>
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Images from the spacecraft orbiting Mars seem to indicate the Red Planet may once have had oceans and lakes, and researchers are still trying to figure out how these bodies of water could have developed. A new explanation is that underground aquifers fed water to the surface, forming the floors of ancient continental-scale basins on Mars. The groundwater emerged through extensive and widespread fractures, leading to the formation of river systems, large-scale regional erosion, sedimentary deposition and water ponding in widespread and long-lasting bodies of water in Mars northern plains.
J. Alexis Palmero Rodriguez, research scientist at the Planetary Science Institute PSI, has been studying the Martian northern lowlands region, finding extensive sedimentary deposits that resemble the abyssal plains of Earth’s ocean floors. It is also like the floors of other basins on Mars where oceans are thought to have developed.
The origin of these deposits and the formation of Martian lakes and seas has been a controversial subject over the years. One theory is that there was a sudden release of large volumes of water and sediment from zones of apparent crustal collapse known as “chaotic terrains.” However, these zones of collapse are on the whole rare on Mars, while the plains deposits are widespread and common within large basin settings, Rodriguez said.
From evidence in the planet’s northern plains (south of Gemini Scopuli in Planum Boreum), Rodriguez’ new model does not require sudden massive groundwater discharges. Instead, it advocates for groundwater discharges being widespread, long-lived and common in the northern plains of Mars.
“With the loss over time of water from the subsurface aquifer, areas of the northern plains ultimately collapsed, creating the rough hilly surfaces we see today. Some plateaus may have avoided this fate and preserved sedimentary plains containing an immense record of hydrologic activity,” Rodriguez said. “The geologic record in the collapsed hilly regions would have been jumbled and largely lost.
“This model implies that groundwater discharges within basin settings on Mars may have been frequent and led to formation of mud pools, lakes and oceans. In addition, our model indicates this could have happened at any point in the planet’s history,” he said. “There could have been many oceans on Mars over time.”
If life existed in Martian underground systems, life forms could have been brought up to the surface via the discharges of these deep-seated fluids. Organisms and their fossils may therefore be preserved within some of these sedimentary strata, Rodriguez said.
His paper was published in the journal Icarus.
Source: Planetary Science Institute<|endoftext|>
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# Math 5 Coordinate Grids - PowerPoint PPT Presentation
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Math 5 Coordinate Grids. Instructor: Mrs. Tew Turner. In this lesson we will how you can locate points on a coordinate grid. Divide each number by 10. 100 1000 230 1200. Math Warm-up. Divide each number by 10. 100 1000 230 1200. Math Warm-up - ANSWERS 10 100 23 120.
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Math 5 Coordinate Grids
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## Math 5Coordinate Grids
Instructor: Mrs. Tew Turner
In this lesson we will how you can locate points on a coordinate grid.
Divide each number by 10.
100
1000
230
1200
Math Warm-up
Divide each number by 10.
100
1000
230
1200
10
100
23
120
In this lesson we will answer the question:
How are ordered pairs of numbers represented on a coordinate grid?
You will need a copy of the print out called M5L#43 coordinates-1 to use for this lesson.
So pause the tape, print the handout, and then restart the tape.
Vocabulary
coordinate plane- is a flat surface formed by two intersecting, perpendicular lines. Each of these lines is called an axis.
x-axis- The horizontal line.
y-axis- The vertical line.
Vocabulary
Quadrants-The x-axis and the y-axis divide the coordinate plane into four areas. These areas are calledquadrants.
Ordered Pairs- Digits that indicate where a point is located on a coordinate grid. Ex: (3,2)
Connecting to Previous Learning
A coordinate grid makes it easy to locate a point on a map. Start at 0.
The ORDERED PAIR names a point on a coordinate grid. You may have used this skill while trying to locate a place on a map.
The numbers are all positive.
How would we locate the ordered pair, (3,2)
### Document Camera
Make sure you have the Lesson # and date on your page.
Use Coordinates handout
Guided Practice
Locate the ordered pairs:
(2, 7)
(6, 9)
(4, 6)
Guided Practice
Write the ordered pairs for each letter:
(R)
(S)
(T)
S
R
T
Independent Practice
Locate the ordered pairs:
(3, 7)
(9, 2)
(8, 6)
Independent Practice
Write the ordered pairs for each letter:
(W)
(X)
(Y)
Y
W
X
Lesson Review
• A coordinate grid makes it easy to locate a point on a map.
• Start at 0. You follow the 0 line to locate the 1st number, the x-coordinate in the ordered pair.
• (3, 6)
• The second number, the y-coordinate names the distance UP from the x-coordinate.
• (3, 6)
Quick Check
Quick Check<|endoftext|>
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Many children do not have a serious speech problems or impairments (phonological disorder or dyspraxia), but just have speech articulation difficulties with a particular sound, or produce a sound incorrectly for no obvious reason. An example of a common articualtion problem would be a “lisp”. When a child has a difficulty with just one or two sounds it is usually not too difficult to improve their speech, although it is advantageous if you try and fix the problem when they are young. If you leave a speech problem till the child gets older, it becomes more difficult to put it right. With speech therapy and practice, these articulation difficulties can soon be fixed. Although there is a general time frame when we expect most children to be able to produce certain sounds correctly, many children acquire some sounds a little later, but if you are concerned about your child’s speech development see your local speech and language therapist / pathologist and discuss the possibility of speech therapy.
Some common speech problems:
A “lisp” occurs when the individual makes a sound (usually an /s/ sound) between their tongue and their teeth. Less commonly, the sound is made because the air is pushed over the sides of the tongue rather than the front, and this is often called a “lateral lisp”.
“Gliding” occurs when sounds such as the /r/ or the /l/ are replaced with a /w/, so “rug” is pronounced and “wug“, or “lake” is pronounced as “wake“.
Another common speech error made by English speaking children is to replace the /th/ sound with a /f/ or a /v/, so “thin” is pronounced as “fin“.
Some children “front” sounds. For instance, a child may use a /t/ instead of a /c/, so “car” is pronounced as “tar“.
Other children may have difficulties articulating /sh/, /ch/, and /j/ so they have difficulties saying words such as “shop“, “chair“, or “jam“.
These difficulties in isolation can be fixed quite easily with help from a speech pathologist / therapist. When the child has a combination of the above problem, or more complex difficulties with speech, it can take longer to improve things and it is important that you seek professional help and practice regularly.
For a wider range of books, click here to see our Bookshop.<|endoftext|>
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April may be National Poetry Month but educators incorporate poetry into their curriculum throughout the school year. Listed below are great open resources to reference now or bookmark for future perusal.
Ways to Celebrate Poetry Month with The New York Times: Mostly contemporary references geared toward middle and high school instruction. A must read to gain engaging ideas on connecting teenagers and poetry. Also check out the Fifth Annual New York Times Student Poetry Contest.
Poetry Out Loud: A national contest that encourages the nation’s youth to learn about great poetry through memorization and recitation. This program helps students master public speaking skills, build self-confidence, and learn about their literary heritage. The National Endowment for the Arts and the Poetry Foundation have partnered with U.S. state arts agencies to support this project. Poetry Out Loud also has a wealth of teaching resources.
Poetry Foundation: Hosts a wealth of information and resources for poetry.
Bringing Poetry to the Classroom: National Education Association gives information about incorporating poetry in grades K-12.
Reading Rockets: Numerous links to resources ranging from book lists to video interviews with poets and writers.<|endoftext|>
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The locations of all 2704 Gamma Ray Bursts detected by BATSE in the 9 year mission.
Click on image for full size
NASA, Compton Gamma Ray Observatory, BATSE Team
Gamma Ray Bursts - The Most Powerful Objects in the Universe?
In the 1960's, the United States launched some satellites
to look for very high energy light, called Gamma Rays.
Gamma Rays are produced whenever a nuclear bomb explodes.
The satellites found many bursts of Gamma Rays, but they
were not coming from explosions on Earth. They were coming
from outer space.
Modern satellites have found thousands of these Gamma
Ray Bursts. They happen about once a day and come from all over
the sky, as the map shows. There are two types of bursts. Some
are short, lasting less than 2 seconds. Others are longer,
bursting for as long as 1000 seconds. We now think that
all Gamma Ray Bursts come from the creation of black holes in
distant galaxies. The two types of bursts come from two different
ways to make a black hole.
Short Gamma Ray Bursts come from two neutron stars orbiting each
other. They slowly lose energy and merge together to form a black
hole. The gamma rays come from debris falling into the black hole.
Long Gamma Ray Bursts come from the deaths of very massive stars.
At the end of their lives, these stars collapse and explode as a
type of supernova. The gamma rays shoot out along jets from these
Shop Windows to the Universe Science Store!
The Fall 2009 issue of The Earth Scientist
, which includes articles on student research into building design for earthquakes and a classroom lab on the composition of the Earth’s ancient atmosphere, is available in our online store
You might also be interested in:
Neutron Stars are the end point of a massive star's life. When a really massive star runs out of nuclear fuel in its core the core begins to collapse under gravity. When the core collapses the entire star...more
Scientists and students have designed a new satellite called Firefly for the CubeSat program. The Firefly satellite is the size of a loaf of bread and consists of three cubes attached end to end in a rectangular...more
In the 1960's, the United States launched some satellites to look for very high energy light, called Gamma Rays. Gamma Rays are produced whenever a nuclear bomb explodes. The satellites found many bursts...more
During the early 1900's, which is not very long ago, astronomers were unaware that there were other galaxies outside our own Milky Way Galaxy. When they saw a small fuzzy patch in the sky through their...more
Spiral galaxies may remind you of a pinwheel. They are rotating disks of mostly hydrogen gas, dust and stars. Through a telescope or binoculars, the bright nucleus of the galaxy may be visible but the...more
When stars like our own sun die they will become white dwarfs. As a star like our sun is running out of fuel in its core it begins to bloat into a red giant. This will happen to our sun in 5 Billion years....more
What's in a Name: Arabic for "head of the demon" Claim to Fame: Represents Medusa's eye in Perseus. A special variable star that "winks" every 3 days. Type of Star: Blue-white Main Sequence Star, and...more<|endoftext|>
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### Regular Polygon
A regular polygon is a plane shape with equal sides and equal angles, which means that all sides and all angles of a regular polygon are in harmony.
Common examples of regular polygons are:
1. Equilateral triangle
2. Square
3. Regular pentagon
4. Regular hexagon
5. Regular octagon
An equilateral triangle is a triangle with three equal angles and three equal sides.
The sum of angles of a triangle is equal to 180°. Since there are three angles in a triangle, each angle in an equilateral triangle is equal to 180°/3 = 60°
We know that the length and breadth of a square are equal.
l = b
Where, l = length and b = breadth
The perimeter (P) of a square is the sum of the four sides of the square.
P = l + b + l + b
Since l = b,
P = l + l + l + l
= 4l
4l implies that the four sides of the square are equal.
A square has four angles and each angle is equal to 90°.
The angles in a polygon can be obtained via the formular of the sum of the angles of an n-sided polygon, where n is equal to the number of sides of the polygon. The formular is (n-2)180°.
A pentagon is a 5-sided polygon, which means that n = 5. A hexagon is a 6-sided polygon where n= 6. And an octagon is an 8-sided polygon where n = 8.
Apart from the equilateral triangle, the other four regular polygons are crucial to the study of the geometric structure of the harmonious universe. The regular hexagon, for instance, denotes the six square facets of the cube representing a balanced and stable world.
Bagua or Pakua is structurally an octagon. It is broken or Yin or two (even number) if it is an irregular octagon, and unbroken or Yang or one (odd number) if it is a regular octagon.
Therefore,
Broken (2) = Irregular Polygon
Unbroken (1) = Regular Polygon<|endoftext|>
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A billion years ago, a star nursery gave birth to about 4000 stars. Ordinarily, this cluster of newborns would have been pulled apart by the galaxy’s gravitational forces and sent scattering through the Milky Way.
Instead, they generated enough gravity between them to hold together – and form a stellar stream, or a river of stars.
What’s relevant to Australia — and people living elsewhere in the Southern Hemisphere – is this river covers most of the southern night sky, flowing together as one mighty body.
Right under our noses
And while it has been streaming under the Earth’s gaze from long before people existed – and is relatively near to our solar system — until a couple of weeks ago, we didn’t know what we were staring at.
Researchers from the University of Vienna used data from the European Space Agency satellite Gaia to make their discovery, which they describe as being “shockingly close to the Sun” at 100 parsecs or 326 light years.
The stream itself is four times as long – it would take 1304 light years to travel from one end to the other. So it’s huge.
João Alves is a professor of stellar astrophysics at the University of Vienna, and co-author of the research paper containing the astronomers’ findings published in Astronomy & Astrophysics.
In a prepared statement from the journal, he said: “Identifying nearby disk streams is like looking for the proverbial needle in a haystack.
“Astronomers have been looking at, and through, this new stream for a long time, as it covers most of the night sky, but only now realise it is there, and it is huge, and shockingly close to the Sun.”
An astronomer’s dream
Dr Alves said finding such a stream “close to home is very useful, it means they are not too faint nor too blurred for further detailed exploration, as astronomers dream”.
The scientists used Gaia data to measure the 3D motion of stars in space.
As they surveyed the distribution of nearby stars moving together, they were drawn to a particular group that was unknown and unstudied – and that “showed precisely the expected characteristics of a cluster of stars born together but being pulled apart” by the gravitational field of the Milky Way.
“Most star clusters in the Galactic disk disperse rapidly after their birth as they do not contain enough stars to create a deep gravitational potential well, or in other words, they do not have enough glue to keep them together,” said Stefan Meingast, lead author of the paper, in a prepared statement.
“Even in the immediate solar neighbourhood there are, however, a few clusters with sufficient stellar mass to remain bound for several hundred million years. So in principle similar, large, stream-like remnants of clusters or associations should also be part of the Milky Way disk.”
Bigger than most known star clusters
In other words, the behaviour of the Milky Way predicted the existence of the stellar stream that is “more massive than most known clusters in the immediate solar neighbourhood”.
At one billion years old, the cluster has completed four full orbits around the Galaxy, time enough to be strung out into the stream-like structure as a consequence of gravitational interaction with the Milky Way disc.
According to Astronomy & Astrophysics, the river of stars can be used as a valuable gravity probe to measure the mass of the Galaxy. It also opens a path to telling us how galaxies get their stars, test the gravitational field of the Milky Way, and, because of its proximity, become “a wonderful target for planet-finding missions”.
Associate Professor Daniel Zucker is an an astronomer and researcher with Macquarie University. He is part of an Australian-led group of astronomers working with European collaborators on a project called the GALAH survey, which has decoded the “DNA” of more than 340,000 stars in the Milky Way, which they hope will help them find the siblings of the Sun – the stars that were born in the same cosmic nursery.
Dr Zucker said the discovery of the stellar stream being so close to our solar system – and containing heavy elements that are comparable to those in our Sun – could help solve the puzzle.
“We think stars form in clusters and over time they dissipate through the Milky Way,” he said.
“The stars in the (newly discovered stellar stream) are relatively young. They could be younger than a billion years … They could tell us how our Sun formed but also what happened to our Sun’s siblings. Like where are they now?”<|endoftext|>
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When it comes to greenhouse gases, carbon dioxide tends to steal the spotlight — but new research in the journal Proceedings of the National Academy of Sciences (PNAS) reveals how scientists have developed a new, predictive tool to estimate nitrous oxide (N2O) emissions from rivers and streams around the world. N2O, a greenhouse gas with 300 times the warming potential of carbon dioxide, persists for over a century in the Earth’s atmosphere and is known to cause significant damage to the Earth’s ozone layer. Rivers and streams can be sources of N2O because they are hotspots for denitrification, a process whereby microbes convert dissolved nitrogen into nitrogenous gas.
While previous research has attempted to quantify where and when N2O is emitted, rivers and streams have posed a significant challenge because accurately measuring N2O from flowing waters is difficult, particularly at the scale of an entire river system. The current study presents a widely applicable predictive model from which to estimate N2O emissions from waterways based on simple metrics including stream size, land use and land cover of adjacent landscape, biome type and varying climatic conditions.
“Rapid land use change, such as the conversion of historic wetlands to agricultural lands, has increased the delivery of bioavailable nitrogen from the landscape to the detriment of receiving streams and rivers,” said Jennifer Tank, Galla Professor in the Department of Biological Sciences at the University of Notre Dame, co-author of the study and director of Notre Dame’s Environmental Change Initiative. “Some of that nitrogen will be converted by microbes into N2O, and because it is a powerful greenhouse gas, where and when that happens in flowing waters is of great interest, both now and into the future.”
Working with an international team of scientists, Tank and her graduate student Martha Dee analyzed previously published emissions data from streams and rivers around the world including Michigan’s Kalamazoo River, New York’s Hudson River, the Swale-Ouse River in the United Kingdom and six large rivers across Africa. In addition, the team collected its own measurements of N2O from two river networks regionally, including the Manistee River in Michigan and the Tippecanoe River in Indiana. The researchers’ analysis of the combined dataset found that N2O emissions are dependent on river size — as it increases, the production of N2O shifts from the streambed to the overlying water.
“The current understanding of nitrous oxide production is limited in stream and river networks in a time of rapid global change,” said Dee co-author of the study. “Our study uses a diverse, global set of data combined with regional measurements to create a model that that can better predict the impact of human activity and environmental drivers on N2O production.”
The new model will be a valuable tool for scientists and water managers alike, as the framework allows for accurate prediction of N2O emissions under a variety of scenarios including water temperature, changes in land use and the influence of climate change on emission outcomes.
Co-authors of the study include Alessandra Marzadri and Daniele Tonina at the Center for Ecohydraulics Research at the University of Idaho, and Alberto Bellin in the Department of Civil, Environmental and Mechanical Engineering at the University of Trento in Italy.
Research was funded through a collaborative grant from the National Science Foundation Hydrologic Sciences Program.
Contact: Jennifer Tank, 574-631-3976, [email protected]<|endoftext|>
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NCERT Solutions for Class 10 Maths Unit 4
Quadratic Equations Class 10
Unit 4 Quadratic Equations Exercise 4.1, 4.2, 4.3, 4.3 4.4, 4.4 Solutions
In elementary algebra, a quadratic equation is any equation having the form
a2 + bx + c = 0
where x represents an unknown, and a, b, and c represent numbers such that a is not equal to 0. If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term. Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation only contains powers of x that are non-negative integers, and therefore it is a polynomial equation, and in particular it is a second degree polynomial equation since the greatest power is two.
Quadratic equations can be solved by factoring, by completing the square, by using the quadratic formula, or by graphing. Solutions to problems equivalent to the quadratic equation were known as early as 2000 BC.
Exercise 4.1 : Solutions of Questions on Page Number : 73
Q1 :
Check whether the following are quadratic equations:
It is of the form .
Hence, the given equation is a quadratic equation.
It is of the form .
Hence, the given equation is a quadratic equation.
It is not of the form .
Hence, the given equation is not a quadratic equation.
It is of the form .
Hence, the given equation is a quadratic equation.
It is of the form .
Hence, the given equation is a quadratic equation.
It is not of the form .
Hence, the given equation is not a quadratic equation.
It is not of the form .
Hence, the given equation is not a quadratic equation.
It is of the form .
Hence, the given equation is a quadratic equation.
Q2 :
Represent the following situations in the form of quadratic equations.
(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan's present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
(i) Let the breadth of the plot be x m.
Hence, the length of the plot is (2x + 1) m.
Area of a rectangle = Length x Breadth
∴ 528 = x (2x + 1)
(ii) Let the consecutive integers be x and x + 1.
It is given that their product is 306.
(iii) Let Rohan’s age be x.
Hence, his mother’s age = x + 26
3 years hence,
Rohan’s age = x + 3
Mother’s age = x + 26 + 3 = x + 29
It is given that the product of their ages after 3 years is 360.
(iv) Let the speed of train be x km/h.
Time taken to travel 480 km =
In second condition, let the speed of train = km/h
It is also given that the train will take 3 hours to cover the same distance.
Therefore, time taken to travel 480 km = hrs
Speed x Time = Distance
⇒480+3x-3840x-24=480
⇒3x-3840x=24
⇒3x2-24x-3840=0
⇒x2-8x-1280=0
Exercise 4.2 : Solutions of Questions on Page Number : 76
Q1 :
Find the roots of the following quadratic equations by factorisation:
Q2 :
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. Find out how many marbles they had to start with.
(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs 750. Find out the number of toys produced on that day.
Q3 :
Find two numbers whose sum is 27 and product is 182.
Q4 :
Find two consecutive positive integers, sum of whose squares is 365.
Q5 :
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
Q6 :
A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.
Exercise 4.3 : Solutions of Questions on Page Number : 87
Q1 :
Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
Q2 :
Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.
Exercise 4.3 4.4 : Solutions of Questions on Page Number : 88
Q1 :
Find the nature of the roots of the following quadratic equations.
If the real roots exist, find them;
(I) 2x2 - 3x + 5 = 0
(II)
(III) 2x2 - 6x + 3 = 0
Q2 :
Find the roots of the following equations:
Q3 :
The sum of the reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is. Find his present age.
Q4 :
In a class test, the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
Q5 :
The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
Q6 :
The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
Q7 :
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Q8 :
Two water taps together can fill a tank in hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Q9 :
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speeds of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.
Q10 :
Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, find the sides of the two squares.
Exercise 4.4 : Solutions of Questions on Page Number : 91
Q1 :
Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(I) 2x2 + kx + 3 = 0
(II) kx (x - 2) + 6 = 0
Q2 :
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2?
If so, find its length and breadth.
Q3 :
Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
Q4 :
Is it possible to design a rectangular park of perimeter 80 and area 400 m2? If so find its length and breadth.<|endoftext|>
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# Line Graph and Pie Chart MCQ Quiz - Objective Question with Answer for Line Graph and Pie Chart - Download Free PDF
Last updated on Feb 20, 2024
## Latest Line Graph and Pie Chart MCQ Objective Questions
#### Comprehension:
Directions: Answer the questions based on the information given below.
The following pie chart represents the total quantity (in ml) of five mixtures 'P', 'Q', 'R', 'S', and 'T'. Each mixture contains honey and water.
The following line graph represents the difference between the quantity of honey and water in each mixture.
Some additional information is also known :
(1) Quantity of honey is more than the quantity of water in mixture 'P', 'Q', and 'T'.
(2) The quantity of water is more than the quantity of honey in mixture 'R' and 'S'.
(3) The quantity of honey and the quantity of water in each mixture is an 'integer' value only.
If mixture 'Q' contains (2z + 5)% of water then find the value of 'z'?
1. 10
2. 15
3. 20
4. 25
5. 30
Option 1 : 10
#### Line Graph and Pie Chart Question 1 Detailed Solution
Let the total quantity of (honey + water) in all five mixture together = 360°
Total quantity of (honey + water) in mixture 'S' = 24% = (24/100) × 360° = 86.4°
ATQ, 54° + 90° + 160 + 86.4° + 57.6° = 360°
⇒ 288° + 160 = 360°
⇒ 72° = 160
⇒ 1° = (20/9)
Total quantity of (honey + water) mixture 'P' = 54° = 54 × (20/9) = 120 ml
Total quantity of (honey + water) mixture 'Q' = 90° = 90 × (20/9) = 200 ml
Total quantity of (honey + water) mixture 'S' = 86.4° = 86.4 × (20/9) = 192 ml
Total quantity of (honey + water) mixture 'T' = 57.6° = 57.6 × (20/9) = 128 ml
For mixture 'P' :
Total quantity of (honey + water) mixture 'P' = 120 ml
Difference between the quantity of honey and water in mixture 'P' = 40 ml
⇒ Honey - water = 40 ml
Quantity of honey in mixture 'P' = (120 + 40)/2 = 80 ml
Quantity of water in mixture 'P' = (120 - 40)/2 = 40 ml
For mixture 'Q' :
Total quantity of (honey + water) mixture 'Q' = 200 ml
Difference between the quantity of honey and water in mixture 'Q' = 100 ml
⇒ Honey - water = 100 ml
Quantity of honey in mixture 'Q' = (200 + 100)/2 = 150 ml
Quantity of water in mixture 'Q' = (200 - 100)/2 = 50 ml
For mixture 'R' :
Total quantity of (honey + water) mixture 'R' = 160 ml
Difference between the quantity of honey and water in mixture 'R' = 70 ml
⇒ water - honey = 70 ml
Quantity of honey in mixture 'R' = (160 - 70)/2 = 45 ml
Quantity of water in mixture 'R' = (160 + 70)/2 = 115 ml
For mixture 'S' :
Total quantity of (honey + water) mixture 'S' = 192 ml
Difference between the quantity of honey and water in mixture 'S' = 16 ml
⇒ water - honey = 16 ml
Quantity of honey in mixture 'S' = (192 - 16)/2 = 88 ml
Quantity of water in mixture 'S' = (192 + 16)/2 = 104 ml
For mixture 'T' :
Total quantity of (honey + water) mixture 'T' = 128 ml
Difference between the quantity of honey and water in mixture 'T' = 28 ml
⇒ Honey - water = 100 ml
Quantity of honey in mixture 'T' = (128 + 28)/2 = 78 ml
Quantity of water in mixture 'T' = (128 - 28)/2 = 50 ml
Putting all the values in a table we get
Mixture Quantity of honey (ml) Quantity of water (ml) Total quantity of (honey + water)
P 80 40 120
Q 150 50 200
R 45 115 160
S 88 104 192
T 78 50 128
Percentage of water in mixture 'Q' = (50/200) × 100 = 25%
ATQ, (2z + 5)% = 25%
⇒ z = 10
Hence, the correct answer is 10.
#### Comprehension:
Directions: Answer the questions based on the information given below.
The following pie chart represents the total quantity (in ml) of five mixtures 'P', 'Q', 'R', 'S', and 'T'. Each mixture contains honey and water.
The following line graph represents the difference between the quantity of honey and water in each mixture.
Some additional information is also known :
(1) Quantity of honey is more than the quantity of water in mixture 'P', 'Q', and 'T'.
(2) The quantity of water is more than the quantity of honey in mixture 'R' and 'S'.
(3) The quantity of honey and the quantity of water in each mixture is an 'integer' value only.
If mixture 'P' and 'R' are mixed together to form a new mixture and 20 ml water is added to it then what will be the ratio of honey to water in the newly formed mixture?
1. 2 : 5
2. 4 : 5
3. 5 : 7
4. 6 : 7
5. 9 : 11
Option 3 : 5 : 7
#### Line Graph and Pie Chart Question 2 Detailed Solution
Let the total quantity of (honey + water) in all five mixture together = 360°
Total quantity of (honey + water) in mixture 'S' = 24% = (24/100) × 360° = 86.4°
ATQ, 54° + 90° + 160 + 86.4° + 57.6° = 360°
⇒ 288° + 160 = 360°
⇒ 72° = 160
⇒ 1° = (20/9)
Total quantity of (honey + water) mixture 'P' = 54° = 54 × (20/9) = 120 ml
Total quantity of (honey + water) mixture 'Q' = 90° = 90 × (20/9) = 200 ml
Total quantity of (honey + water) mixture 'S' = 86.4° = 86.4 × (20/9) = 192 ml
Total quantity of (honey + water) mixture 'T' = 57.6° = 57.6 × (20/9) = 128 ml
For mixture 'P' :
Total quantity of (honey + water) mixture 'P' = 120 ml
Difference between the quantity of honey and water in mixture 'P' = 40 ml
⇒ Honey - water = 40 ml
Quantity of honey in mixture 'P' = (120 + 40)/2 = 80 ml
Quantity of water in mixture 'P' = (120 - 40)/2 = 40 ml
For mixture 'Q' :
Total quantity of (honey + water) mixture 'Q' = 200 ml
Difference between the quantity of honey and water in mixture 'Q' = 100 ml
⇒ Honey - water = 100 ml
Quantity of honey in mixture 'Q' = (200 + 100)/2 = 150 ml
Quantity of water in mixture 'Q' = (200 - 100)/2 = 50 ml
For mixture 'R' :
Total quantity of (honey + water) mixture 'R' = 160 ml
Difference between the quantity of honey and water in mixture 'R' = 70 ml
⇒ water - honey = 70 ml
Quantity of honey in mixture 'R' = (160 - 70)/2 = 45 ml
Quantity of water in mixture 'R' = (160 + 70)/2 = 115 ml
For mixture 'S' :
Total quantity of (honey + water) mixture 'S' = 192 ml
Difference between the quantity of honey and water in mixture 'S' = 16 ml
⇒ water - honey = 16 ml
Quantity of honey in mixture 'S' = (192 - 16)/2 = 88 ml
Quantity of water in mixture 'S' = (192 + 16)/2 = 104 ml
For mixture 'T' :
Total quantity of (honey + water) mixture 'T' = 128 ml
Difference between the quantity of honey and water in mixture 'T' = 28 ml
⇒ Honey - water = 100 ml
Quantity of honey in mixture 'T' = (128 + 28)/2 = 78 ml
Quantity of water in mixture 'T' = (128 - 28)/2 = 50 ml
Putting all the values in a table we get
Mixture Quantity of honey (ml) Quantity of water (ml) Total quantity of (honey + water)
P 80 40 120
Q 150 50 200
R 45 115 160
S 88 104 192
T 78 50 128
If mixture 'P' and 'R' are mixed together to form a new mixture and 20 ml water is added to it
The total quantity of honey in final mixture = 80 + 45 = 125 ml
The total quantity of water in final mixture = 40 + 115 + 20 = 175 ml
Required ratio = 125 : 175 = 5 : 7
Hence, the correct answer is 5 : 7.
#### Comprehension:
Directions: Answer the questions based on the information given below.
The following pie chart represents the total quantity (in ml) of five mixtures 'P', 'Q', 'R', 'S', and 'T'. Each mixture contains honey and water.
The following line graph represents the difference between the quantity of honey and water in each mixture.
Some additional information is also known :
(1) Quantity of honey is more than the quantity of water in mixture 'P', 'Q', and 'T'.
(2) The quantity of water is more than the quantity of honey in mixture 'R' and 'S'.
(3) The quantity of honey and the quantity of water in each mixture is an 'integer' value only.
If 60% of mixture 'P', 50% of mixture 'T', and 40% of mixture 'U' are added to form a mixture 'V' then the quantity of honey in mixture 'V' becomes 109 ml. If the total quantity of mixture 'U' is 20 ml less than that of mixture 'P' then find the quantity of water in mixture 'U'?
1. 35 ml
2. 45 ml
3. 50 ml
4. 55 ml
5. 60 ml
Option 2 : 45 ml
#### Line Graph and Pie Chart Question 3 Detailed Solution
Let the total quantity of (honey + water) in all five mixture together = 360°
Total quantity of (honey + water) in mixture 'S' = 24% = (24/100) × 360° = 86.4°
ATQ, 54° + 90° + 160 + 86.4° + 57.6° = 360°
⇒ 288° + 160 = 360°
⇒ 72° = 160
⇒ 1° = (20/9)
Total quantity of (honey + water) mixture 'P' = 54° = 54 × (20/9) = 120 ml
Total quantity of (honey + water) mixture 'Q' = 90° = 90 × (20/9) = 200 ml
Total quantity of (honey + water) mixture 'S' = 86.4° = 86.4 × (20/9) = 192 ml
Total quantity of (honey + water) mixture 'T' = 57.6° = 57.6 × (20/9) = 128 ml
For mixture 'P' :
Total quantity of (honey + water) mixture 'P' = 120 ml
Difference between the quantity of honey and water in mixture 'P' = 40 ml
⇒ Honey - water = 40 ml
Quantity of honey in mixture 'P' = (120 + 40)/2 = 80 ml
Quantity of water in mixture 'P' = (120 - 40)/2 = 40 ml
For mixture 'Q' :
Total quantity of (honey + water) mixture 'Q' = 200 ml
Difference between the quantity of honey and water in mixture 'Q' = 100 ml
⇒ Honey - water = 100 ml
Quantity of honey in mixture 'Q' = (200 + 100)/2 = 150 ml
Quantity of water in mixture 'Q' = (200 - 100)/2 = 50 ml
For mixture 'R' :
Total quantity of (honey + water) mixture 'R' = 160 ml
Difference between the quantity of honey and water in mixture 'R' = 70 ml
⇒ water - honey = 70 ml
Quantity of honey in mixture 'R' = (160 - 70)/2 = 45 ml
Quantity of water in mixture 'R' = (160 + 70)/2 = 115 ml
For mixture 'S' :
Total quantity of (honey + water) mixture 'S' = 192 ml
Difference between the quantity of honey and water in mixture 'S' = 16 ml
⇒ water - honey = 16 ml
Quantity of honey in mixture 'S' = (192 - 16)/2 = 88 ml
Quantity of water in mixture 'S' = (192 + 16)/2 = 104 ml
For mixture 'T' :
Total quantity of (honey + water) mixture 'T' = 128 ml
Difference between the quantity of honey and water in mixture 'T' = 28 ml
⇒ Honey - water = 100 ml
Quantity of honey in mixture 'T' = (128 + 28)/2 = 78 ml
Quantity of water in mixture 'T' = (128 - 28)/2 = 50 ml
Putting all the values in a table we get
Mixture Quantity of honey (ml) Quantity of water (ml) Total quantity of (honey + water)
P 80 40 120
Q 150 50 200
R 45 115 160
S 88 104 192
T 78 50 128
Total quantity of mixture 'U' = 120 - 20 = 100 ml
Let quantity of honey in 'U' = k ml
If 60% of mixture 'P', 50% of mixture 'T', and 40% of mixture 'U' are added to form a mixture 'V' then the quantity of honey in mixture 'V' becomes 109 ml.
ATQ, 60% × 80 + 50% × 78 + 40% × k = 109
⇒ k = 55
Quantity of water in mixture 'U' = 100 - 55 = 45 ml
Hence, the correct answer is 45 ml.
#### Comprehension:
Directions: Answer the questions based on the information given below.
The following pie chart represents the total quantity (in ml) of five mixtures 'P', 'Q', 'R', 'S', and 'T'. Each mixture contains honey and water.
The following line graph represents the difference between the quantity of honey and water in each mixture.
Some additional information is also known :
(1) Quantity of honey is more than the quantity of water in mixture 'P', 'Q', and 'T'.
(2) The quantity of water is more than the quantity of honey in mixture 'R' and 'S'.
(3) The quantity of honey and the quantity of water in each mixture is an 'integer' value only.
If 36 ml of mixture 'S' and 30 ml of mixture 'P' is replaced by water then find the difference between the new quantity of honey in mixture 'P' and mixture 'S'?
1. 7.5 ml
2. 9.5 ml
3. 11.5 ml
4. 13.5 ml
5. 17.5 ml
Option 3 : 11.5 ml
#### Line Graph and Pie Chart Question 4 Detailed Solution
Let the total quantity of (honey + water) in all five mixture together = 360°
Total quantity of (honey + water) in mixture 'S' = 24% = (24/100) × 360° = 86.4°
ATQ, 54° + 90° + 160 + 86.4° + 57.6° = 360°
⇒ 288° + 160 = 360°
⇒ 72° = 160
⇒ 1° = (20/9)
Total quantity of (honey + water) mixture 'P' = 54° = 54 × (20/9) = 120 ml
Total quantity of (honey + water) mixture 'Q' = 90° = 90 × (20/9) = 200 ml
Total quantity of (honey + water) mixture 'S' = 86.4° = 86.4 × (20/9) = 192 ml
Total quantity of (honey + water) mixture 'T' = 57.6° = 57.6 × (20/9) = 128 ml
For mixture 'P' :
Total quantity of (honey + water) mixture 'P' = 120 ml
Difference between the quantity of honey and water in mixture 'P' = 40 ml
⇒ Honey - water = 40 ml
Quantity of honey in mixture 'P' = (120 + 40)/2 = 80 ml
Quantity of water in mixture 'P' = (120 - 40)/2 = 40 ml
For mixture 'Q' :
Total quantity of (honey + water) mixture 'Q' = 200 ml
Difference between the quantity of honey and water in mixture 'Q' = 100 ml
⇒ Honey - water = 100 ml
Quantity of honey in mixture 'Q' = (200 + 100)/2 = 150 ml
Quantity of water in mixture 'Q' = (200 - 100)/2 = 50 ml
For mixture 'R' :
Total quantity of (honey + water) mixture 'R' = 160 ml
Difference between the quantity of honey and water in mixture 'R' = 70 ml
⇒ water - honey = 70 ml
Quantity of honey in mixture 'R' = (160 - 70)/2 = 45 ml
Quantity of water in mixture 'R' = (160 + 70)/2 = 115 ml
For mixture 'S' :
Total quantity of (honey + water) mixture 'S' = 192 ml
Difference between the quantity of honey and water in mixture 'S' = 16 ml
⇒ water - honey = 16 ml
Quantity of honey in mixture 'S' = (192 - 16)/2 = 88 ml
Quantity of water in mixture 'S' = (192 + 16)/2 = 104 ml
For mixture 'T' :
Total quantity of (honey + water) mixture 'T' = 128 ml
Difference between the quantity of honey and water in mixture 'T' = 28 ml
⇒ Honey - water = 100 ml
Quantity of honey in mixture 'T' = (128 + 28)/2 = 78 ml
Quantity of water in mixture 'T' = (128 - 28)/2 = 50 ml
Putting all the values in a table we get
Mixture Quantity of honey (ml) Quantity of water (ml) Total quantity of (honey + water)
P 80 40 120
Q 150 50 200
R 45 115 160
S 88 104 192
T 78 50 128
If 36 ml of mixture 'S' and 30 ml of mixture 'P' is replaced by water.
The new quantity of honey in mixture 'P' = 80 × [(120 - 30)/120] = 60 ml
The new quantity of honey in mixture 'S' = 88 × [(192 - 36)/192] = 71.5 ml
Required difference = 71.5 - 60 = 11.5 ml
Hence, the correct answer is 11.5 ml
#### Comprehension:
Directions: Answer the questions based on the information given below.
The following pie chart represents the total quantity (in ml) of five mixtures 'P', 'Q', 'R', 'S', and 'T'. Each mixture contains honey and water.
The following line graph represents the difference between the quantity of honey and water in each mixture.
Some additional information is also known :
(1) Quantity of honey is more than the quantity of water in mixture 'P', 'Q', and 'T'.
(2) The quantity of water is more than the quantity of honey in mixture 'R' and 'S'.
(3) The quantity of honey and the quantity of water in each mixture is an 'integer' value only.
If 20% of the total mixture of 'P' is replaced by water and this process is repeated one more time then find quantity of water in the final mixture of 'P'?
1. 62.8 ml
2. 64.8 ml
3. 65.8 ml
4. 66.8 ml
5. 68.8 ml
Option 5 : 68.8 ml
#### Line Graph and Pie Chart Question 5 Detailed Solution
Let the total quantity of (honey + water) in all five mixture together = 360°
Total quantity of (honey + water) in mixture 'S' = 24% = (24/100) × 360° = 86.4°
ATQ, 54° + 90° + 160 + 86.4° + 57.6° = 360°
⇒ 288° + 160 = 360°
⇒ 72° = 160
⇒ 1° = (20/9)
Total quantity of (honey + water) mixture 'P' = 54° = 54 × (20/9) = 120 ml
Total quantity of (honey + water) mixture 'Q' = 90° = 90 × (20/9) = 200 ml
Total quantity of (honey + water) mixture 'S' = 86.4° = 86.4 × (20/9) = 192 ml
Total quantity of (honey + water) mixture 'T' = 57.6° = 57.6 × (20/9) = 128 ml
For mixture 'P' :
Total quantity of (honey + water) mixture 'P' = 120 ml
Difference between the quantity of honey and water in mixture 'P' = 40 ml
⇒ Honey - water = 40 ml
Quantity of honey in mixture 'P' = (120 + 40)/2 = 80 ml
Quantity of water in mixture 'P' = (120 - 40)/2 = 40 ml
For mixture 'Q' :
Total quantity of (honey + water) mixture 'Q' = 200 ml
Difference between the quantity of honey and water in mixture 'Q' = 100 ml
⇒ Honey - water = 100 ml
Quantity of honey in mixture 'Q' = (200 + 100)/2 = 150 ml
Quantity of water in mixture 'Q' = (200 - 100)/2 = 50 ml
For mixture 'R' :
Total quantity of (honey + water) mixture 'R' = 160 ml
Difference between the quantity of honey and water in mixture 'R' = 70 ml
⇒ water - honey = 70 ml
Quantity of honey in mixture 'R' = (160 - 70)/2 = 45 ml
Quantity of water in mixture 'R' = (160 + 70)/2 = 115 ml
For mixture 'S' :
Total quantity of (honey + water) mixture 'S' = 192 ml
Difference between the quantity of honey and water in mixture 'S' = 16 ml
⇒ water - honey = 16 ml
Quantity of honey in mixture 'S' = (192 - 16)/2 = 88 ml
Quantity of water in mixture 'S' = (192 + 16)/2 = 104 ml
For mixture 'T' :
Total quantity of (honey + water) mixture 'T' = 128 ml
Difference between the quantity of honey and water in mixture 'T' = 28 ml
⇒ Honey - water = 100 ml
Quantity of honey in mixture 'T' = (128 + 28)/2 = 78 ml
Quantity of water in mixture 'T' = (128 - 28)/2 = 50 ml
Putting all the values in a table we get
Mixture Quantity of honey (ml) Quantity of water (ml) Total quantity of (honey + water)
P 80 40 120
Q 150 50 200
R 45 115 160
S 88 104 192
T 78 50 128
If 20% of the total mixture of 'P' is replaced by water and this process is repeated one more time.
The final quantity of honey in mixture 'P' = 80 × (100 - 20)% × (100 - 20)% = 51.2 ml
The final quantity of water in mixture 'P' = 120 - 51.2 = 68.8 ml
Hence, the correct answer is 68.8 ml.
## Top Line Graph and Pie Chart MCQ Objective Questions
#### Line Graph and Pie Chart Question 6
Two line charts are given below. Line chart 1 shows the ratio of number of males to the number of females in two companies A and B for the 5 years. Line chart 2 shows the total number of males (both companies A and B) and total number of females (both companies A and B) for the 5 years.
What is the ratio of number of males of company B in Y1 to the total number of females of company A in Y3 and Y5 ?
1. 117 ∶ 218
2. 117 ∶ 215
3. 129 ∶ 215
4. 119 ∶ 218
Option 2 : 117 ∶ 215
#### Line Graph and Pie Chart Question 6 Detailed Solution
Concept used:
If Ratio of two numbers is given as 1.1 , then the numbers would be written as 11x and 10x where x is HCF of both numbers.
Calculations:
Let the number of males and females in company A in Y1 be 11k and 10k respectively and the number of males and females in company B in Y2 be 9m and 10m respectively.Then,
11k + 9m = 21100 ---------(1)
And 10k + 10m = 20600 ----(2)
On solving (1) and (2), we get:
m = 780
So, The number of males in Company B in Y1 is 9m = 9(780) = 7020
Similarly let is assume the number of males and females in company B in Y3 be 27t and 20t respectively and the number of males and females in company A in Y3 be 16l and 20l respectively. Then,
27t + 16l = 18025
And 20t + 20l = 16000
On solving (3) and (4), we get:
t = 475 and l = 325
So, Total number of females in company A in Y3 = 20l = 20(325) = 6500
Now, Let the number of males and females in company A in Y5 be 28x and 20x respectively and the number of males and females in company B in Y5 be 17y and 20y respectively.Then,
28x + 17y = 13550 ------------(5)
And 20x + 20y = 11800 ------(6)
On solving equations (5) and (6) , we get :
x = 320
So, The number of females in company A in Y5
= 20x
= 20(320)
= 6400
Now, required ratio
= 7020 : (6500 + 6400)
= 7020 : 12900
= 117 : 215
Hence, The Required value is 117 : 215.
#### Line Graph and Pie Chart Question 7
The number of students passing of failing in an exam for a particular subject is presented in the bar chart above. Students who pass the exam cannot appear for the exam again. Students who fail the exam in the first attempt must appear for the exam in the following year. Students always pass the exam in their second attempt.
The number of students who took the exam for the first time in the year year 2 and the year 3 respectively, are ______
1. 55 and 48
2. 60 and 50
3. 65 and 53
4. 55 and 53
Option 1 : 55 and 48
#### Line Graph and Pie Chart Question 7 Detailed Solution
Given data from the chart,
Year Pass Fail Total 1 50 10 60 2 60 5 65 2 50 3 53
Fail in the first year will give an exam in the second year.
Students will pass on 2nd attempt.
It means fail in 1st year students will become pass in 2nd year.
The number of students who took the exam for the first time in the year 2nd = 50 + 5 = 55
The number of students who took the exam for the first time in the year 3rd = 45 + 3 = 48
#### Line Graph and Pie Chart Question 8
The pie-diagram below shows various expenditures incurred in publishing a book:
If for an edition of the book, the cost of paper is Rs. 20,000, then what is the promotion cost for this edition?
1. Rs, 5,000
2. Rs. 6,000
3. Rs. 7,000
4. Rs. 8,000
Option 4 : Rs. 8,000
#### Line Graph and Pie Chart Question 8 Detailed Solution
Given -
cost of paper = Rs. 20,000
Solution -
Let the total price be Rs. x
⇒ the cost of the paper = Rs. 20,000
⇒ 25% of x = 20,000
⇒ x =Rs. 80,000
⇒ promotion cost = 10% of 80,000 = Rs. 8000
∴ promotion cost = Rs. 8000
#### Line Graph and Pie Chart Question 9:
Directions : Study the following line and pie chart carefully and answer the questions given beside.
The following graphs show the distribution of total voters in Six Different Cities A, B, C, D, E and F and details of female voters in these cities.
Number of total Voters = 66000
Percentage distribution of voters(Male + Female)
Percentage of women voters
The number of female voters in city C is what per cent of male voters in city E.
1. 33.33%
2. 9.09%
3. 27.27%
4. 11.11%
Option 3 : 27.27%
#### Line Graph and Pie Chart Question 9 Detailed Solution
Given:
Female voters in city C = 20%
Total voters in city E = 20%
Total voters in city C = 18%
Calculation:
Number of female voters in city C= 20% of 18% of 66000
Number of male voters in city E = 66% of 20% of 66000
∴ The required percentage = (20% of 18% of 66000)/(66% of 20% of 66000) × 100
⇒ 3/11 × 100 = 27.27%
∴ The required percentage is 27.27%.
#### Comprehension:
Direction: Study the data carefully and answer the question. The pie chart below shows the percentage of total cars sold in Delhi of different brands.
Total number of cars sold is 300000.
The line graph shows the ratio of petrol and diesel cars sold of different brands.
Note: Car is Either Petrol based or Diesel-based.
Numbers represented in vertical axis like a & b can be written as a : 1 and b : 1
Find the ratio of the number of petrol cars of Honda and Diesel cars of Toyota.
1. 64 ∶ 49
2. 80 ∶ 57
3. 40 ∶ 37
4. 81 ∶ 50
5. 93 ∶ 64
Option 4 : 81 ∶ 50
#### Line Graph and Pie Chart Question 10 Detailed Solution
Percentage of cars of Honda and Toyota is 18% & 10% respectively;
Since the ratio of petrol and diesel cars of Honda and Toyota is 1.5(3/2) and 0.5(1/2) respectively;
∴ Number of petrol cars of Honda = 300000 × 0.18 × 3/5 = 32400
Number of Diesel cars of Toyota = 300000 × 0.10 × 2/3 = 20000
∴ Required ratio = 32400 ∶ 20000 = 81 ∶ 50
#### Line Graph and Pie Chart Question 11:
Two line charts are given below. Line chart 1 shows the ratio of number of males to the number of females in two companies A and B for the 5 years. Line chart 2 shows the total number of males (both companies A and B) and total number of females (both companies A and B) for the 5 years.
What is the ratio of number of males of company B in Y1 to the total number of females of company A in Y3 and Y5 ?
1. 117 ∶ 218
2. 117 ∶ 215
3. 129 ∶ 215
4. 119 ∶ 218
Option 2 : 117 ∶ 215
#### Line Graph and Pie Chart Question 11 Detailed Solution
Concept used:
If Ratio of two numbers is given as 1.1 , then the numbers would be written as 11x and 10x where x is HCF of both numbers.
Calculations:
Let the number of males and females in company A in Y1 be 11k and 10k respectively and the number of males and females in company B in Y2 be 9m and 10m respectively.Then,
11k + 9m = 21100 ---------(1)
And 10k + 10m = 20600 ----(2)
On solving (1) and (2), we get:
m = 780
So, The number of males in Company B in Y1 is 9m = 9(780) = 7020
Similarly let is assume the number of males and females in company B in Y3 be 27t and 20t respectively and the number of males and females in company A in Y3 be 16l and 20l respectively. Then,
27t + 16l = 18025
And 20t + 20l = 16000
On solving (3) and (4), we get:
t = 475 and l = 325
So, Total number of females in company A in Y3 = 20l = 20(325) = 6500
Now, Let the number of males and females in company A in Y5 be 28x and 20x respectively and the number of males and females in company B in Y5 be 17y and 20y respectively.Then,
28x + 17y = 13550 ------------(5)
And 20x + 20y = 11800 ------(6)
On solving equations (5) and (6) , we get :
x = 320
So, The number of females in company A in Y5
= 20x
= 20(320)
= 6400
Now, required ratio
= 7020 : (6500 + 6400)
= 7020 : 12900
= 117 : 215
Hence, The Required value is 117 : 215.
#### Line Graph and Pie Chart Question 12:
The following line graph shows the number of workers working in 5 different companies on a particular day.
If we convert this line graph into a pi chart then what angle (in degrees) is made by B?
1. 150 °
2. 75 °
3. 60 °
4. 50 °
Option 4 : 50 °
#### Line Graph and Pie Chart Question 12 Detailed Solution
Calculation:
We have to convert this line graph into a pie chart.
The total number of workers is
⇒ 230 + 150 + 170 + 250 + 280 = 1080
As per the question,
Value of 1080 = 360°
So value of 150 = (360 /1080) × 150
⇒ 150 / 3 = 50°
If we convert this line graph into a pi chart then the angle made by B is 50°
#### Line Graph and Pie Chart Question 13:
Directions : Study the following line and pie chart carefully and answer the questions given beside.
The following graphs show the distribution of total voters in Six Different Cities A, B, C, D, E and F and details of female voters in these cities.
Number of total Voters = 66000
Percentage distribution of voters(Male + Female)
Percentage of women voters
Find the ratio of the number of female voters of city E and the number of male voters of city D.
1. 18 : 35
2. 34 : 35
3. 31 : 47
4. 19 : 35
Option 2 : 34 : 35
#### Line Graph and Pie Chart Question 13 Detailed Solution
Given:
Female voters of city E = 34%
Total voters of city D = 30%
Calculation:
Number of female voters of city E = 34% of 20 % of 66000 =34 × 132 = 4488
Number of male voters of city D = 70% of 10 % of 66000 = 7 × 660 = 4620
∴ Ratio = 4488 : 4620 = 34 : 35
#### Comprehension:
Directions: The following line-graph shows the number of defected mobiles manufactured by a company in four years, and the pie-chart shows the percentage of various defects in mobiles in the year 2000. Study the line-graph and the pie-chart and answer the questions that follow.
In 2000, if the mobiles with miscellaneous defects are declared as defect-free, what percentage of the defected mobiles has a dent?
1. 36.84%
2. 32.72%
3. 35%
4. 46.84%
Option 1 : 36.84%
#### Line Graph and Pie Chart Question 14 Detailed Solution
Defected mobiles with dents = 35% of 640 = 0.35 × 640 = 224
Mobiles declared defect-free = 5% of 640 = 0.05 × 640 = 32
Defected mobiles = 640 - 32 = 608
Hence, % of defected mobiles with dent = (224/608) × 100 = 36.84%
#### Comprehension:
Direction: Study the data carefully and answer the question. The pie chart below shows the percentage of total cars sold in Delhi of different brands.
Total number of cars sold is 300000.
The line graph shows the ratio of petrol and diesel cars sold of different brands.
Note: Car is Either Petrol based or Diesel-based.
Numbers represented in vertical axis like a & b can be written as a : 1 and b : 1
By what percentage the Ford petrol cars sold are less than the Maruti petrol cars?
1. 6.25%
2. 7.5%
3. 12.5%
4. 10.25%
5. 5%
Option 1 : 6.25%
#### Line Graph and Pie Chart Question 15 Detailed Solution
Percentage of cars of Ford and Maruti is 20% & 24% respectively;
Since the ratio of petrol and diesel cars of Ford and Maruti is 3 ∶ 1 and 2 ∶ 1 respectively;
∴ Number of petrol cars of Ford = 300000 × 0.2 × 3/4 = 45000
Number of petrol cars of Maruti = 300000 × 0.24 × 2/3 = 48000
∴ Required percentage = [48000 – 45000]/48000 = 6.25%<|endoftext|>
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# Chapter 9 PowerPoint PPT Presentation
Chapter 9. Section 4. Complex Numbers. Write complex numbers as multiples of i . Add and subtract complex numbers. Multiply complex numbers. Divide complex numbers. Solve quadratic equations with complex number solutions. 9.4. 2. 3. 4. 5.
Chapter 9
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Chapter 9
Section 4
## Complex Numbers
Write complex numbers as multiples of i.
Add and subtract complex numbers.
Multiply complex numbers.
Divide complex numbers.
Solve quadratic equations with complex number solutions.
9.4
2
3
4
5
Some quadratic equations have no real number solutions. For example, the numbers
are notreal numbers because – 4 appears in the radicand. To ensure that every quadratic equation has a solution, we need a new set of numbers that includes the real numbers. This new set of numbers is defined with a new number i, call the imaginary unit, such that
and
Slide 9.4-3
### Objective 1
Write complex numbers as multiples of i.
Slide 9.4-4
For any positive real number b,
### Write complex numbers as multiples if i.
We can write numbers such as and as multiples of i, using the properties of ito define any square root of a negative number as follows.
Slide 9.4-5
Write as a multiple of i.
It is easy to mistake for with the i under the radical. For this reason, it is customary to write the factor i first when it is multiplied by a radical. For example, we usually write rather than
EXAMPLE 1
Simplifying Square Roots of Negative Numbers
Solution:
Slide 9.4-6
Write complex numbers as multiples if i. (cont’d)
Numbers that are nonzero multiples of iare pure imaginary numbers. The complex numbers include all real numbers and all imaginary numbers.
Complex Number
A complex number is a number of the form a + bi, where a and bare real numbers. If a = 0 and b≠ 0, then the number bi is a pure imaginary number.
In the complex number a+ bi,a is called the real part and b is called the imaginary part. A complex number written in the form a + bi (or a + ib) is in standard form. See the figure on the following slide which shows the relationship among the various types of numbers discussed in this course.
Slide 9.4-7
Slide 9.4-8
### Objective 2
Add and subtract complex numbers.
Slide 9.4-9
Adding and subtracting complex numbers is similar to adding and subtracting binomials.
### Add and subtract complex numbers.
To add complex numbers, add their real parts and add their imaginary parts.
To subtract complex numbers, add the additive inverse (or opposite).
Slide 9.4-10
EXAMPLE 2
Adding and Subtracting Complex Numbers
Solution:
Slide 9.4-11
### Objective 3
Multiply complex numbers.
Slide 9.4-12
### Multiply complex numbers.
We multiply complex numbers as we do polynomials. Since i2 = –1 by definition, whenever i2appears, we replace it with –1.
Slide 9.4-13
Find each product.
EXAMPLE 3
Multiplying Complex Numbers
Solution:
Slide 9.4-14
### Objective 4
Divide complex numbers.
Slide 9.4-15
### Write complex number quotients in standard form.
The quotient of two complex numbers is expressed in standard form by changing the denominator into a real number.
The complex numbers 1 + 2i and 1 – 2i are conjugates. That is, the conjugate of the complex number a + bi is a – bi. Multiplying the complex number a + bi by its conjugate a – bi gives the real number a2 + b2.
Product of Conjugates
That is, the product of a complex number and its conjugate is the sum of the squares of the real and imaginary part.
Slide 9.4-16
Write the quotient in standard form.
EXAMPLE 4
Dividing Complex Numbers
Solution:
Slide 9.4-17
### Objective 5
Solve quadratic equations with complex number solutions.
Slide 9.4-18
### Solve quadratic equations with complex solutions.
Quadratic equations that have no real solutions do have complex solutions.
Slide 9.4-19
Solve (x– 2)2 = –64.
EXAMPLE 5
Solving a Quadratic Equation with Complex Solutions (Square Root Property)
Solution:
Slide 9.4-20
Solve x2– 2x = –26.
EXAMPLE 6
Solving a Quadratic Equation with Complex Solutions (Quadratic Formula)
Solution:
Slide 9.4-21<|endoftext|>
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Being able to understand speech is essential to our evolution as humans. Hearing lets us perceive the same word even when spoken at different speeds or pitches, and also gives us extra sensitivity to unexpected sounds. Now, new studies from the Perelman School of Medicine at the University of Pennsylvania clarify how these two crucial features of audition are managed by the brain.
In the first study, published online in eLife this week, Maria N. Geffen, PhD, an assistant professor in the departments of Neuroscience and Otorhinolaryngology and Head and Neck Surgery, and her team, including first co-authors Ryan G. Natan and John J. Briguglio, both doctoral candidates, discovered how different neurons work together in the brain to reduce responses to frequent sounds, and enhance responses to rare sounds.
When navigating through complex acoustic environments, we hear both important and unimportant sounds, and an essential task for our brain is to separate out the important sounds from unimportant ones.
“In everyday conversations, you want to be able to carry on a discussion, yet simultaneously perceive when someone else calls your name,” Geffen said. ”Similarly, a mouse running through the forest wants to be able to detect the sound of an owl approaching even though there are many other, more ordinary sounds around him.”
“It’s really important to understand the mechanisms underlying these basic auditory processes, given how much we depend on them in everyday life,” she added.
Researchers found that a perceptual phenomenon known as “stimulus-specific adaptation” in the brain might help with this complex task. This feature of perception occurs across all our senses. In the context of hearing, it is a reduction of auditory cortical neurons’ responses to the frequently-heard, “expected” sounds of any given environment. This desensitization to expected sounds creates a relatively heightened sensitivity to unexpected sounds—which is desirable because unexpected sounds often carry extra significance.
While this phenomenon has been studied for decades, there were limited tools available previously to examine the function of specific cell types in stimulus-specific adaptation. In the study, Geffen’s team used recently discovered optogenetics techniques—which enable a given type of neuron to be switched on or off at will with bursts of light delivered to a lab mouse’s brain through optical fibers.
The team found that, surprisingly, two major types of cortical neurons provided two separate mechanisms for this type of adaptation.<|endoftext|>
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A cool new way to cool (or warm)!
The cooling or warming of patients being treated for certain medical conditions can greatly increase their chance of survival. For example, survival after cardiac arrest can be almost doubled if the patient is gently cooled for about 24 hours. Patients with stroke, trauma, and spinal cord injuries can also heal better and suffer less brain damage if they are maintained at a normal, or lower than normal, temperature for a certain period while also avoiding fevers. Patients undergoing surgery, however, need to be maintained at normal body temperature. If patients are too cold during surgery they tend to lose more blood, develop infections, and suffer heart damage. Unplanned decreases in body temperature before, during, or after surgery can increase the time patients must stay in the hospital and add $7,000 or more, per patient, to the costs of hospitalization.
Traditionally, patient temperature is controlled by means of water blankets, air blankets, or catheters placed in blood vessels, but these methods do not allow for easy temperature adjustment and tend to introduce complications. Water blankets are easy to put on or under a patient, but they touch the skin and can lead to excessive shivering, which in turn may increase stress and shivering and thus counteract the cooling that is desired. Air blankets are generally used for warming, but to be effective they need to cover most of the patient’s body, which is difficult when a surgical procedure exposes tissues and organs to heat loss. Special intravascular catheters are used for cooling or warming, but they must be inserted by a physician and add the risks of infection and blood clots.
A new device is now available that cools or warms a patient internally by way of the esophagus and avoids many of the complications encountered with traditional devices. Known as the EnsoETM, for esophageal temperature management, this device is a multi-lumen silicone tube that is closed at one end and connected to an external heat exchanger (standard water blanket console, or chiller) at the other. (Fig. 1) A central lumen is available to allow for gastric decompression and drainage. (Fig. 2)
The device is inserted in the mouth and passed along the esophagus until it reaches the stomach. Once it is in place, the temperature of the water circulating inside the tube is adjusted safely and efficiently by means of the control system in the heat exchanger. Warming or cooling through the core of the patient in this fashion is very efficient, since skin contact is avoided, and because much of the body’s blood flow surrounds the esophagus. Placement of this new device is easy and similar to many of the other types of tubes inserted in the stomach while in the hospital. Patients do not feel the placement because, typically, they are not awake, either because they have been anesthetized or because of their medical condition. As highlighted in this article, numerous studies show that this new device provides a safe and effective means of temperature management. For patients being cooled, the EnsoETM results in significantly less shivering than in patients cooled with surface blankets, thus reducing the costs of sedatives and anti-shivering medications. A further advantage of the device is that since it is placed in the gastrointestinal tract, placement involves a non-sterile procedure, further enhancing the ease of use and allowing a wide variety of healthcare provider to take control of their patient’s temperature to provide optimum care.
Medical Intensive Care Unit, University Medical Centre Maribor, Ljubljanska 5, 2000 Maribor, Slovenia
PublicationEsophageal Heat Transfer for Patient Temperature Control and Targeted Temperature Management.
Naiman MI, Gray M, Haymore J, Hegazy AF, Markota A, Badjatia N, Kulstad EB
J Vis Exp. 2017 Nov 21<|endoftext|>
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The supermassive black hole, that can devour anything in its path, was spotted by the Hubble Space Telescope.
It should be found at the centre of its galaxy, but, this image shows it being pushed around by gravitational waves.
It will eventually break free from its galaxy and then roam the universe.
Fortunately for us, it is currently eight billion light-years from Earth.
It is the first time the suspected happening has been confirmed.
A NASA spokesman said: "Though there have been several other suspected, similarly booted black holes elsewhere, none has been confirmed so far.
"Astronomers think this object is a very strong case.
A CGI of a planet being swallowed by a supermassive black hole.
"Weighing more than 1 billion suns, the rogue black hole is the most massive black hole ever detected to have been kicked out of its central home."
The Hubble Space Telescope captured an image of a quasar named 3C 186 that is offset from the centre of its galaxy.
Astronomers hypothesize that this supermassive black hole was jettisoned from the centre of its galaxy by the recoil from gravitational waves produced by the merging of two supermassive black holes.
Researchers estimate that it took the equivalent energy of 100 million supernovas exploding simultaneously to jettison the black hole.
The spokesman added: "The most plausible explanation for this propulsive energy is that the monster object was given a kick by gravitational waves unleashed by the merger of two hefty black holes at the centre of the host galaxy."
First predicted by Albert Einstein, gravitational waves are ripples in space that are created when two massive objects collide.
The ripples are similar to the concentric circles produced when a hefty rock is thrown into a pond.
Last year, the Laser Interferometer Gravitational-Wave Observatory (LIGO) helped astronomers prove that gravitational waves exist by detecting them emanating from the union of two stellar-mass black holes, which are several times more massive than the sun.
Research team leader Marco Chiaberge of the Space Telescope Science Institute (STScI) and Johns Hopkins University, in Baltimore, Maryland, said: "When I first saw this, I thought we were seeing something very peculiar.
"When we combined observations from Hubble, the Chandra X-ray Observatory, and the Sloan Digital Sky Survey, it all pointed towards the same scenario. The amount of data we collected, from X-rays to ultraviolet to near-infrared light, is definitely larger than for any of the other candidate rogue black holes."
This image, taken by NASA's Hubble Space Telescope, reveals an unusual sight: a runaway quasar.
Though there have been several other suspected, similarly booted black holes elsewhere, none has been confirmed so far.
Chiaberge's paper will appear in the March 30 issue of Astronomy & Astrophysics.
Black holes cannot be observed directly, but they are the energy source at the heart of quasars - intense, compact gushers of radiation that can outshine an entire galaxy.
The quasar, named 3C 186, and its host galaxy reside 8 billion light-years away in a galaxy cluster.
The team discovered the galaxy's peculiar features while conducting a Hubble survey of distant galaxies unleashing powerful blasts of radiation in the throes of galaxy mergers.
Mr Chiaberge added: "I was anticipating seeing a lot of merging galaxies, and I was expecting to see messy host galaxies around the quasars, but I wasn't really expecting to see a quasar that was clearly offset from the core of a regularly shaped galaxy.
"Black holes reside in the center of galaxies, so it's unusual to see a quasar not in the center."
The astronomers calculated that the black hole is moving so fast it would travel from Earth to the moon in three minutes. That's fast enough for the black hole to escape the galaxy in 20 million years and roam through the universe forever.
Based on this visible evidence, along with theoretical work, the researchers developed a scenario to describe how the behemoth black hole could be expelled from its central home. According to their theory, two galaxies merge, and their black holes settle into the centre of the newly formed elliptical galaxy.
As the black holes whirl around each other, gravity waves are flung out like water from a lawn sprinkler.
The hefty objects move closer to each other over time as they radiate away gravitational energy.
If the two black holes do not have the same mass and rotation rate, they emit gravitational waves more strongly along one direction.
When the two black holes collide, they stop producing gravitational waves.
This illustration shows how gravitational waves can propel a black hole from the center of a galaxy.
The scenario begins in the first panel with the merger of two galaxies, each with a central black hole.
In the second panel, the two black holes in the newly merged galaxy settle into the center and begin whirling around each other.
This energetic action produces gravitational waves. As the two hefty objects continue to radiate away gravitational energy, they move closer to each other over time, as seen in the third panel.
If the black holes do not have the same mass and rotation rate, they emit gravitational waves more strongly in one direction, as shown by the bright area at upper left.
The black holes finally merge in the fourth panel, forming one giant black hole.
The energy emitted by the merger propels the black hole away from the center in the opposite direction of the strongest gravitational waves.
The newly merged black hole then recoils in the opposite direction of the strongest gravitational waves and shoots off like a rocket.
An alternative explanation for the offset quasar, although unlikely, proposes that the bright object does not reside within the galaxy.
Instead, the quasar is located behind the galaxy, but the Hubble image gives the illusion that it is at the same distance as the galaxy.
If this were the case, the researchers should have detected a galaxy in the background hosting the quasar.<|endoftext|>
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Divisibility Rules for 3, 6, and 9
Start Practice
## Divisibility Rules for 3, 6, and 9
Divisible means a number can be divided by another evenly, without remainder.
In the last lesson, you learned that a number that ends in 0 is divisible by 2, 5, and 10.Ā
Letās learn some more divisibility rules!
### Divisible by 3
Thereās a trick to figuring out if a number is divisible by 3! š
A number is divisible by 3 if the sum of its digits is divisible by 3.Ā
What does this mean?!
Letās try it out!
342
This number has three digits! What are they?
3, 4, and 2
3 + 4 + 2 = 9
Is 9 divisible by 3?Ā
Yes! This means 342 is also divisible by 3! ā
Tip: The only single-digit numbers divisible by 3 are: 3, 6, and 9.Ā
Letās try this again!
Is this divisible by 3? š
8,217
Letās find the sum of the digits!
8 + 2 + 1 + 7 = 18
Great! Now, hereās another tip. If you add and find a two-digit number, add the digits again!Ā
Find the sum of the digits in 18.
1 + 8 = 9
Is 9 divisible by 3? š¤
Yes! That means 8,217 is divisible by 3.Ā
One more!
6,701
6 + 7 + 0 + 1 = 14
Now, add the digits of 14.
1 + 4 = 5
Is 5 divisible by 3? š¤
No, itās not! That means 6,701 is not divisible by 3.Ā
### Divisible by 6
You already know half of the divisibility rule for 6!
If a number is divisible by 6, it's divisible by 3 and 2.
A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.Ā
Is this number divisible by 6? š
1,836
Let's check. Is it divisible by 3?
š Find the sum of the digits!
1 + 8 + 3 + 6 = 18
Try it once more!
1 + 8 = 9
Yes, 1,836 is divisible by 3.
Weāre not done yet! Is it also divisible by 2?
Yes, it ends in 6!
So 1,836 is divisible by 6. ā
Letās try one more!
Is 42,129 divisible by 6?
Start by checking if it's divisible by 3:
4 + 2 + 1 + 2 + 9 = 18
Once more!
1 + 8 = 9
Yes, 42,129 is divisible by 3.
Is it also divisible by 2?
No, it does not end in 0, 2, 4, 6, or 8.
This means 42,129 is not divisible by 6.
### Divisible by 9
Figuring out whether a number is divisible by 9 means finding the sum of the digits again!
A number is divisible by 9 if the sum of its digits is also divisible by 9.
Is 567 divisible by 9?
Letās find the sum of the digits!
5 + 6 + 7 = 18
Do you remember whether 18 is divisible by 9? š¤
Itās ok if you donāt. We can find the sum of the digits of 18.
1 + 8 = 9
Since it equals 9, we know 567 is divisible by 9.
Letās try another one!
Is 48,933 divisible by 9?
Find the sum of the digits:
4 + 8 + 9 + 3 + 3 = 27
Is 27 divisible by 9?
It is! But if you didnāt know that, you could find the sum of the digits again:
2 + 7 = 9
It equals 9! That means, 48,933 is divisible by 9.<|endoftext|>
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The microprocessor is a device that can be programmed to perform arithmetic and logical operations and other functions in a preordered sequence. The microprocessor is usually used as the central processing unit (CPU) in today’s computer systems when it is connected to other components, such as memory chips and input/ output circuits. The basic arrangement and design of the circuits residing in the microprocessor is called the architecture.
Theory of Operation
In the study of alternating current, basic generator principles were introduced to explain the generation of an AC voltage by a coil rotating in a magnetic field. Since this is the basis for all generator operation, it is necessary to review the principles of generation of electrical energy.
When lines of magnetic force are cut by a conductor passing through them, voltage is induced in the conductor. The strength of the induced voltage is dependent upon the speed of the conductor and the strength of the magnetic field. If the ends of the conductor are connected to form a complete circuit, a current is induced in the conductor. The conductor and the magnetic field make up an elementary generator.
This simple generator is illustrated in Figure 10-255, together with the components of an external generator circuit which collect and use the energy produced by the simple generator. The loop of wire (A and B of Figure 10-255) is arranged to rotate in a magnetic field. When the plane of the loop of wire is parallel to the magnetic lines of force, the voltage induced in the loop causes a current to flow in the direction indicated by the arrows in Figure 10-255. The voltage induced at this position is maximum, since the wires are cutting the lines of force at right angles and are thus cutting more lines of force per second than in any other position relative to the magnetic field. As the loop approaches the vertical position shown in Figure 10-256, the induced voltage
decreases because both sides of the loop (A and B) are approximately parallel to the lines of force and the rate of cutting is reduced. When the loop is vertical, no lines of force are cut since the wires are momentarily traveling parallel to the magnetic lines of force, and there is no induced voltage. As the rotation of the loop continues, the number of lines of force cut increases until the loop has rotated an additional 90° to a horizontal plane. As shown in Figure 10-257, the number of lines of force cut and the induced voltage once again are maximum.
The direction of cutting, however, is in the opposite direction to that occurring in Figures 10-255 and10-256, so the direction (polarity) of the induced voltage is reversed. As rotation of the loop continues, the number of lines of force having been cut again decreases, and the induced voltage becomes zero at the position shown in Figure 10-258, since the wires A and B are again parallel to the magnetic lines of force.
If the voltage induced throughout the entire 360° of rotation is plotted, the curve shown in Figure 10-259 results. This voltage is called an alternating voltage because of its reversal from positive to negative value — first in one direction and then in the other.
To use the voltage generated in the loop for producing a current flow in an external circuit, some means must be provided to connect the loop of wire in series with the external circuit. Such an electrical connection can be effected by opening the loop of wire and connecting its two ends to two metal rings, called slip rings, against which two metal or carbon brushes ride. The brushes are connected to the external circuit. By replacing the slip rings of the basic AC generator with two half cylinders, called a commutator, a basic DC generator is obtained.
[Figure 10-260] In this illustration, the black side of the coil is connected to the black segment, and the white side of the coil to the white segment. The segments are insulated from each other. The two stationary brushes are placed on opposite sides of the commutator and are so mounted that each brush contacts each segment of the commutator as the latter revolves simultaneously with the loop. The rotating parts of a DC generator (coil and commutator) are called an armature. The generation of an emf by the loop rotating in the magnetic field is the same for both AC and DC generators, but the action of the commutator produces a DC voltage.
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Derivative of Tan x & Proof in Easy Steps
Derivatives > Derivative of Tan x
What is the Derivative of Tan x?
The derivative of tan x is sec2x:
How to Take the Derivative of Tan x
You can take the derivative of tan x using the quotient rule. That’s because of a basic trig identity, which is a quotient of the sine function and cosine function:
tan(x) = sin(x) / cos(x).
Step 1: Name the numerator (top term) in the quotient g(x) and the denominator (bottom term) h(x). You could use any names you like, as it won’t make a difference to the algebra. However, g(x) and h(x) are common choices.
• g(x) = sin(x)
• h(x) = cos(x)
Step 2: Put g(x) and h(x) into the quotient rule formula.
Note that I used d/dx here to denote a derivative (Leibniz Notation) instead of g(x)′ or h(x)′ (Prime Notation (Lagrange), Function & Numbers). You can use either notation: they mean the same thing.
Step 3: Differentiate the functions from Step 2. There are two parts to differentiate:
1. The derivative of the first part of the function (sin(x)) is cos(x)
2. The derivative of cos(x) is -sin(x).
Placing those derivatives into the formula from Step 3, we get:
Which we can rewrite as:
f′(x) = cos2(x) + sin2(x) / cos(x)2.
Step 4: Use algebra / trig identities to simplify.
• Specifically, start by using the identity cos2(x) + sin2(x) = 1
• This gives you 1/cos2(x), which is equivalent in trigonometry to sec2(x).
Proof of the Derivative of Tan x
There are a couple of ways to prove the derivative tan x. You could start with the definition of a derivative and prove the rule using trigonometric identities. But there’s actually a much easier way, and is basically the steps you took above to solve for the derivative. As it relies only on trig identities and a little algebra, it is valid as a proof. Plus, it skips the need for using the definition of a derivative at all.
Steps
Example problem: Prove the derivative tan x is sec2x.
Step 1: Write out the derivative tan x as being equal to the derivative of the trigonometric identity sin x / cos x:
Step 2: Use the quotient rule to get:
Step 3: Use algebra to simplify:
Step 4: Substitute the trigonometric identity sin(x) + cos2(x) = 1:
Step 5: Substitute the trigonometric identity 1/cos2x = sec2x to get the final answer:
d/dx tan x = sec2x
That’s it!
References
Nicolaides, A. (2007). Pure mathematics: Differential calculus and applications, Volume 4. Pass Publications.<|endoftext|>
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Global Warming and Extreme Weather
Patterns of extreme weather are changing in the United States, and climate science predicts that further changes are in store. Extreme weather events lead to billions of dollars in economic damage and loss of life each year. Scientists project that global warming could affect the frequency, timing, location and severity of many types of extreme weather events in the decades to come.
Over the last five years, science has continued to make progress in exploring the connections between global warming and extreme weather. Meanwhile, the United States has experienced a string of extreme events – including massive floods in the Midwest, Tennessee and Northeast, intense hurricanes in Florida and along the Gulf Coast, drought and wildfire in the Southeast and Southwest, and others – that serve as a reminder of the damage that extreme weather can cause to people, the economy and the environment.
This report reviews recent trends in several types of extreme weather, the impacts caused by notable events that have occurred since 2005, and the most recent scientific projections of future changes in extreme weather.
To protect the nation from the damage to property and ecosystems that results from changes in extreme weather patterns – as well as other consequences of global warming – the United States must move quickly to reduce emissions of global warming pollutants.The worldwide scientific consensus that the earth is warming and that human activities are largely responsible has solidified in recent years.
· A recent report published by the U.S. National Academy of Sciences stated that “the conclusion that the Earth system is warming and that much of this warming is very likely due to human activities” is “so thoroughly examined and tested, and supported by so many independent observations and results,” that its “likelihood of subsequently being found to be wrong is vanishingly small.”
· The national academies of sciences of 13 leading nations issued a joint statement in 2009 stating that “climate change is happening even faster than previously estimated.”
· A 2009 study of the work of more than 1,300 climate researchers actively publishing in the field found that 97 to 98 percent of those researchers agree with the central theories behind global warming.
The consequences of global warming are already beginning to be experienced in the United States, and are likely to grow in the years to come, particularly if emissions of global warming pollutants continue unabated.
· Average temperature in the United States has increased by more than 2° Fahrenheit over the last 50 years. Temperatures are projected to rise by as much as an additional 7° F to 11° F on average by the end of the century, should emissions of global warming pollutants continue to increase.
· The United States has experienced an increase in heavy precipitation events, with the amount of precipitation falling in the top 1 percent of rainfall events increasing by 20 percent over the course of the 20th century. The trend toward extreme precipitation is projected to continue, even as higher temperatures and drier summers increase the risk of drought in much of the country.
· Snow cover has decreased over the past three decades in the Northern Hemisphere, and the volume of spring snowpack in the Mountain West and Pacific Northwest has declined significantly since the mid-20th century.
· Sea level has risen by nearly 8 inches globally since 1870. Global sea level is currently projected to rise by as much as 2.5 to 6.25 feet by the end of the century if global warming pollution continues unabated. Parts of the northeastern United States could experience an additional 8 inches of sea-level rise due to changes in ocean circulation patterns.
Several types of extreme weather events have occurred more frequently or with greater intensity in recent years. Global warming may drive changes in the frequency, timing, location or severity of such events in the future.
· The strongest tropical cyclones have been getting stronger around the globe over the last several decades, with a documented increase in the number of severe Category 4 and 5 hurricanes in the Atlantic Ocean since 1980.
· Scientists project that global warming may bring fewer – but more intense – hurricanes worldwide, and that those hurricanes will bring increased precipitation. The number of intense Category 4 and 5 hurricanes in the Atlantic may nearly double over the course of the next century.
· Estimated total damages from the seven most costly hurricanes to strike the United States since the beginning of 2005 exceed $200 billion. That includes damages from Hurricane Katrina, which was not only the most costly weather-related disaster of all time in the United States, but which also caused major changes to important ecosystems, including a massive loss of land on barrier islands along the Gulf Coast.
Sea Level Rise and Coastal Storms
· Sea level at many locations along the East Coast has been rising at a rate of nearly 1 foot per century due to the expansion of sea water as it has warmed and due to the melting of glaciers. Relative sea level has risen faster along the Gulf Coast, where land has been subsiding, and less along the northern Pacific Coast.
· In addition to sea-level rise, wave heights have been rising along the northern Pacific coast in recent years, possibly indicating an increase in the intensity of Pacific winter storms. In the 1990s, scientists estimated that the height of a “100-year wave” (one expected to occur every 100 years) off the coast of the Pacific Northwest was approximately 33 feet; now it is estimated to be 46 feet.
· Projected future sea-level rise of 2.5 to 6.25 feet by the end of the century would put more of the nation’s coastline at risk of erosion or inundation by even today’s typical coastal storms.
o In the mid-Atlantic region alone, between 900,000 and 3.4 million people live in areas that would be threatened by a 3.3 foot (1 meter) rise in sea level.
o Along the Gulf Coast from Galveston, Texas, to Mobile, Alabama, more than half the highways, nearly all the rail miles, 29 airports and almost all existing port infrastructure are at risk of flooding in the future due to higher seas and storm surges.
Had New York City experienced a 20-inch (0.5 meter) rise in sea level over the 1997 to 2007 period (at the low end of current projections for sea level rise by the end of the century), the number of moderate coastal flooding events would have increased from zero to 136 – the equivalent of a coastal flood warning every other week.
Rainfall, Floods and Extreme Snowstorms
· The number of heavy precipitation events in the United States increased by 24 percent between 1948 and 2006, with the greatest increases in New England and the Midwest. (See page [x].) In much of the eastern part of the country, a storm so intense that once it would have been expected to occur every 50 years can now be expected to occur every 40 years.
· The largest increases in heavy rainfall events in the United States are projected to occur in the Northeast and Midwest. The timing of overall precipitation is also projected to change, with increases in precipitation during the winter and spring in much of the north, but drier summers across most of the country.
· Global warming is projected to bring more frequent intense precipitation events, since warmer air is capable of holding more water vapor. Changing precipitation patterns could lead to increased risk of floods
. What is now a 100-year flood in the Columbia River basin could occur once every three years under an extreme global warming scenario, due to the combination of wetter winters and accelerated snowmelt. This change is projected to occur even as the region experiences an increase in summer drought due to reduced summer precipitation and declining availability of snowmelt in the summer.
· Flooding is the most common weather-related disaster in the United States. Recent years have seen a string of incredibly destructive floods, including the 2008 Midwest flood that inundated Cedar Rapids, Iowa, and caused an estimated $8 to $10 billion in damage, and the massive 2010 floods in New England and Tennessee.
· Projections of more frequent heavy precipitation apply to both rain and snow storms (although warming will bring a shift in precipitation from snow to rain over time). The 2010 record snowfalls in the mid-Atlantic region (dubbed “Snowmageddon”) are fully consistent with projections of increased extreme precipitation in a warming world – and with the string of massive flooding events elsewhere in the country during 2010.
Heat Waves, Drought and Wildfires
· Over the past century, drought has become more common in parts of the northern Rockies, the Southwest and the Southeast. Periods of extreme heat have also become more common since 1960.
· Large wildfires have become more frequent in the American West since the mid-1980s, with the greatest increases in large wildfires coming in the northern Rockies and northern California.
· Heat waves are projected to be more frequent, more intense, and last longer in a warming world. Much of the United States – especially the Southwest – is projected to experience more frequent or severe drought.
· Scientists project that a warmer climate could lead to a 54 percent increase in the average area burned by western wildfires annually, with the greatest increases in the Pacific Northwest and Rocky Mountains.
· Heat waves are among the most lethal of extreme weather events. A 2006 heat wave that affected the entire contiguous United States was blamed for at least 147 deaths in California and another 140 deaths in New York City.
· Wildfire is capable of causing great damage to property, while the cost of fighting wildfires is a significant drain on public resources. In 2008, California spent $200 million in a single month fighting a series of wildfires in the northern part of the state.
Avoiding the potential increased risks from extreme weather events—and their costs to the economy and society—is one of many reasons for the United States and the world to reduce emissions of global warming pollution.
· The United States and the world should adopt measures designed to prevent an increase in global average temperatures of more than 2° C (3.6° F) above pre-industrial levels – a commitment that would enable the world to avoid the most damaging impacts of global warming.
· The United States should commit to emission reductions equivalent to a 35 percent reduction in global warming pollution from 2005 levels by 2020 and an 83 percent reduction by 2050, with the majority of near-term emission reductions coming from the U.S. economy. A variety of policy measures can be used to achieve this goal, including:
o A cap-and-trade system that puts a price on emissions of global warming pollutants.
o A renewable energy standard to promote the use of clean renewable energy.
o A strong energy efficiency resource standard for utilities that maximizes the use of cost-effective energy efficiency improvements.
o Enhanced energy efficiency standards for appliances and vehicles and stronger energy codes for new or renovated commercial and residential buildings.
o Investments in low-carbon transportation infrastructure – including transit and passenger rail – and support for a transition to plug-in and other alternative fuel vehicles.
o Retention of the EPA’s authority to require reductions in global warming pollution at power plants, as well as retention of state authority to go beyond federal minimum standards in reducing global warming pollution.
· State and local governments should adopt similar measures to reduce global warming pollution and encourage a transition to clean energy.
In addition, federal, state and local officials should take steps to better protect the public from the impact of extreme weather events. Government officials should explicitly factor the potential for global warming-induced changes in extreme weather patterns into the design of public infrastructure and revise policies that encourage construction in areas likely to be at risk of flooding in a warming climate.<|endoftext|>
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3.7-billion-year-old fossil makes life on Mars less of a long shot
The wavelike mounds of sediment called stromatolites (marked by dashed lines) embedded inside this cross section of a 3.7-billion-year-old rock may be the oldest known fossilized evidence of life on Earth. USA TODAY
And you thought that slime in the bottom of your fridge was ancient.
Scientists have found the oldest known remnant of life, a fossil dating back a staggering 3.7 billion years. If confirmed, the date would support the theory that life took root in just a blink of an eye after the planet’s birth. Such early life would also make life on Mars seem less of a long shot.
The newfound remains consist of a layer of rock that, to the untutored eye, looks, well ... like a layer of rock. Scientists say it’s actually a stromatolite, a mineral structure created by the busywork of countless microorganisms. These microbes thrived in a shallow sea bathing a still young and fresh Earth, according to a study in this week’s Nature.
Previous chemical analysis of old rocks hinted life arose by 3.7 billion years ago, but that evidence was open to question, says study co-author Allen Nutman of Australia’s University of Wollongong.
“What we’ve done is produce something tangible,” Nutman says, “an actual fossil record (that is) evidence for life at those times.”
The researchers “were able to see evidence for life in a way that I had never expected,” says Texas A&M University’s Michael Tice, who was not associated with the study. “We have a much better window back in time, thanks to what these folks did.”
The fossil was discovered in a barren stretch of Greenland that researchers have patiently explored for some three decades. Unusually heavy spring rains recently melted a longstanding snow patch, exposing a distinct layer of rock.
The rock layer contained a level bottom, but the top was jagged. Standing less than two inches high, the layer resembled a miniature mountain range in profile.
After having read extensively about such objects, “we immediately knew what we were looking at,” Nutman says. That jagged top suggested not just a rock but a rock born of biology.
The rock’s structure mimics that of a 2-billion-year-old object widely accepted as a stromatolite, says study co-author Martin Van Kranendonk of Australia’s University of New South Wales. Chemical clues also hint that microbes played a role in the object’s formation.
The structure of the rocks around the fossil suggests it was bathed in seawater. Beyond that, the team knows nothing of the stromatolite’s origins. Modern stromatolites — such as the famous mushroom-shaped examples at Australia’s Shark Bay — are built by crowds of bacteria and other single-celled microbes. But the identity of whatever was living 3.7 billion years ago is obscured by the altered state of rock from that era, which over the eons has been cooked and twisted beyond recognition.
More study of the stromatolite is needed to confirm the fossil’s identity, says Abigail Allwood of the Jet Propulsion Laboratory, author of an accompanying analysis for Nature.
But if it is confirmed, “that says to me that life is not a fussy, reluctant, unlikely kind of thing,” she says. “Give it half a chance, and it’ll run with it.” After all, asteroids had bombarded the Earth only 100 million years earlier, but life still blossomed, spread and grew sophisticated by the time the fossil formed.
Parts of Mars looked much like the area where the stromatolite took shape at roughly the same time, Allwood says. Until now, scientists wondered whether the lakes on Mars persisted long enough for life to spring up, but now, she says, “what we’re potentially seeing is evidence that life can get a foothold in such a short time frame.”<|endoftext|>
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# 1.3: Basic Classes of Functions
We have studied the general characteristics of functions, so now let’s examine some specific classes of functions. We finish the section with examples of piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.
## Linear Functions and Slope
The easiest type of function to consider is a linear function. Linear functions have the form (f(x)=ax+b), where (a) and (b) are constants. In Figure (PageIndex{1}), we see examples of linear functions when a is positive, negative, and zero. Note that if (a>0), the graph of the line rises as (x) increases. In other words, (f(x)=ax+b) is increasing on ((−∞, ∞)). If (a<0), the graph of the line falls as (x) increases. In this case, (f(x)=ax+b) is decreasing on ((−∞, ∞)). If (a=0), the line is horizontal.
Figure (PageIndex{1}): These linear functions are increasing or decreasing on ((∞, ∞)) and one function is a horizontal line.
As suggested by Figure, the graph of any linear function is a line. One of the distinguishing features of a line is its slope. The slope is the change in (y) for each unit change in (x). The slope measures both the steepness and the direction of a line. If the slope is positive, the line points upward when moving from left to right. If the slope is negative, the line points downward when moving from left to right. If the slope is zero, the line is horizontal. To calculate the slope of a line, we need to determine the ratio of the change in (y) versus the change in (x). To do so, we choose any two points ((x_1,y_1)) and ((x_2,y_2)) on the line and calculate (dfrac{y_2−y_1}{x_2−x_1}). In Figure (PageIndex{2}), we see this ratio is independent of the points chosen.
Figure (PageIndex{2}): For any linear function, the slope ((y_2−y_1)/(x_2−x_1)) is independent of the choice of points ((x_1,y_1)) and ((x_2,y_2)) on the line.
Definition: Linear Functions
Consider line (L) passing through points ((x_1,y_1)) and ((x_2,y_2)). Let (Δy=y_2−y_1) and (Δx=x_2−x_1) denote the changes in (y) and (x),respectively. The slope of the line is
[m=dfrac{y_2−y_1}{x_2−x_1}=dfrac{Δy}{Δx}]
We now examine the relationship between slope and the formula for a linear function. Consider the linear function given by the formula (f(x)=ax+b). As discussed earlier, we know the graph of a linear function is given by a line. We can use our definition of slope to calculate the slope of this line. As shown, we can determine the slope by calculating ((y_2−y_1)/(x_2−x_1)) for any points ((x_1,y_1)) and ((x_2,y_2)) on the line. Evaluating the function (f) at (x=0), we see that ((0,b)) is a point on this line. Evaluating this function at (x=1), we see that ((1,a+b)) is also a point on this line. Therefore, the slope of this line is
[dfrac{(a+b)−b}{1−0}=a.]
We have shown that the coefficient (a) is the slope of the line. We can conclude that the formula (f(x)=ax+b) describes a line with slope (a). Furthermore, because this line intersects the (y)-axis at the point ((0,b)), we see that the y-intercept for this linear function is ((0,b)). We conclude that the formula (f(x)=ax+b) tells us the slope, a, and the (y)-intercept, ((0,b)), for this line. Since we often use the symbol (m) to denote the slope of a line, we can write
[f(x)=mx+b]
to denote the slope-intercept form of a linear function.
Sometimes it is convenient to express a linear function in different ways. For example, suppose the graph of a linear function passes through the point ((x_1,y_1)) and the slope of the line is (m). Since any other point ((x,f(x))) on the graph of (f) must satisfy the equation
[m=dfrac{f(x)−y_1}{x−x_1},]
this linear function can be expressed by writing
[f(x)−y_1=m(x−x_1).]
We call this equation the point-slope equation for that linear function.
Since every nonvertical line is the graph of a linear function, the points on a nonvertical line can be described using the slope-intercept or point-slope equations. However, a vertical line does not represent the graph of a function and cannot be expressed in either of these forms. Instead, a vertical line is described by the equation (x=k) for some constant (k). Since neither the slope-intercept form nor the point-slope form allows for vertical lines, we use the notation
[ax+by=c,]
where (a,b) are both not zero, to denote the standard form of a line.
Definition: point-slope equation, point-slope equation and the standard form of a line
Consider a line passing through the point ((x_1,y_1)) with slope (m). The equation
[y−y_1=m(x−x_1)]
is the point-slope equation for that line.
Consider a line with slope (m) and (y)-intercept ((0,b).) The equation
[y=mx+b]
is an equation for that line in point-slope equation.
The standard form of a line is given by the equation
[ax+by=c,]
where (a) and (b) are both not zero. This form is more general because it allows for a vertical line, (x=k).
Example (PageIndex{1}): Finding the Slope and Equations of Lines
Consider the line passing through the points ((11,−4)) and ((−4,5)), as shown in Figure.
Figure (PageIndex{3}): Finding the equation of a linear function with a graph that is a line between two given points.
1. Find the slope of the line.
2. Find an equation for this linear function in point-slope form.
3. Find an equation for this linear function in slope-intercept form.
Solution
1. The slope of the line is
[m=dfrac{y_2−y_1}{x_2−x_1}=dfrac{5−(−4)}{−4−11}=−dfrac{9}{15}=−dfrac{3}{5}.]
2. To find an equation for the linear function in point-slope form, use the slope (m=−3/5) and choose any point on the line. If we choose the point ((11,−4)), we get the equation
[f(x)+4=−dfrac{3}{5}(x−11).]
3. To find an equation for the linear function in slope-intercept form, solve the equation in part b. for (f(x)). When we do this, we get the equation
[f(x)=−dfrac{3}{5}x+dfrac{13}{5}.]
Exercise (PageIndex{1})
Consider the line passing through points ((−3,2)) and ((1,4)).
1. Find the slope of the line.
2. Find an equation of that line in point-slope form.
3. Find an equation of that line in slope-intercept form.
Hint
The slope (m=Δy/Δx).
(m=1/2).
The point-slope form is (y−4=dfrac{1}{2}(x−1)).
The slope-intercept form is (y=dfrac{1}{2}x+dfrac{7}{2}).
Example (PageIndex{2}):
Jessica leaves her house at 5:50 a.m. and goes for a 9-mile run. She returns to her house at 7:08 a.m. Answer the following questions, assuming Jessica runs at a constant pace.
1. Describe the distance (D) (in miles) Jessica runs as a linear function of her run time (t) (in minutes).
2. Sketch a graph of (D).
3. Interpret the meaning of the slope.
Solution:
a. At time (t=0), Jessica is at her house, so (D(0)=0). At time (t=78) minutes, Jessica has finished running (9) mi, so (D(78)=9). The slope of the linear function is
(m=dfrac{9−0}{78−0}=dfrac{3}{26}.)
The (y)-intercept is ((0,0)), so the equation for this linear function is
(D(t)=dfrac{3}{26}t.)
b. To graph (D), use the fact that the graph passes through the origin and has slope (m=3/26.)
c. The slope (m=3/26≈0.115) describes the distance (in miles) Jessica runs per minute, or her average velocity.
## Polynomials
A linear function is a special type of a more general class of functions: polynomials. A polynomial function is any function that can be written in the form
[f(x)=a_nx^n+a_{n−1}x^{n−1}+…+a_1x+a_0]
for some integer (n≥0) and constants (a_n,a+{n−1},…,a_0), where (a_n≠0). In the case when (n=0), we allow for (a_0=0); if (a_0=0), the function (f(x)=0) is called the zero function. The value (n) is called the degree of the polynomial; the constant an is called the leading coefficient. A linear function of the form (f(x)=mx+b) is a polynomial of degree 1 if (m≠0) and degree 0 if (m=0). A polynomial of degree 0 is also called a constant function. A polynomial function of degree 2 is called a quadratic function. In particular, a quadratic function has the form (f(x)=ax^2+bx+c), where (a≠0). A polynomial function of degree (3) is called a cubic function.
## Power Functions
Some polynomial functions are power functions. A power function is any function of the form (f(x)=ax^b), where (a) and (b) are any real numbers. The exponent in a power function can be any real number, but here we consider the case when the exponent is a positive integer. (We consider other cases later.) If the exponent is a positive integer, then (f(x)=ax^n) is a polynomial. If (n) is even, then (f(x)=ax^n) is an even function because (f(−x)=a(−x)^n=ax^n) if (n) is even. If (n) is odd, then (f(x)=ax^n) is an odd function because (f(−x)=a(−x)^n=−ax^n) if (n) is odd (Figure (PageIndex{3})).
Figure (PageIndex{4}): (a) For any even integer (n),(f(x)=ax^n) is an even function. (b) For any odd integer (n),(f(x)=ax^n) is an odd function.
## Behavior at Infinity
To determine the behavior of a function (f) as the inputs approach infinity, we look at the values (f(x)) as the inputs, (x), become larger. For some functions, the values of (f(x)) approach a finite number. For example, for the function (f(x)=2+1/x), the values (1/x) become closer and closer to zero for all values of (x) as they get larger and larger. For this function, we say “(f(x)) approaches two as x goes to infinity,” and we write f(x)→2 as x→∞. The line y=2 is a horizontal asymptote for the function (f(x)=2+1/x) because the graph of the function gets closer to the line as (x) gets larger.
For other functions, the values (f(x)) may not approach a finite number but instead may become larger for all values of x as they get larger. In that case, we say “(f(x)) approaches infinity as (x) approaches infinity,” and we write (f(x)→∞) as (x→∞). For example, for the function (f(x)=3x^2), the outputs (f(x)) become larger as the inputs (x) get larger. We can conclude that the function (f(x)=3x^2) approaches infinity as (x) approaches infinity, and we write (3x^2→∞) as (x→∞). The behavior as (x→−∞) and the meaning of (f(x)→−∞) as (x→∞) or (x→−∞) can be defined similarly. We can describe what happens to the values of (f(x)) as (x→∞) and as (x→−∞) as the end behavior of the function.
To understand the end behavior for polynomial functions, we can focus on quadratic and cubic functions. The behavior for higher-degree polynomials can be analyzed similarly. Consider a quadratic function (f(x)=ax^2+bx+c). If (a>0), the values (f(x)→∞) as (x→±∞). If (a<0), the values (f(x)→−∞) as (x→±∞). Since the graph of a quadratic function is a parabola, the parabola opens upward if (a>0).; the parabola opens downward if (a<0) (Figure (PageIndex{4a})).
Now consider a cubic function (f(x)=ax^3+bx^2+cx+d). If (a>0), then (f(x)→∞) as (x→∞) and (f(x)→−∞) as (x→−∞). If (a<0), then (f(x)→−∞) as (x→∞) and (f(x)→∞) as (x→−∞). As we can see from both of these graphs, the leading term of the polynomial determines the end behavior (Figure (PageIndex{4b})).
Figure (PageIndex{5}): (a) For a quadratic function, if the leading coefficient (a>0),the parabola opens upward. If (a<0), the parabola opens downward. (b) For a cubic function (f), if the leading coefficient (a>0), the values (f(x)→∞) as (x→∞) and the values (f(x)→−∞) as (x→−∞). If the leading coefficient (a<0), the opposite is true.
## Zeros of Polynomial Functions
Another characteristic of the graph of a polynomial function is where it intersects the (x)-axis. To determine where a function f intersects the (x)-axis, we need to solve the equation (f(x)=0) for (n) the case of the linear function (f(x)=mx+b), the (x)-intercept is given by solving the equation (mx+b=0). In this case, we see that the (x)-intercept is given by ((−b/m,0)). In the case of a quadratic function, finding the (x)-intercept(s) requires finding the zeros of a quadratic equation: (ax^2+bx+c=0). In some cases, it is easy to factor the polynomial (ax^2+bx+c) to find the zeros. If not, we make use of the quadratic formula.
[ax^2+bx+c=0,]
where (a≠0). The solutions of this equation are given by the quadratic formula
If the discriminant (b^2−4ac>0), Equation ef{quad} tells us there are two real numbers that satisfy the quadratic equation. If (b^2−4ac=0), this formula tells us there is only one solution, and it is a real number. If (b^2−4ac<0), no real numbers satisfy the quadratic equation.
In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the x-axis. In some instances, it is possible to find the (x)-intercepts by factoring the polynomial to find its zeros. In other cases, it is impossible to calculate the exact values of the (x)-intercepts. However, as we see later in the text, in cases such as this, we can use analytical tools to approximate (to a very high degree) where the (x)-intercepts are located. Here we focus on the graphs of polynomials for which we can calculate their zeros explicitly.
Example (PageIndex{3}): Graphing Polynomial Functions
For the following functions,
1. (f(x)=−2x^2+4x−1)
2. (f(x)=x^3−3x^2−4x)
1. describe the behavior of (f(x)) as (x→±∞),
2. find all zeros of (f), and
3. sketch a graph of (f).
Solution
1.The function (f(x)=−2x^2+4x−1) is a quadratic function.
1.Because (a=−2<0),as (x→±∞,f(x)→−∞.)
2. To find the zeros of (f), use the quadratic formula. The zeros are
(x=−4±dfrac{sqrt{4^2−4(−2)(−1)}}{2(−2)}=dfrac{−4±sqrt{8}}{−4}=dfrac{−4±2sqrt{2}}{−4}=dfrac{2±2sqrt{2}}{2}.)
3.To sketch the graph of (f),use the information from your previous answers and combine it with the fact that the graph is a parabola opening downward.
2. The function (f(x)=x^3−3x^2−4x) is a cubic function.
1.Because (a=1>0),as (x→∞), (f(x)→∞). As (x→−∞), (f(x)→−∞).
2.To find the zeros of (f), we need to factor the polynomial. First, when we factor (x|) out of all the terms, we find
(f(x)=x(x^2−3x−4).)
Then, when we factor the quadratic function (x^2−3x−4), we find
(f(x)=x(x−4)(x+1).)
Therefore, the zeros of f are (x=0,4,−1).
3. Combining the results from parts i. and ii., draw a rough sketch of (f).
Exercise (PageIndex{2})
Consider the quadratic function (f(x)=3x^2−6x+2.) Find the zeros of (f). Does the parabola open upward or downward?
Hint
The zeros are (x=1±sqrt{3}/3). The parabola opens upward.
## Mathematical Models
A large variety of real-world situations can be described using mathematical models. A mathematical model is a method of simulating real-life situations with mathematical equations. Physicists, engineers, economists, and other researchers develop models by combining observation with quantitative data to develop equations, functions, graphs, and other mathematical tools to describe the behavior of various systems accurately. Models are useful because they help predict future outcomes. Examples of mathematical models include the study of population dynamics, investigations of weather patterns, and predictions of product sales.
As an example, let’s consider a mathematical model that a company could use to describe its revenue for the sale of a particular item. The amount of revenue (R) a company receives for the sale of n items sold at a price of (p) dollars per item is described by the equation (R=p⋅n). The company is interested in how the sales change as the price of the item changes. Suppose the data in Table show the number of units a company sells as a function of the price per item.
(p) (n) 6 8 10 12 14 19.4 18.5 16.2 13.8 12.2
In Figure, we see the graph the number of units sold (in thousands) as a function of price (in dollars). We note from the shape of the graph that the number of units sold is likely a linear function of price per item, and the data can be closely approximated by the linear function (n= −1.04p+26) for (0≤p≤25), where (n) predicts the number of units sold in thousands. Using this linear function, the revenue (in thousands of dollars) can be estimated by the quadratic function
[R(p)=p⋅ (−1.04p+26)=−1.04p^2+26p]
for (0≤p≤25) In Example (PageIndex{1}), we use this quadratic function to predict the amount of revenue the company receives depending on the price the company charges per item. Note that we cannot conclude definitively the actual number of units sold for values of (p), for which no data are collected. However, given the other data values and the graph shown, it seems reasonable that the number of units sold (in thousands) if the price charged is (p) dollars may be close to the values predicted by the linear function (n=−1.04p+26.)
Figure (PageIndex{6}): The data collected for the number of items sold as a function of price is roughly linear. We use the linear function (n=−1.04p+26) to estimate this function.
Example (PageIndex{4}): Maximizing Revenue
A company is interested in predicting the amount of revenue it will receive depending on the price it charges for a particular item. Using the data from Table, the company arrives at the following quadratic function to model revenue (R) as a function of price per item (p:)
[R(p)=p⋅(−1.04p+26)=−1.04p^2+26p]
for 0≤p≤25.
1. Predict the revenue if the company sells the item at a price of (p=$5) and (p=$17).
2. Find the zeros of this function and interpret the meaning of the zeros.
3. Sketch a graph of (R).
4. Use the graph to determine the value of (p) that maximizes revenue. Find the maximum revenue.
Solution
a. Evaluating the revenue function at (p=5) and (p=17), we can conclude that
(R(5)=−1.04(5)^2+26(5)=104,so revenue=$104,000;) (R(17)=−1.04(17)^2+26(17)=141.44,so revenue=$144,440.)
b. The zeros of this function can be found by solving the equation (−1.04p^2+26p=0). When we factor the quadratic expression, we get (p(−1.04p+26)=0). The solutions to this equation are given by (p=0,25). For these values of (p), the revenue is zero. When (p=$0), the revenue is zero because the company is giving away its merchandise for free. When (p=$25),the revenue is zero because the price is too high, and no one will buy any items.
c. Knowing the fact that the function is quadratic, we also know the graph is a parabola. Since the leading coefficient is negative, the parabola opens downward. One property of parabolas is that they are symmetric about the axis, so since the zeros are at (p=0) and (p=25), the parabola must be symmetric about the line halfway between them, or (p=12.5).
d. The function is a parabola with zeros at (p=0) and (p=25), and it is symmetric about the line (p=12.5), so the maximum revenue occurs at a price of (p=$12.50) per item. At that price, the revenue is (R(p)=−1.04(12.5)^2+26(12.5)=$162,500.)
## Algebraic Functions
By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.
Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function is any function of the form (f(x)=p(x)/q(x)),where (p(x)) and (q(x)) are polynomials. For example,
(f(x)=dfrac{3x−1}{5x+2}) and (g(x)=dfrac{4}{x^2+1})
are rational functions. A root function is a power function of the form (f(x)=x^{1/n}), where n is a positive integer greater than one. For example, f(x)=x1/2=x√ is the square-root function and (g(x)=x^{1/3}=sqrt[3]{x})) is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, (f(x)=sqrt{4−x^2}) is an algebraic function.
Example (PageIndex{5}): Finding Domain and Range for Algebraic Functions
For each of the following functions, find the domain and range.
1. (f(x)=dfrac{3x−1}{5x+2})
2. (f(x)=sqrt{4−x^2})
Solution
1.It is not possible to divide by zero, so the domain is the set of real numbers (x) such that (x≠−2/5). To find the range, we need to find the values (y) for which there exists a real number (x) such that
(y=dfrac{3x−1}{5x+2})
When we multiply both sides of this equation by (5x+2), we see that (x) must satisfy the equation
(5xy+2y=3x−1.)
From this equation, we can see that (x) must satisfy
(2y+1=x(3−5y).)
If y=(3/5), this equation has no solution. On the other hand, as long as (y≠3/5),
(x=dfrac{2y+1}{3−5y})
satisfies this equation. We can conclude that the range of (f) is ({y|y≠3/5}).
2. To find the domain of (f), we need (4−x^2≥0). When we factor, we write (4−x^2=(2−x)(2+x)≥0). This inequality holds if and only if both terms are positive or both terms are negative. For both terms to be positive, we need to find (x) such that
(2−x≥0) and (2+x≥0.)
These two inequalities reduce to (2≥x) and (x≥−2). Therefore, the set ({x|−2≤x≤2}) must be part of the domain. For both terms to be negative, we need
(2−x≤0) and (2+x≥0.)
These two inequalities also reduce to (2≤x) and (x≥−2). There are no values of (x) that satisfy both of these inequalities. Thus, we can conclude the domain of this function is ({x|−2≤x≤2}.)
If (−2≤x≤2), then (0≤4−x^2≤4). Therefore, (0≤sqrt{4−x2}≤2), and the range of (f) is ({y|0≤y≤2}.)
Exercise (PageIndex{3})
Find the domain and range for the function (f(x)=(5x+2)/(2x−1).)
Hint
The denominator cannot be zero. Solve the equation (y=(5x+2)/(2x−1)) for (x) to find the range.
The domain is the set of real numbers (x) such that (x≠1/2). The range is the set ({y|y≠5/2}).
The root functions (f(x)=x^{1/n}) have defining characteristics depending on whether (n) is odd or even. For all even integers (n≥2), the domain of (f(x)=x^{1/n}) is the interval ([0,∞)). For all odd integers (n≥1), the domain of (f(x)=x^{1/n}) is the set of all real numbers. Since (x^{1/n}=(−x)^{1/n}) for odd integers (n) ,(f(x)=x^{1/n}) is an odd function if(n) is odd. See the graphs of root functions for different values of (n) in Figure.
Figure (PageIndex{7}): (a) If (n) is even, the domain of (f(x)=sqrt[n]{x}) is ([0,∞)). (b) If (n) is odd, the domain of (f(x)=dfrac[n]{x}) is ((−∞,∞)) and the function (f(x)=dfrac[n]{x}) is an odd function.
Example (PageIndex{6}): Finding Domains for Algebraic Functions
For each of the following functions, determine the domain of the function.
1. (f(x)=dfrac{3}{x^2−1})
2. (f(x)=dfrac{2x+5}{3x^2+4})
3. (f(x)=sqrt{4−3x})
4. (f(x)=sqrt[3]{2x−1})
Solution
1. You cannot divide by zero, so the domain is the set of values (x) such that (x^2−1≠0). Therefore, the domain is ({x|x≠±1}).
2. You need to determine the values of (x) for which the denominator is zero. Since (3x^2+4≥4) for all real numbers (x), the denominator is never zero. Therefore, the domain is ((−∞,∞).)
3. Since the square root of a negative number is not a real number, the domain is the set of values (x) for which (4−3x≥0). Therefore, the domain is ({x|x≤4/3}.)
4. The cube root is defined for all real numbers, so the domain is the interval ((−∞, ∞).)
Exercise (PageIndex{4})
Find the domain for each of the following functions: (f(x)=(5−2x)/(x^2+2)) and (g(x)=sqrt{5x−1}).
Hint
Determine the values of (x) when the expression in the denominator of (f) is nonzero, and find the values of (x) when the expression inside the radical of (g) is nonnegative.
The domain of (f) is ((−∞, ∞)). The domain of (g) is ({x|x≥1/5}.)
## Transcendental Functions
Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. They are (sinx, cosx, tanx, cotx, secx, and cscx.) (We discuss trigonometric functions later in the chapter.) An exponential function is a function of the form (f(x)=b^x), where the base (b>0,b≠1). A logarithmic function is a function of the form (f(x)=log_b(x)) for some constant (b>0,b≠1,) where (log_b(x)=y) if and only if (b^y=x). (We also discuss exponential and logarithmic functions later in the chapter.)
Example (PageIndex{7}): Classifying Algebraic and Transcendental Functions
Classify each of the following functions, a. through c., as algebraic or transcendental.
1. (f(x)=dfrac{sqrt{x^3+1}}{4x+2})
2. (f(x)=2^{x^2})
3. ( f(x)=sin(2x))
Solution
1. Since this function involves basic algebraic operations only, it is an algebraic function.
2. This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.)
3. As in part b, this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental.
Exercise (PageIndex{5}):
Is (f(x)=x/2) an algebraic or a transcendental function?
Algebraic
## Piecewise-Defined Functions
Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function. The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of (x):
[f(x)=egin{cases}−x & x<0x & x≥0end{cases}.]
Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for (xa), we need to pay special attention to what happens at (x=a) when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at (x=a). We examine this in the next example.
Example (PageIndex{8}): Graphing a Piecewise-Defined Function
Sketch a graph of the following piecewise-defined function:
[f(x)=egin{cases}x+3,&x<1(x−2)^2,&x≥1end{cases}.]
Solution
Graph the linear function (y=x+3) on the interval ((−∞,1)) and graph the quadratic function (y=(x−2)^2) on the interval ([1,∞)). Since the value of the function at (x=1) is given by the formula (f(x)=(x−2)^2), we see that (f(1)=1). To indicate this on the graph, we draw a closed circle at the point ((1,1)). The value of the function is given by (f(x)=x+2) for all (x<1), but not at (x=1). To indicate this on the graph, we draw an open circle at ((1,4)).
Figure (PageIndex{8}): This piecewise-defined function is linear for (x<1) and quadratic for (x≥1.)
2) Sketch a graph of the function
(f(x)=egin{cases}2−x,&x≤2x+2,&x>2end{cases}.)
Solution:
Example (PageIndex{9}): Parking Fees Described by a Piecewise-Defined Function
In a big city, drivers are charged variable rates for parking in a parking garage. They are charged $10 for the first hour or any part of the first hour and an additional$2 for each hour or part thereof up to a maximum of $30 for the day. The parking garage is open from 6 a.m. to 12 midnight. 1. Write a piecewise-defined function that describes the cost (C) to park in the parking garage as a function of hours parked (x). 2. Sketch a graph of this function (C(x).) Solution 1.Since the parking garage is open 18 hours each day, the domain for this function is ({x|0 [C(x)=egin{cases}10,&0 2.The graph of the function consists of several horizontal line segments. Exercise (PageIndex{6}) The cost of mailing a letter is a function of the weight of the letter. Suppose the cost of mailing a letter is (49¢) for the first ounce and (21¢) for each additional ounce. Write a piecewise-defined function describing the cost (C) as a function of the weight (x) for (0 Hint The piecewise-defined function is constant on the intervals (0,1],(1,2],…. Answer [C(x)=egin{cases}49,&0 ## Transformations of Functions We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function (y=x^2) to get the function( f(x)=(x−2)^2). This subtraction represents a shift of the function (y=x^2) two units to the right. A shift, horizontally or vertically, is a type of transformation of a function. Other transformations include horizontal and vertical scalings, and reflections about the axes. A vertical shift of a function occurs if we add or subtract the same constant to each output (y). For (c>0), the graph of (f(x)+c) is a shift of the graph of (f(x)) up c units, whereas the graph of (f(x)−c) is a shift of the graph of (f(x)) down (c) units. For example, the graph of the function (f(x)=x^3+4) is the graph of (y=x^3) shifted up (4) units; the graph of the function (f(x)=x^3−4) is the graph of (y=x^3) shifted down (4) units (Figure (PageIndex{6})). Figure (PageIndex{9}): (a) For (c>0), the graph of (y=f(x)+c) is a vertical shift up (c) units of the graph of (y=f(x)). (b) For (c>0), the graph of (y=f(x)−c) is a vertical shift down c units of the graph of (y=f(x)). A horizontal shift of a function occurs if we add or subtract the same constant to each input (x). For (c>0), the graph of (f(x+c)) is a shift of the graph of (f(x)) to the left (c) units; the graph of (f(x−c)) is a shift of the graph of (f(x)) to the right (c) units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let’s look at an example. Consider the function (f(x)=|x+3|) and evaluate this function at (x−3). Since (f(x−3)=|x|) and (x−3 Figure (PageIndex{10}): (a) For (c>0), the graph of (y=f(x+c)) is a horizontal shift left (c) units of the graph of (y=f(x)). (b) For (c>0), the graph of (y=f(x−c)) is a horizontal shift right (c) units of the graph of (y=f(x).) A vertical scaling of a graph occurs if we multiply all outputs (y) of a function by the same positive constant. For (c>0), the graph of the function (cf(x)) is the graph of (f(x)) scaled vertically by a factor of (c). If (c>1), the values of the outputs for the function (cf(x)) are larger than the values of the outputs for the function (f(x)); therefore, the graph has been stretched vertically. If (0 Figure (PageIndex{11}): (a) If (c>1), the graph of (y=cf(x)) is a vertical stretch of the graph of (y=f(x)). (b) If (0 The horizontal scaling of a function occurs if we multiply the inputs (x) by the same positive constant. For (c>0), the graph of the function (f(cx)) is the graph of (f(x)) scaled horizontally by a factor of (c). If (c>1), the graph of (f(cx)) is the graph of (f(x)) compressed horizontally. If (0 Figure (PageIndex{12}): (a) If (c>1), the graph of (y=f(cx)) is a horizontal compression of the graph of (y=f(x)). (b) If (0 We have explored what happens to the graph of a function (f) when we multiply (f) by a constant (c>0) to get a new function (cf(x)). We have also discussed what happens to the graph of a function (f)when we multiply the independent variable (x) by (c>0) to get a new function (f(cx)). However, we have not addressed what happens to the graph of the function if the constant (c) is negative. If we have a constant (c<0), we can write (c) as a positive number multiplied by (−1); but, what kind of transformation do we get when we multiply the function or its argument by (−1?) When we multiply all the outputs by (−1), we get a reflection about the (x)-axis. When we multiply all inputs by (−1), we get a reflection about the (y)-axis. For example, the graph of (f(x)=−(x^3+1)) is the graph of (y=(x^3+1)) reflected about the (x)-axis. The graph of (f(x)=(−x)^3+1) is the graph of (y=x^3+1) reflected about the (y)-axis (Figure (PageIndex{10})). Figure (PageIndex{13}): (a) The graph of (y=−f(x)) is the graph of (y=f(x)) reflected about the (x)-axis. (b) The graph of (y=f(−x)) is the graph of (y=f(x)) reflected about the (y)-axis. If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function (f(x)), the graph of the related function (y=cf(a(x+b))+d) can be obtained from the graph of (y=f(x))by performing the transformations in the following order. • Horizontal shift of the graph of (y=f(x)). If (b>0), shift left. If (b<0) shift right. • Horizontal scaling of the graph of (y=f(x+b)) by a factor of (|a|). If (a<0), reflect the graph about the (y)-axis. • Vertical scaling of the graph of (y=f(a(x+b))) by a factor of (|c|). If (c<0), reflect the graph about the (x) -axis. • Vertical shift of the graph of (y=cf(a(x+b))). If (d>0), shift up. If (d<0), shift down. We can summarize the different transformations and their related effects on the graph of a function in the following table. Transformation of (f (c>0))Effect of the graph of (f) (f(x)+c)Vertical shift up (c) units (f(x)-c)Vertical shift down (c) units (f(x+c))Shift left by (c) units (f(x-c))Shift right by (c) units (cf(x)) Vertical stretch if (c>1); vertical compression if (0 (f(cx)) Horizontal stretch if (0 horizontal compression if (c>1) (-f(x))Reflection about the (x)-axis (-f(x))Reflection about the (y)-axis Example (PageIndex{10}): Transforming a Function For each of the following functions, a. and b., sketch a graph by using a sequence of transformations of a well-known function. 1. (f(x)=−|x+2|−3) 2. (f(x)=sqrt[3]{x}+1) Solution: 1.Starting with the graph of (y=|x|), shift (2) units to the left, reflect about the (x)-axis, and then shift down (3) units. Figure (PageIndex{14}): The function (f(x)=−|x+2|−3) can be viewed as a sequence of three transformations of the function (y=|x|). 2. Starting with the graph of y=x√, reflect about the y-axis, stretch the graph vertically by a factor of 3, and move up 1 unit. Figure (PageIndex{15}): The function (f(x)=sqrt[3]{x}+1)can be viewed as a sequence of three transformations of the function (y=sqrt{x}). Exercise (PageIndex{7}) Describe how the function (f(x)=−(x+1)^2−4) can be graphed using the graph of (y=x^2) and a sequence of transformations Answer Shift the graph (y=x^2) to the left 1 unit, reflect about the (x)-axis, then shift down 4 units. ### Key Concepts • The power function (f(x)=x^n) is an even function if n is even and (n≠0), and it is an odd function if (n) is odd. • The root function (f(x)=x^{1/n}) has the domain ([0,∞)) if n is even and the domain ((−∞,∞)) if (n) is odd. If (n) is odd, then (f(x)=x^{1/n}) is an odd function. • The domain of the rational function (f(x)=p(x)/q(x)), where (p(x)) and (q(x)) are polynomial functions, is the set of x such that (q(x)≠0). • Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions. • A polynomial function (f) with degree (n≥1) satisfies (f(x)→±∞) as (x→±∞). The sign of the output as (x→∞) depends on the sign of the leading coefficient only and on whether (n) is even or odd. • Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the (x)- and (y)-axes are examples of transformations of functions. ## Key Equations • Point-slope equation of a line (y−y1=m(x−x_1)) • Slope-intercept form of a line (y=mx+b) • Standard form of a line (ax+by=c) • Polynomial function (f(x)=a_n^{x^n}+a_{n−1}x^{n−1}+⋯+a_1x+a_0) ## Glossary algebraic function a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable (x) cubic function a polynomial of degree 3; that is, a function of the form (f(x)=ax^3+bx^2+cx+d), where (a≠0) degree for a polynomial function, the value of the largest exponent of any term linear function a function that can be written in the form (f(x)=mx+b) logarithmic function a function of the form (f(x)=log_b(x)) for some base (b>0,b≠1) such that (y=log_b(x)) if and only if (b^y=x) mathematical model A method of simulating real-life situations with mathematical equations piecewise-defined function a function that is defined differently on different parts of its domain point-slope equation equation of a linear function indicating its slope and a point on the graph of the function polynomial function a function of the form (f(x)=a_nx^n+a_{n−1}x^{n−1}+…+a_1x+a_0) power function a function of the form (f(x)=x^n) for any positive integer (n≥1) quadratic function a polynomial of degree 2; that is, a function of the form (f(x)=ax^2+bx+c) where (a≠0) rational function a function of the form (f(x)=p(x)/q(x)), where (p(x)) and (q(x)) are polynomials root function a function of the form (f(x)=x^{1/n}) for any integer (n≥2) slope the change in y for each unit change in x slope-intercept form equation of a linear function indicating its slope and y-intercept transcendental function a function that cannot be expressed by a combination of basic arithmetic operations transformation of a function a shift, scaling, or reflection of a function ### Contributors • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org. ## Complexity class In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. counting problems and function problems) and using other models of computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and quantum computers). The study of the relationships between complexity classes is a major area of research in theoretical computer science. There are often general hierarchies of complexity classes for example, it is known that a number of fundamental time and space complexity classes relate to each other in the following way: NLPNPPSPACEEXPTIMEEXPSPACE (where ⊆ denotes the subset relation). However, many relationships are not yet known for example, one of the most famous open problems in computer science concerns whether or not P equals NP. The relationships between classes often answer questions about the fundamental nature of computation. The P versus NP problem, for instance, is directly related to questions of whether nondeterminism adds any computational power to computers and whether problems having a solution that can be quickly checked for correctness can also be quickly solved. ## 1.3: Basic Classes of Functions data Bool = False | True deriving (Read, Show, Eq, Ord, Enum, Bounded) The boolean type Bool is an enumeration. The basic boolean functions are && (and), || (or), and not. The name otherwise is defined as True to make guarded expressions more readable. #### 6.1.2 Characters and Strings The character type Char is an enumeration whose values represent Unicode characters [11]. The lexical syntax for characters is defined in Section 2.6 character literals are nullary constructors in the datatype Char. Type Char is an instance of the classes Read, Show, Eq, Ord, Enum, and Bounded. The toEnum and fromEnum functions, standard functions from class Enum, map characters to and from the Int type. Note that ASCII control characters each have several representations in character literals: numeric escapes, ASCII mnemonic escapes, and the ^ X notation. In addition, there are the following equivalences: a and BEL, and BS, f and FF, and CR, and HT, v and VT, and and LF. A string is a list of characters: Strings may be abbreviated using the lexical syntax described in Section 2.6. For example, "A string" abbreviates #### 6.1.3 Lists data [a] = [] | a : [a] deriving (Eq, Ord) Lists are an algebraic datatype of two constructors, although with special syntax, as described in Section 3.7. The first constructor is the null list, written []' ("nil"), and the second is :' ("cons"). The module PreludeList (see Section 8.1) defines many standard list functions. Arithmetic sequences and list comprehensions, two convenient syntaxes for special kinds of lists, are described in Sections 3.10 and 3.11, respectively. Lists are an instance of classes Read, Show, Eq, Ord, Monad, Functor, and MonadPlus. #### 6.1.4 Tuples Tuples are algebraic datatypes with special syntax, as defined in Section 3.8. Each tuple type has a single constructor. All tuples are instances of Eq, Ord, Bounded, Read, and Show (provided, of course, that all their component types are). There is no upper bound on the size of a tuple, but some Haskell implementations may restrict the size of tuples, and limit the instances associated with larger tuples. However, every Haskell implementation must support tuples up to size 15, together with the instances for Eq, Ord, Bounded, Read, and Show. The Prelude and libraries define tuple functions such as zip for tuples up to a size of 7. The constructor for a tuple is written by omitting the expressions surrounding the commas thus (x,y) and (,) x y produce the same value. The same holds for tuple type constructors thus, (Int,Bool,Int) and (,,) Int Bool Int denote the same type. The following functions are defined for pairs (2-tuples): fst, snd, curry, and uncurry. Similar functions are not predefined for larger tuples. #### 6.1.5 The Unit Datatype data () = () deriving (Eq, Ord, Bounded, Enum, Read, Show) The unit datatype () has one non- _|_ member, the nullary constructor (). See also Section 3.9. #### 6.1.6 Function Types #### 6.1.7 The IO and IOError Types IOError is an abstract type representing errors raised by I/O operations. It is an instance of Show and Eq. Values of this type are constructed by the various I/O functions and are not presented in any further detail in this report. The Prelude contains a few I/O functions (defined in Section 8.3), and Part II contains many more. #### 6.1.8 Other Types data Maybe a = Nothing | Just a deriving (Eq, Ord, Read, Show) data Either a b = Left a | Right b deriving (Eq, Ord, Read, Show) data Ordering = LT | EQ | GT deriving (Eq, Ord, Bounded, Enum, Read, Show) The Maybe type is an instance of classes Functor, Monad, and MonadPlus. The Ordering type is used by compare in the class Ord. The functions maybe and either are found in the Prelude. ### 6.2 Strict Evaluation The function seq is defined by the equations: seq _|_b = _|_ seq a b = b, if a /=_|_ seq is usually introduced to improve performance by avoiding unneeded laziness. Strict datatypes (see Section 4.2.1) are defined in terms of the$! operator. However, the provision of seq has important semantic consequences, because it is available at every type . As a consequence, _|_ is not the same as x -> _|_ , since seq can be used to distinguish them. For the same reason, the existence of seq weakens Haskell's parametricity properties.
The operator $! is strict (call-by-value) application, and is defined in terms of seq. The Prelude also defines the$ operator to perform non-strict application.
infixr 0 $,$!
($), ($!) :: (a -> b) -> a -> b
f $x = f x f$! x = x seq f x
It is also useful in higher-order situations, such as map ($0) xs, or zipWith ($) fs xs.
#### Figure 5
Default class method declarations (Section 4.3) are provided for many of the methods in standard classes. A comment with each class declaration in Chapter 8 specifies the smallest collection of method definitions that, together with the default declarations, provide a reasonable definition for all the class methods. If there is no such comment, then all class methods must be given to fully specify an instance.
#### 6.3.1 The Eq Class
class Eq a where
(==), (/=) :: a -> a -> Bool
x /= y = not (x == y)
x == y = not (x /= y)
The Eq class provides equality (==) and inequality (/=) methods. All basic datatypes except for functions and IO are instances of this class. Instances of Eq can be derived for any user-defined datatype whose constituents are also instances of Eq.
This declaration gives default method declarations for both /= and ==, each being defined in terms of the other. If an instance declaration for Eq defines neither == nor /=, then both will loop. If one is defined, the default method for the other will make use of the one that is defined. If both are defined, neither default method is used.
#### 6.3.2 The Ord Class
class (Eq a) => Ord a where
compare :: a -> a -> Ordering
(<), (<=), (>=), (>) :: a -> a -> Bool
max, min :: a -> a -> a
compare x y | x == y = EQ
| x <= y = LT
| otherwise = GT
x <= y = compare x y /= GT
x < y = compare x y == LT
x >= y = compare x y /= LT
x > y = compare x y == GT
-- Note that (min x y, max x y) = (x,y) or (y,x)
max x y | x <= y = y
| otherwise = x
min x y | x <= y = x
| otherwise = y
The Ord class is used for totally ordered datatypes. All basic datatypes except for functions, IO, and IOError, are instances of this class. Instances of Ord can be derived for any user-defined datatype whose constituent types are in Ord. The declared order of the constructors in the data declaration determines the ordering in derived Ord instances. The Ordering datatype allows a single comparison to determine the precise ordering of two objects.
The default declarations allow a user to create an Ord instance either with a type-specific compare function or with type-specific == and <= functions.
#### 6.3.3 The Read and Show Classes
type ReadS a = String -> [(a,String)]
type ShowS = String -> String
-- . default decl for readList given in Prelude
class Show a where
showsPrec :: Int -> a -> ShowS
show :: a -> String
showList :: [a] -> ShowS
showsPrec _ x s = show x ++ s
show x = showsPrec 0 x ""
-- . default decl for showList given in Prelude
The Read and Show classes are used to convert values to or from strings. The Int argument to showsPrec and readsPrec gives the operator precedence of the enclosing context (see Section 10.4).
showsPrec and showList return a String-to-String function, to allow constant-time concatenation of its results using function composition. A specialised variant, show, is also provided, which uses precedence context zero, and returns an ordinary String. The method showList is provided to allow the programmer to give a specialised way of showing lists of values. This is particularly useful for the Char type, where values of type String should be shown in double quotes, rather than between square brackets.
Derived instances of Read and Show replicate the style in which a constructor is declared: infix constructors and field names are used on input and output. Strings produced by showsPrec are usually readable by readsPrec.
All Prelude types, except function types and IO types, are instances of Show and Read. (If desired, a programmer can easily make functions and IO types into (vacuous) instances of Show, by providing an instance declaration.)
For convenience, the Prelude provides the following auxiliary functions:
shows :: (Show a) => a -> ShowS
shows = showsPrec 0
read s = case [x | (x,t) <- reads s, ("","") <- lex t] of
[x] -> x
[] -> error "PreludeText.read: no parse"
_ -> error "PreludeText.read: ambiguous parse"
shows and reads use a default precedence of 0. The read function reads input from a string, which must be completely consumed by the input process.
The function lex :: ReadS String, used by read, is also part of the Prelude. It reads a single lexeme from the input, discarding initial white space, and returning the characters that constitute the lexeme. If the input string contains only white space, lex returns a single successful "lexeme" consisting of the empty string. (Thus lex "" = [("","")].) If there is no legal lexeme at the beginning of the input string, lex fails (i.e. returns []).
#### 6.3.4 The Enum Class
class Enum a where
succ, pred :: a -> a
toEnum :: Int -> a
enumFrom :: a -> [a] -- [n..]
enumFromThen :: a -> a -> [a] -- [n,n'..]
enumFromTo :: a -> a -> [a] -- [n..m]
enumFromThenTo :: a -> a -> a -> [a] -- [n,n'..m]
-- Default declarations given in Prelude
Class Enum defines operations on sequentially ordered types. The functions succ and pred return the successor and predecessor, respectively, of a value. The functions fromEnum and toEnum map values from a type in Enum to and from Int. The enumFrom. methods are used when translating arithmetic sequences (Section 3.10).
Instances of Enum may be derived for any enumeration type (types whose constructors have no fields) see Chapter 10.
The calls succ maxBound and pred minBound should result in a runtime error.
enumFrom x = enumFromTo x maxBound
enumFromThen x y = enumFromThenTo x y bound
where
| otherwise = minBound
Enumeration types: (), Bool, and Ordering. The semantics of these instances is given by Chapter 10. For example, [LT..] is the list [LT,EQ,GT].
For all four of these Prelude numeric types, all of the enumFrom family of functions are strict in all their arguments.
#### 6.3.5 The Functor Class
class Functor f where
fmap :: (a -> b) -> f a -> f b
The Functor class is used for types that can be mapped over. Lists, IO, and Maybe are in this class.
Instances of Functor should satisfy the following laws:
fmap id = id fmap (f . g) = fmap f . fmap g
All instances of Functor defined in the Prelude satisfy these laws.
(>>=) :: m a -> (a -> m b) -> m b
(>>) :: m a -> m b -> m b
return :: a -> m a
fail :: String -> m a
m >> k = m >>= \_ -> k
fail s = error s
"do" expressions provide a convenient syntax for writing monadic expressions (see Section 3.14). The fail method is invoked on pattern-match failure in a do expression.
In the Prelude, lists, Maybe, and IO are all instances of Monad. The fail method for lists returns the empty list [], for Maybe returns Nothing, and for IO raises a user exception in the IO monad (see Section 7.3).
Instances of Monad should satisfy the following laws:
return a >>= k = k a m >>= return = m m >>= (x -> k x >>= h) = (m >>= k) >>= h
All instances of Monad defined in the Prelude satisfy these laws.
The Prelude provides the following auxiliary functions:
sequence :: Monad m => [m a] -> m [a]
sequence_ :: Monad m => [m a] -> m ()
mapM :: Monad m => (a -> m b) -> [a] -> m [b]
mapM_ :: Monad m => (a -> m b) -> [a] -> m ()
(=<<) :: Monad m => (a -> m b) -> m a -> m b
#### 6.3.7 The Bounded Class
The Bounded class is used to name the upper and lower limits of a type. Ord is not a superclass of Bounded since types that are not totally ordered may also have upper and lower bounds. The types Int, Char, Bool, (), Ordering, and all tuples are instances of Bounded. The Bounded class may be derived for any enumeration type minBound is the first constructor listed in the data declaration and maxBound is the last. Bounded may also be derived for single-constructor datatypes whose constituent types are in Bounded.
### 6.4 Numbers
Haskell provides several kinds of numbers the numeric types and the operations upon them have been heavily influenced by Common Lisp and Scheme. Numeric function names and operators are usually overloaded, using several type classes with an inclusion relation shown in Figure 6.1. The class Num of numeric types is a subclass of Eq, since all numbers may be compared for equality its subclass Real is also a subclass of Ord, since the other comparison operations apply to all but complex numbers (defined in the Complex library). The class Integral contains integers of both limited and unlimited range the class Fractional contains all non-integral types and the class Floating contains all floating-point types, both real and complex.
The Prelude defines only the most basic numeric types: fixed sized integers (Int), arbitrary precision integers (Integer), single precision floating (Float), and double precision floating (Double). Other numeric types such as rationals and complex numbers are defined in libraries. In particular, the type Rational is a ratio of two Integer values, as defined in the Ratio library.
The default floating point operations defined by the Haskell Prelude do not conform to current language independent arithmetic (LIA) standards. These standards require considerably more complexity in the numeric structure and have thus been relegated to a library. Some, but not all, aspects of the IEEE floating point standard have been accounted for in Prelude class RealFloat.
The standard numeric types are listed in Table 6.1. The finite-precision integer type Int covers at least the range [ - 2 29 , 2 29 - 1] . As Int is an instance of the Bounded class, maxBound and minBound can be used to determine the exact Int range defined by an implementation. Float is implementation-defined it is desirable that this type be at least equal in range and precision to the IEEE single-precision type. Similarly, Double should cover IEEE double-precision. The results of exceptional conditions (such as overflow or underflow) on the fixed-precision numeric types are undefined an implementation may choose error ( _|_ , semantically), a truncated value, or a special value such as infinity, indefinite, etc.
Type Class Description Integer Integral Arbitrary-precision integers Int Integral Fixed-precision integers (Integral a) => Ratio a RealFrac Rational numbers Float RealFloat Real floating-point, single precision Double RealFloat Real floating-point, double precision (RealFloat a) => Complex a Floating Complex floating-point
### Standard Numeric Types
The standard numeric classes and other numeric functions defined in the Prelude are shown in Figures 6.2--6.3. Figure 6.1 shows the class dependencies and built-in types that are instances of the numeric classes.
class (Num a, Ord a) => Real a where
toRational :: a -> Rational
class (Real a, Enum a) => Integral a where
quot, rem, div, mod :: a -> a -> a
quotRem, divMod :: a -> a -> (a,a)
toInteger :: a -> Integer
class (Num a) => Fractional a where
(/) :: a -> a -> a
recip :: a -> a
fromRational :: Rational -> a
### Standard Numeric Classes and Related Operations, Part 1
class (RealFrac a, Floating a) => RealFloat a where
floatDigits :: a -> Int
floatRange :: a -> (Int,Int)
decodeFloat :: a -> (Integer,Int)
encodeFloat :: Integer -> Int -> a
exponent :: a -> Int
significand :: a -> a
scaleFloat :: Int -> a -> a
isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE
:: a -> Bool
atan2 :: a -> a -> a
gcd, lcm :: (Integral a) => a -> a-> a
(^) :: (Num a, Integral b) => a -> b -> a
(^^) :: (Fractional a, Integral b) => a -> b -> a
### Standard Numeric Classes and Related Operations, Part 2
#### 6.4.1 Numeric Literals
The syntax of numeric literals is given in Section 2.5. An integer literal represents the application of the function fromInteger to the appropriate value of type Integer. Similarly, a floating literal stands for an application of fromRational to a value of type Rational (that is, Ratio Integer). Given the typings:
fromInteger :: (Num a) => Integer -> a
fromRational :: (Fractional a) => Rational -> a
integer and floating literals have the typings (Num a) => a and (Fractional a) => a, respectively. Numeric literals are defined in this indirect way so that they may be interpreted as values of any appropriate numeric type. See Section 4.3.4 for a discussion of overloading ambiguity.
#### 6.4.2 Arithmetic and Number-Theoretic Operations
The infix class methods (+), (*), (-), and the unary function negate (which can also be written as a prefix minus sign see section 3.4) apply to all numbers. The class methods quot, rem, div, and mod apply only to integral numbers, while the class method (/) applies only to fractional ones. The quot, rem, div, and mod class methods satisfy these laws if y is non-zero:
(x quot y)*y + (x rem y) == x (x div y)*y + (x mod y) == x
quot is integer division truncated toward zero, while the result of div is truncated toward negative infinity. The quotRem class method takes a dividend and a divisor as arguments and returns a (quotient, remainder) pair divMod is defined similarly:
even x = x rem 2 == 0
odd = not . even
Finally, there are the greatest common divisor and least common multiple functions. gcd x y is the greatest (positive) integer that divides both x and y for example gcd (-3) 6 = 3, gcd (-3) (-6) = 3, gcd 0 4 = 4. gcd 0 0 raises a runtime error.
lcm x y is the smallest positive integer that both x and y divide.
#### 6.4.3 Exponentiation and Logarithms
The one-argument exponential function exp and the logarithm function log act on floating-point numbers and use base e . logBase a x returns the logarithm of x in base a . sqrt returns the principal square root of a floating-point number. There are three two-argument exponentiation operations: (^) raises any number to a nonnegative integer power, (^^) raises a fractional number to any integer power, and (**) takes two floating-point arguments. The value of x ^0 or x ^^0 is 1 for any x , including zero 0** y is undefined.
#### 6.4.4 Magnitude and Sign
A number has a magnitude and a sign . The functions abs and signum apply to any number and satisfy the law:
signum x | x > 0 = 1
| x == 0 = 0
| x < 0 = -1
#### 6.4.5 Trigonometric Functions
Class Floating provides the circular and hyperbolic sine, cosine, and tangent functions and their inverses. Default implementations of tan, tanh, logBase, **, and sqrt are provided, but implementors are free to provide more accurate implementations.
Class RealFloat provides a version of arctangent taking two real floating-point arguments. For real floating x and y , atan2 y x computes the angle (from the positive x-axis) of the vector from the origin to the point (x,y) . atan2 y x returns a value in the range [-pi, pi]. It follows the Common Lisp semantics for the origin when signed zeroes are supported. atan2 y 1, with y in a type that is RealFloat, should return the same value as atan y . A default definition of atan2 is provided, but implementors can provide a more accurate implementation.
The precise definition of the above functions is as in Common Lisp, which in turn follows Penfield's proposal for APL [9]. See these references for discussions of branch cuts, discontinuities, and implementation.
#### 6.4.6 Coercions and Component Extraction
The ceiling, floor, truncate, and round functions each take a real fractional argument and return an integral result. ceiling x returns the least integer not less than x , and floor x , the greatest integer not greater than x . truncate x yields the integer nearest x between 0 and x , inclusive. round x returns the nearest integer to x , the even integer if x is equidistant between two integers.
The function properFraction takes a real fractional number x and returns a pair (n,f) such that x = n+f , and: n is an integral number with the same sign as x and f is a fraction f with the same type and sign as x , and with absolute value less than 1. The ceiling, floor, truncate, and round functions can be defined in terms of properFraction.
Two functions convert numbers to type Rational: toRational returns the rational equivalent of its real argument with full precision approxRational takes two real fractional arguments x and e and returns the simplest rational number within e of x , where a rational p/q in reduced form is simpler than another p ' / q ' if |p| <=|p ' | and q <=q ' . Every real interval contains a unique simplest rational in particular, note that 0/1 is the simplest rational of all.
The class methods of class RealFloat allow efficient, machine-independent access to the components of a floating-point number. The functions floatRadix, floatDigits, and floatRange give the parameters of a floating-point type: the radix of the representation, the number of digits of this radix in the significand, and the lowest and highest values the exponent may assume, respectively. The function decodeFloat applied to a real floating-point number returns the significand expressed as an Integer and an appropriately scaled exponent (an Int). If decodeFloat x yields ( m , n ), then x is equal in value to mb n , where b is the floating-point radix, and furthermore, either m and n are both zero or else b d-1 <=m<b d , where d is the value of floatDigits x. encodeFloat performs the inverse of this transformation. The functions significand and exponent together provide the same information as decodeFloat, but rather than an Integer, significand x yields a value of the same type as x, scaled to lie in the open interval (-1,1) . exponent 0 is zero. scaleFloat multiplies a floating-point number by an integer power of the radix.
The functions isNaN, isInfinite, isDenormalized, isNegativeZero, and isIEEE all support numbers represented using the IEEE standard. For non-IEEE floating point numbers, these may all return false.
Also available are the following coercion functions:
fromIntegral :: (Integral a, Num b) => a -> b
realToFrac :: (Real a, Fractional b) => a -> b
top | back | next | contents | function index
December 2002
## Functions
To make life easier, MATLAB includes many standard functions. Each function is a block of code that accomplishes a specific task. MATLAB contains all of the standard functions such as sin, cos, log, exp, sqrt, as well as many others. Commonly used constants such as pi, and i or j for the square root of -1, are also incorporated into MATLAB.
To determine the usage of any function, type help [function name] at the MATLAB command window.
MATLAB even allows you to write your own functions with the function command follow the link to learn how to write your own functions and see a listing of the functions we created for this tutorial.
## What are the types of Discontinuities?
The graph of $f(x)$ below shows a function that is discontinuous at $x = a$.
In this graph, you can easily see that $limlimits_ f(x) = L> % mbox < and >% limlimits_ f(x) = M>.$
The function is approaching different values depending on the direction $x$ is coming from. When this happens, we say the function has a jump discontinuity at $x=a$.
### Infinite Discontinuities
The graph below shows a function that is discontinuous at $x=a$.
The arrows on the function indicate it will grow infinitely large as $x$ approaches $a$. Since the function doesn't approach a particular finite value, the limit does not exist. This is an infinite discontinuity.
The following two graphs are also examples of infinite discontinuities at $x = a$. Notice that in all three cases, both of the one-sided limits are infinite.
### Removable Discontinuities
In the graphs below, there is a hole in the function at $x=a$. These holes are called removable discontinuities
Notice that for both graphs, even though there are holes at $x = a$, the limit value at $x=a$ exists.
### Removable Discontinuities can be Fixed
Removable discontinuities can be fixed by redefining the function, as shown in the following example.
##### Example
The function below has a removable discontinuity at $x = 2$. Redefine the function so that it becomes continuous at $x=2$.
The graph of the function is shown below for reference.
In order to fix the discontinuity, we need to know the $y$-value of the hole in the graph. To determine this, we find the value of $limlimits_ f(x)$.
Examining the form of the limit we see
The division by zero in the $frac 0 0$ form tells us there is definitely a discontinuity at this point.
Next, using the techniques covered in previous lessons (see Indeterminate Limits---Factorable) we can easily determine
$displaystylelim_ f(x) = frac 1 2$
The limit value is also the $y$-value of the hole in the graph. Now we can redefine the original function in a piecewise form:
The first piece preserves the overall behavior of the function, while the second piece plugs the hole.
### Endpoint Discontinuities
When a function is defined on an interval with a closed endpoint, the limit cannot exist at that endpoint. This is because the limit has to examine the function values as $x$ approaches from both sides.
For example, consider finding $displaystylelimlimits_ sqrt x$ (see the graph below).
Note that $x=0$ is the left-endpoint of the functions domain: $[0,infty)$, and the function is technically not continuous there because the limit doesn't exist (because $x$ can't approach from both sides).
We should note that the function is right-hand continuous at $x=0$ which is why we don't see any jumps, or holes at the endpoint.
### Mixed Discontinuities
Consider the graph shown below.
The function is obviously discontinuous at $x = 3$. From the left, the function has an infinite discontinuity, but from the right, the discontinuity is removable. Since there is more than one reason why the discontinuity exists, we say this is a mixed discontinuity
## Contents
Glycosaminoglycans vary greatly in molecular mass, disaccharide construction, and sulfation. This is because GAG synthesis is not template driven like proteins or nucleic acids, but constantly altered by processing enzymes. [4]
GAGs are classified into four groups based on core disaccharide structures. [5] Heparin/heparan sulfate (HSGAGs) and chondroitin sulfate/dermatan sulfate (CSGAGs) are synthesized in the Golgi apparatus, where protein cores made in the rough endoplasmic reticulum are post-translationally modified with O-linked glycosylations by glycosyltransferases forming proteoglycans. Keratan sulfate may modify core proteins through N-linked glycosylation or O-linked glycosylation of the proteoglycan. The fourth class of GAG, hyaluronic acid is synthesized by integral membrane synthases which immediately secrete the dynamically elongated disaccharide chain.
### HSGAG and CSGAG Edit
HSGAG and CSGAG modified proteoglycans first begin with a consensus Ser-Gly/Ala-X-Gly motif in the core protein. Construction of a tetrasaccharide linker that consists of -GlcAβ1–3Galβ1–3Galβ1–4Xylβ1-O-(Ser)-, where xylosyltransferase, β4-galactosyl transferase (GalTI),β3-galactosyl transferase (GalT-II), and β3-GlcA transferase (GlcAT-I) transfer the four monosaccharides, begins synthesis of the GAG modified protein. The first modification of the tetrasaccharide linker determines whether the HSGAGs or CSGAGs will be added. Addition of a GlcNAc promotes the addition of HSGAGs while addition of GalNAc to the tetrasaccharide linker promotes CSGAG development. [5] GlcNAcT-I transfers GlcNAc to the tetrasaccahride linker, which is distinct from glycosyltransferase GlcNAcT-II, the enzyme that is utilized to build HSGAGs. EXTL2 and EXTL3, two genes in the EXT tumor suppressor family, have been shown to have GlcNAcT-I activity. Conversely, GalNAc is transferred to the linker by the enzyme GalNAcT to initiate synthesis of CSGAGs, an enzyme which may or may not have distinct activity compared to the GalNAc transferase activity of chondroitin synthase. [5]
With regard to HSGAGs, a multimeric enzyme encoded by EXT1 and EXT2 of the EXT family of genes, transfers both GlcNAc and GlcA for HSGAG chain elongation. While elongating, the HSGAG is dynamically modified, first by N-deacetylase, N-sulfotransferase (NDST1), which is a bifunctional enzyme that cleaves the N-acetyl group from GlcNAc and subsequently sulfates the N-position. Next, C-5 uronyl epimerase coverts d-GlcA to l-IdoA followed by 2-O sulfation of the uronic acid sugar by 2-O sulfotransferase (Heparan sulfate 2-O-sulfotransferase). Finally, the 6-O and 3-O positions of GlcNAc moities are sulfated by 6-O (Heparan sulfate 6-O-sulfotransferase) and 3-O (3-OST) sulfotransferases.
Chondroitin sulfate and dermatan sulfate, which comprise CSGAGs, are differentiated from each other by the presence of GlcA and IdoA epimers respectively. Similar to the production of HSGAGs, C-5 uronyl epimerase converts d-GlcA to l-IdoA to synthesize dermatan sulfate. Three sulfation events of the CSGAG chains occur: 4-O and/or 6-O sulfation of GalNAc and 2-O sulfation of uronic acid. Four isoforms of the 4-O GalNAc sulfotransferases (C4ST-1, C4ST-2, C4ST-3, and D4ST-1) and three isoforms of the GalNAc 6-O sulfotransferases (C6ST, C6ST-2, and GalNAc4S-6ST) are responsible for the sulfation of GalNAc. [6]
### Keratan sulfate types Edit
Unlike HSGAGs and CSGAGs, the third class of GAGs, those belonging to keratan sulfate types, are driven towards biosynthesis through particular protein sequence motifs. For example, in the cornea and cartilage, the keratan sulfate domain of aggrecan consists of a series of tandemly repeated hexapeptides with a consensus sequence of E(E/L)PFPS. [7] Additionally, for three other keratan sulfated proteoglycans, lumican, keratocan, and mimecan (OGN), the consensus sequence NX(T/S) along with protein secondary structure was determined to be involved in N-linked oligosaccharide extension with keratan sulfate. [7] Keratan sulfate elongation begins at the nonreducing ends of three linkage oligosaccharides, which define the three classes of keratan sulfate. Keratan sulfate I (KSI) is N -linked via a high mannose type precursor oligosaccharide. Keratan sulfate II (KSII) and keratan sulfate III (KSIII) are O-linked, with KSII linkages identical to that of mucin core structure, and KSIII linked to a 2-O mannose. Elongation of the keratan sulfate polymer occurs through the glycosyltransferase addition of Gal and GlcNAc. Galactose addition occurs primarily through the β-1,4-galactosyltransferase enzyme (β4Gal-T1) while the enzymes responsible for β-3-Nacetylglucosamine have not been clearly identified. Finally, sulfation of the polymer occurs at the 6-position of both sugar residues. The enzyme KS-Gal6ST (CHST1) transfers sulfate groups to galactose while N-acetylglucosaminyl-6-sulfotransferase (GlcNAc6ST) (CHST2) transfers sulfate groups to terminal GlcNAc in keratan sulfate. [8]
### Hyaluronic acid Edit
The fourth class of GAG, hyaluronic acid, is not sulfated and is synthesized by three transmembrane synthase proteins HAS1, HAS2, and HAS3. HA, a linear polysaccharide, is composed of repeating disaccharide units of →4)GlcAβ(1→3)GlcNAcβ(1→ and has a very high molecular mass, ranging from 10 5 to 10 7 Da. Each HAS enzyme is capable of transglycosylation when supplied with UDP-GlcA and UDP-GlcNAc. [9] [10] HAS2 is responsible for very large hyaluronic acid polymers, while smaller sizes of HA are synthesized by HAS1 and HAS3. While each HAS isoform catalyzes the same biosynthetic reaction, each HAS isoform is independently active. HAS isoforms have also been shown to have differing Km values for UDP-GlcA and UDPGlcNAc. [11] It is believed that through differences in enzyme activity and expression, the wide spectrum of biological functions mediated by HA can be regulated, such as its involvement with neural stem cell regulation in the subgranular zone of the brain.
CSGAGs interact with heparin binding proteins, specifically dermatan sulfate interactions with fibroblast growth factor FGF-2 and FGF-7 have been implicated in cellular proliferation and wound repair [15] while interactions with hepatic growth factor/scatter factor (HGF/SF) activate the HGF/SF signaling pathway (c-Met) through its receptor. CASGAGs are important in providing support and adhesiveness in bone, skin, and cartilage. Other biological functions for which CSGAGs are known to play critical functions in include inhibition of axonal growth and regeneration in CNS development, roles in brain development, neuritogenic activity, and pathogen infection. [16]
Keratan sulfates One of the main functions of the third class of GAGs, keratan sulfates, is the maintenance of tissue hydration. [17] Keratan sulfates are in the bone, cartilage, and the cornea of the eye. [18] Within the normal cornea, dermatan sulfate is fully hydrated whereas keratan sulfate is only partially hydrated suggesting that keratan sulfate may behave as a dynamically controlled buffer for hydration. [17] In disease states such as macular corneal dystrophy, in which GAGs levels such as KS are altered, loss of hydration within the corneal stroma is believed to be the cause of corneal haze, thus supporting the long-held hypothesis that corneal transparency is a dependent on proper levels of keratan sulfate. Keratan sulfate GAGs are found in many other tissues besides the cornea, where they are known to regulate macrophage adhesion, form barriers to neurite growth, regulate embryo implantation in the endometrial uterine lining during menstrual cycles, and affect the motility of corneal endothelial cells. [17] In summary, KS plays an anti-adhesive role, which suggests very important functions of KS in cell motility and attachment as well as other potential biological processes.
Dermatan sulfates function in the skin, tendons, blood vessels, and heart valves. [18]
Hyaluronic acid Hyaluronic acid is a major component of synovial tissues and fluid, as well as the ground substance of other connective tissues. Hyaluronic acid binds cells together, lubricates joints, and helps maintain the shape of the eyeballs. [18] :The viscoelasticity of hyaluronic acid make it ideal for lubricating joints and surfaces that move along each other, such as cartilage. A solution of hyaluronic acid under low shear stress has a much higher viscosity than while under high shear stress. [19] Hyaluronidase, an enzyme produced by white blood cells, sperms cells, and some bacteria, breaks apart the hyaluronic acid, causing the solution to become more liquid. [18] In vivo, hyaluronic acid forms randomly kinked coils that entangle to form a hyaluronan network, slowing diffusion and forming a diffusion barrier that regulates transport of substances between cells. For example, hyaluronan helps partition plasma proteins between vascular and extravascular spaces, which affects solubility of macromolecules in the interstitium, changes chemical equilibria, and stabilizes the structure of collagen fibers. [19] Other functions include matrix interactions with hyaluronan binding proteins such as hyaluronectin, glial hyaluronan binding protein, brain enriched hyaluronan binding protein, collagen VI, TSG-6, and inter-alpha-trypsin inhibitor. Cell surface interactions involving hyaluronan are its well-known coupling with CD44, which may be related to tumor progression, and also with RHAMM (Hyaluronan-mediated motility receptor), which has been implicated in developmental processes, tumor metastasis, and pathological reparative processes. Fibroblasts, mesothelial cells, and certain types of stem cells surround themselves in a pericellular "coat", part of which is constructed from hyaluronan, in order to shield themselves from bacteria, red blood cells, or other matrix molecules. For example, with regards to stem cells, hyaluronan, along with chondroitin sulfate, helps to form the stem cell niche. Stem cells are protected from the effects of growth factors by a shield of hyaluronan and minimally sulfated chondroitin sulfate. During progenitor division, the daughter cell moves outside of this pericellular shield where it can then be influenced by growth factors to differentiate even further.
## The Nature of Managerial Work
Managers are responsible for the processes of getting activities completed efficiently with and through other people and setting and achieving the firm’s goals through the execution of four basic management functions: planning, organizing, leading, and controlling. Both sets of processes utilize human, financial, and material resources.
Of course, some managers are better than others at accomplishing this! There have been a number of studies on what managers actually do, the most famous of those conducted by Professor Henry Mintzberg in the early 1970s (Mintzberg, 1973). One explanation for Mintzberg’s enduring influence is perhaps that the nature of managerial work has changed very little since that time, aside from the shift to an empowered relationship between top managers and other managers and employees, and obvious changes in technology, and the exponential increase in information overload.
After following managers around for several weeks, Mintzberg concluded that, to meet the many demands of performing their functions, managers assume multiple roles. A role is an organized set of behaviors, and Mintzberg identified 10 roles common to the work of all managers. As summarized in the following figure, the 10 roles are divided into three groups: interpersonal, informational, and decisional. The informational roles link all managerial work together. The interpersonal roles ensure that information is provided. The decisional roles make significant use of the information. The performance of managerial roles and the requirements of these roles can be played at different times by the same manager and to different degrees, depending on the level and function of management. The 10 roles are described individually, but they form an integrated whole.
The three interpersonal roles are primarily concerned with interpersonal relationships. In the figurehead role, the manager represents the organization in all matters of formality. The top-level manager represents the company legally and socially to those outside of the organization. The supervisor represents the work group to higher management and higher management to the work group. In the liaison role, the manager interacts with peers and people outside the organization. The top-level manager uses the liaison role to gain favors and information, while the supervisor uses it to maintain the routine flow of work. The leader role defines the relationships between the manager and employees.
The direct relationships with people in the interpersonal roles place the manager in a unique position to get information. Thus, the three informational roles are primarily concerned with the information aspects of managerial work. In the monitor role, the manager receives and collects information. In the role of disseminator, the manager transmits special information into the organization. The top-level manager receives and transmits more information from people outside the organization than the supervisor. In the role of spokesperson, the manager disseminates the organization’s information into its environment. Thus, the top-level manager is seen as an industry expert, while the supervisor is seen as a unit or departmental expert.
The unique access to information places the manager at the center of organizational decision making. There are four decisional roles managers play. In the entrepreneur role, the manager initiates change. In the disturbance handler role, the manager deals with threats to the organization. In the resource allocator role, the manager chooses where the organization will expend its efforts. In the negotiator role, the manager negotiates on behalf of the organization. The top-level manager makes the decisions about the organization as a whole, while the supervisor makes decisions about his or her particular work unit.
The supervisor performs these managerial roles but with different emphasis than higher managers. Supervisory management is more focused and short-term in outlook. Thus, the figurehead role becomes less significant and the disturbance handler and negotiator roles increase in importance for the supervisor. Since leadership permeates all activities, the leader role is among the most important of all roles at all levels of management.
So what do Mintzberg’s conclusions about the nature of managerial work mean for you? On the one hand, managerial work is the lifeblood of most organizations because it serves to choreograph and motivate individuals to do amazing things. Managerial work is exciting, and it is hard to imagine that there will ever be a shortage of demand for capable, energetic managers. On the other hand, managerial work is necessarily fast-paced and fragmented, where managers at all levels express the opinion that they must process much more information and make more decisions than they could have ever possibly imagined. So, just as the most successful organizations seem to have well-formed and well-executed strategies, there is also a strong need for managers to have good strategies about the way they will approach their work. This is exactly what you will learn through principles of management.
### Key Takeaway
Managers are responsible for getting work done through others. We typically describe the key managerial functions as planning, organizing, leading, and controlling. The definitions for each of these have evolved over time, just as the nature of managing in general has evolved over time. This evolution is best seen in the gradual transition from the traditional hierarchical relationship between managers and employees, to a climate characterized better as an upside-down pyramid, where top executives support middle managers and they, in turn, support the employees who innovate and fulfill the needs of customers and clients. Through all four managerial functions, the work of managers ranges across 10 roles, from figurehead to negotiator. While actual managerial work can seem challenging, the skills you gain through principles of management—consisting of the functions of planning, organizing, leading, and controlling—will help you to meet these challenges.
### Exercises
1. Why do organizations need managers?
2. What are some different types of managers and how do they differ?
3. What are Mintzberg’s 10 managerial roles?
4. What three areas does Mintzberg use to organize the 10 roles?
5. What four general managerial functions do principles of management include?
## The Existence of a Limit
As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist.
### Evaluating a Limit That Fails to Exist
Evaluate using a table of values.
#### Solution
(Figure) lists values for the function for the given values of .
Table of Functional Values for
−0.1 0.544021110889 0.1 −0.544021110889
−0.01 0.50636564111 0.01 −0.50636564111
−0.001 −0.8268795405312 0.001 0.826879540532
−0.0001 0.305614388888 0.0001 −0.305614388888
−0.00001 −0.035748797987 0.00001 0.035748797987
−0.000001 0.349993504187 0.000001 −0.349993504187
After examining the table of functional values, we can see that the -values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let’s take a more systematic approach. Take the following sequence of -values approaching 0:
The corresponding -values are
At this point we can indeed conclude that does not exist. (Mathematicians frequently abbreviate “does not exist” as DNE. Thus, we would write DNE.) The graph of is shown in (Figure) and it gives a clearer picture of the behavior of as approaches 0. You can see that oscillates ever more wildly between −1 and 1 as approaches 0.
Figure 6. The graph of oscillates rapidly between −1 and 1 as x approaches 0.
Use a table of functional values to evaluate , if possible.
#### Solution
does not exist.
Use -values 1.9, 1.99, 1.999, 1.9999, 1.9999 and 2.1, 2.01, 2.001, 2.0001, 2.00001 in your table.
As the name suggests, here the functions are defined outside the class however they are declared inside the class.
Functions should be declared inside the class to bound it to the class and indicate it as it’s member but they can be defined outside of the class.
To define a function outside of a class, scope resolution operator :: is used.
#### Syntax for declaring function outside of class
Here is this program, the functions showdata() and getdata() are declared inside the class and defined outside the class. This is achieved by using scope resolution operator :: .
## Basic Excel Formulas
Mastering the basic Excel formulas is critical for beginners to become highly proficient in financial analysis Financial Analyst Job Description The financial analyst job description below gives a typical example of all the skills, education, and experience required to be hired for an analyst job at a bank, institution, or corporation. Perform financial forecasting, reporting, and operational metrics tracking, analyze financial data, create financial models . Microsoft Excel Excel Resources Learn Excel online with 100's of free Excel tutorials, resources, guides & cheat sheets! CFI's resources are the best way to learn Excel on your own terms. is considered the industry standard piece of software in data analysis. Microsoft&rsquos spreadsheet program also happens to be one of the most preferred software by investment bankers Investment Banking Job Description This Investment Banking Job description outlines the main skills, education, and work experience required to become an IB analyst or associate and financial analysts in data processing, financial modeling What is Financial Modeling Financial modeling is performed in Excel to forecast a company's financial performance. Overview of what is financial modeling, how & why to build a model. , and presentation. This guide will provide an overview and list of basic Excel functions.
Once you&rsquove mastered this list, move on to CFI&rsquos advanced Excel formulas guide Advanced Excel Formulas Must Know These advanced Excel formulas are critical to know and will take your financial analysis skills to the next level. Download our free Excel ebook! !
### Basic Terms in Excel
There are two basic ways to perform calculations in Excel: Formulas and Functions Formula vs Function A Formula is an equation designed by a user in Excel, while a Function is a predefined calculation in the spreadsheet application. This guide will walk you through Formula vs Function in Excel so you know exactly what the similarities and differences are. Excel allows users to perform simple calculations such .
#### 1. Formulas
In Excel, a formula is an expression that operates on values in a range of cells or a cell. For example, =A1+A2+A3, which finds the sum of the range of values from cell A1 to cell A3.
#### 2. Functions
Functions are predefined formulas in Excel. They eliminate laborious manual entry of formulas while giving them human-friendly names. For example: =SUM(A1:A3). The function sums all the values from A1 to A3.
### Five Time-saving Ways to Insert Data into Excel
When analyzing data, there are five common ways of inserting basic Excel formulas. Each strategy comes with its own advantages. Therefore, before diving further into the main formulas, we&rsquoll clarify those methods, so you can create your preferred workflow earlier on.
#### 1. Simple insertion: Typing a formula inside the cell
Typing a formula in a cell or the formula bar is the most straightforward method of inserting basic Excel formulas. The process usually starts by typing an equal sign, followed by the name of an Excel function.
Excel is quite intelligent in that when you start typing the name of the function, a pop-up function hint will show. It&rsquos from this list you&rsquoll select your preference. However, don&rsquot press the Enter key. Instead, press the Tab key so that you can continue to insert other options. Otherwise, you may find yourself with an invalid name error, often as &lsquo#NAME?&rsquo. To fix it, just re-select the cell, and go to the formula bar to complete your function.
#### 2. Using Insert Function Option from Formulas Tab
If you want full control of your functions insertion, using the Excel Insert Function dialogue box is all you ever need. To achieve this, go to the Formulas tab and select the first menu labeled Insert Function. The dialogue box will contain all the functions you need to complete your financial analysis Types of Financial Analysis Financial analysis involves using financial data to assess a company&rsquos performance and make recommendations about how it can improve going forward. Financial Analysts primarily carry out their work in Excel, using a spreadsheet to analyze historical data and make projections Types of Financial Analysis .
#### 3. Selecting a Formula from One of the Groups in Formula Tab
This option is for those who want to delve into their favorite functions quickly. To find this menu, navigate to the Formulas tab and select your preferred group. Click to show a sub-menu filled with a list of functions. From there, you can select your preference. However, if you find your preferred group is not on the tab, click on the More Functions option &ndash it&rsquos probably just hidden there.
#### 4. Using AutoSum Option
For quick and everyday tasks, the AutoSum function Autosum The autosum Excel formula is a shortcut that can save time in financial modeling in Excel. Type "ALT lazy" src="" srcset="" sizes="" data-src="https://cdn.corporatefinanceinstitute.com/assets/Basic-Excel-Function-AutoSum-1-1024x481.png" data-srcset="//cdn.corporatefinanceinstitute.com/assets/Basic-Excel-Function-AutoSum-1-1024x481.png 1024w, //cdn.corporatefinanceinstitute.com/assets/Basic-Excel-Function-AutoSum-1-300x141.png 300w, //cdn.corporatefinanceinstitute.com/assets/Basic-Excel-Function-AutoSum-1-768x361.png 768w, //cdn.corporatefinanceinstitute.com/assets/Basic-Excel-Function-AutoSum-1-1200x564.png 1200w, //cdn.corporatefinanceinstitute.com/assets/Basic-Excel-Function-AutoSum-1-600x282.png 600w, //cdn.corporatefinanceinstitute.com/assets/Basic-Excel-Function-AutoSum-1.png 1952w" data-sizes="(max-width: 860px) 100vw, 860px">
#### 5. Quick Insert: Use Recently Used Tabs
If you find re-typing your most recent formula a monotonous task, then use the Recently Used menu. It&rsquos on the Formulas tab, a third menu option just next to AutoSum.
### Free Excel Formulas YouTube Tutorial
Watch CFI&rsquos FREE YouTube video tutorial to quickly learn the most important Excel formulas. By watching the video demonstration you&rsquoll quickly learn the most important formulas and functions.
### Seven Basic Excel Formulas For Your Workflow
Since you&rsquore now able to insert your preferred formulas and function correctly, let&rsquos check some fundamental Excel functions to get you started.
#### 1. SUM
The SUM function SUM Function The SUM function is categorized under Math and Trigonometry functions. The function will sum up cells that are supplied as multiple arguments. It is the most popular and widely used function in Excel. SUM helps users perform a quick summation of specified cells in MS Excel. For example, we are given the cost of 100 is the first must-know formula in Excel. It usually aggregates values from a selection of columns or rows from your selected range.
=SUM(number1, [number2], &hellip)
=SUM(B2:G2) &ndash A simple selection that sums the values of a row.
=SUM(A2:A8) &ndash A simple selection that sums the values of a column.
=SUM(A2:A7, A9, A12:A15) &ndash A sophisticated collection that sums values from range A2 to A7, skips A8, adds A9, jumps A10 and A11, then finally adds from A12 to A15.
=SUM(A2:A8)/20 &ndash Shows you can also turn your function into a formula.
#### 2. AVERAGE
The AVERAGE function AVERAGE Function Calculate Average in Excel. The AVERAGE function is categorized under Statistical functions. It will return the average of the arguments. It is used to calculate the arithmetic mean of a given set of arguments. As a financial analyst, the function is useful in finding out the average of numbers. should remind you of simple averages of data such as the average number of shareholders in a given shareholding pool.
=AVERAGE(number1, [number2], &hellip)
=AVERAGE(B2:B11) &ndash Shows a simple average, also similar to (SUM(B2:B11)/10)
#### 3. COUNT
The COUNT function COUNT Function The COUNT Function is an Excel Statistical function. This function helps count the number of cells that contain a number, as well as the number of arguments that contain numbers. It will also count numbers in any given array. It was introduced in Excel in 2000. As a financial analyst, it is useful in analyzing data counts all cells in a given range that contain only numeric values.
=COUNT(value1, [value2], &hellip)
COUNT(A:A) &ndash Counts all values that are numerical in A column. However, you must adjust the range inside the formula to count rows.
COUNT(A1:C1) &ndash Now it can count rows.
#### 4. COUNTA
Like the COUNT function, COUNTA COUNTA Function The COUNTA Function will calculate the number of cells that are not blank within a given set of values. The =counta() function is also commonly referred to as the Excel Countif Not Blank formula. As a financial analyst, the function is useful count cells that are not blank or empty in a given range. counts all cells in a given rage. However, it counts all cells regardless of type. That is, unlike COUNT that only counts numerics, it also counts dates, times, strings, logical values, errors, empty string, or text.
=COUNTA(value1, [value2], &hellip)
COUNTA(C2:C13) &ndash Counts rows 2 to 13 in column C regardless of type. However, like COUNT, you can&rsquot use the same formula to count rows. You must make an adjustment to the selection inside the brackets &ndash for example, COUNTA(C2:H2) will count columns C to H
The IF function IF Function The Excel IF Statement function tests a given condition and returns one value for a TRUE result, and another for a FALSE result. For example, if sales total more than \$5,000, then return a "Yes" for Bonus, else, return a "No". We can also create nested IF statements is often used when you want to sort your data according to a given logic. The best part of the IF formula is that you can embed formulas and function in it.
=IF(logical_test, [value_if_true], [value_if_false])
=IF(C2<D3, &lsquoTRUE,&rsquo &lsquoFALSE&rsquo) &ndash Checks if the value at C3 is less than the value at D3. If the logic is true, let the cell value be TRUE, else, FALSE
=IF(SUM(C1:C10) > SUM(D1:D10), SUM(C1:C10), SUM(D1:D10)) &ndash An example of a complex IF logic. First, it sums C1 to C10 and D1 to D10, then it compares the sum. If the sum of C1 to C10 is greater than the sum of D1 to D10, then it makes the value of a cell equal to the sum of C1 to C10. Otherwise, it makes it the SUM of C1 to C10.
#### 6. TRIM
The TRIM function TRIM Function The TRIM function is categorized under Excel Text functions. TRIM helps remove the extra spaces in data and thus clean up the cells in the worksheet. In financial analysis, the TRIM function can be useful in removing irregular makes sure your functions do not return errors due to unruly spaces. It ensures that all empty spaces are eliminated. Unlike other functions that can operate on a range of cells, TRIM only operates on a single cell. Therefore, it comes with the downside of adding duplicated data in your spreadsheet.
TRIM(A2) &ndash Removes empty spaces in the value in cell A2.
#### 7. MAX & MIN
The MAX MAX Function The MAX Function is categorized under Excel Statistical functions. MAX will return the largest value in a given list of arguments. From a given set of numeric values, it will return the highest value. Unlike MAXA function, the MAX function will count numbers but ignore empty cells and MIN MIN Function The MIN function is categorized under Excel Statistical functions. MIN will return the minimum value in a given list of arguments. From a given set of numeric values, it will return the smallest value. Unlike the MINA function functions help in finding the maximum number and the minimum number in a range of values.
=MIN(number1, [number2], &hellip)
=MIN(B2:C11) &ndash Finds the minimum number between column B from B2 and column C from C2 to row 11 in both columns B and C.
=MAX(number1, [number2], &hellip)
=MAX(B2:C11) &ndash Similarly, it finds the maximum number between column B from B2 and column C from C2 to row 11 in both columns B and C.
### More Resources
Thank you for reading CFI&rsquos guide to basic Excel formulas. To continue your development as a world-class financial analyst Become a Certified Financial Modeling & Valuation Analyst (FMVA)® CFI's Financial Modeling and Valuation Analyst (FMVA)® certification will help you gain the confidence you need in your finance career. Enroll today! , these additional CFI resources will be helpful:<|endoftext|>
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Developing Questions That Promote Discussion
Co-teachers Addie Male and Raeann McElveen discuss how to get students to develop questions that generate good classroom discussion.
Teacher: Addie Male and Raeann McElveen
School: Millennium Brooklyn High School, Brooklyn, NY
Discipline: History (Humanities 11)
Lesson Topic: Voices from the field: Perspectives of the soldiers
Lesson Month: March
Number of Students: 25
Other: Collaborative team teaching. Humanities is a two-year course combining English and social studies. The course culminates in a Global History NYS Regents exam.
Featured Lesson’s Student Goals:
- Content objectives – Utilize student-generated questions to explore and discuss primary and secondary resource material via a Socratic seminar; utilize student-facilitated and -generated discussion to further delve into essential questions surrounding causes of WWI and realities for soldiers during WWI
- Literacy/language objectives – Cite textual evidence to support ideas and to engage in student-centered discussion; ask questions of a text once annotated; develop descriptive language with historical accuracy
- Engagement/interaction objectives – Initiate and participate effectively in a range of collaborative discussions with diverse partners; propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; clarify, verify, or challenge ideas and conclusions; respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify views and understanding and make new connections in light of the evidence and reasoning presented
Common Core Standards for English Language Arts
Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 9–10 topics, texts, and issues, building on others' ideas and expressing their own clearly and persuasively.
Evaluate a speaker's point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence.
Present information, findings, and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning and the organization, development, substance, and style are appropriate to purpose, audience, and task.
The focus of this 10-day unit was nationalism, World War I, and transformation. It covered the Russian Revolution, communism, the collapse of Imperial China, and nationalism in India/Southwest Asia and the Ottoman Empire. The essential questions for the unit were: What were the causes and impacts of a global depression? and What are the circumstances of violent conflicts and how have they affected countries/specific groups? This unit was sandwiched between two much longer units, both including literary texts—the Industrial Revolution (students read The Jungle) and the Rise and Fall of Fascism/World War II Book Club (students read Maus, Maus II, Night, Slaughterhouse V, The Diary of Anne Frank). This shorter unit was couched in historical points of view and first-person narrative. It was taught in March, toward the end of this two-year-long curriculum during which students explored historical concepts from the dawn of man to the present and literature written during or about the time periods. The lesson in the video occurred on the third day of the unit.
Before the Video
Students began the unit by learning about the “isms” that caused the Great War (nationalism, imperialism, and militarism). They explored different kinds of military strategies, did some mapping, looked at photographs, and viewed a clip of trench warfare from the film Paths of Glory. The object was for students to understand the context of World War1 through the lens of a soldier. Their homework was to read and annotate two primary documents (one of which was “A Suffolk Farmhand at Gallipoli”) and submit via email an open-ended question for the following day’s lesson.
During the Video
This lesson began with a “Do Now,” in which students took out their homework templates with guiding questions and set and shared personal goals for themselves for a Socratic seminar. Ms. Male and Ms. McElveen reviewed the guidelines and expectations and introduced the student-generated open-ended questions that would guide a Socratic seminar. Led by two student facilitators, the class discussed the assigned readings and the historical content explored thus far in the unit. They shared ideas, responded to questions aloud and in writing, cited evidence to support their arguments, took notes, and actively listened as others contributed. The Socratic seminar ended with students reflecting on their own participation and their peers' feedback.
After the Video
During the rest of the unit, students learned about the Treaty of Versailles—the events that led to it and its effects on European powers. Following was a mini-unit that focused on speeches and rhetoric.
Ms. Male and Ms. McElveen identified the source documents and media for student review.
The foundation for this lesson was established in the first year of the two-year-long Humanities course when students were first introduced to the meaning, structure, and methodology behind the Socratic seminar. In order to participate, students needed to know how to cite evidence; craft appropriate questions (that invite multiple answers); understand the difference between analytical, interpretative, and factual questions; and have experience charting. On the day of the lesson, students needed to have brought with them a personal goal, an open-ended question, and a piece of evidence.
At the start of the unit, Ms. Male and Ms. McElveen used photographs and mapping to engage the kinesthetic and visual learners. The different roles assigned to students in the pre-seminar lit circles catered to differing needs of students (such as the travel tracer, who tracks what happens in reading by drawing pictures—ideal for a kinesthetic learner.) The teachers used the gradual release of responsibility model to scaffold for students who needed help with annotating by 1) modeling the process; 2) doing it together with students; 3) having students lead the process, and offering support where needed; and 4) having students work independently. They used a similar method for teaching students how to lead the Socratic seminar (beginning the process the first year).
For this lesson, Ms. Male and Ms. McElveen provided guiding questions for the reading homework assignment and had students set personal goals to support development of executive functioning skills. By writing down their goals, students were held accountable and could refer back to their goals during seminar. The teachers used the “It says, I say…” graphic organizer to help students ask and respond to interpretative questions. Throughout the year, the teachers incorporated many supports to foster student accountability: student-written and signed contracts, color-coded deadline calendars, and student-accessible hanging folders containing work from previous lessons (when students missed class, it was up to them to get the work they missed).
Students were seated in a circle for the Socratic seminar and engaged in whole-group discussion. Two students were assigned the role of moderator. Ms. Male and Ms. McElveen chose their seats strategically—placing themselves between students they knew would need additional support. To help students “warm up” and build confidence for participation in a Socratic seminar, Ms. Male and Ms. McElveen grouped students into small “lit circles” of four to eight students each. Each student had a very prescribed role (e.g., discussion director, literary luminary, vocabulary enricher, the summarizer, travel tracer, connector). Throughout the year, they modified the seating arrangements based on their objectives. Sometimes they incorporated a “fishbowl seminar,” where students sat in two circles, one inside and one outside. The teachers could manipulate the circles in different ways. (For example, the inside circle held quieter student, while the outside circle held those students who tended to dominate conversations. This encouraged the students in the inner circle to feel empowered to speak out more. Each student in the outer circle was paired with a student on the inner circle to coach and give advice.)
Resources and Tools
- Smart Board
- A Suffolk Farmhand at Gallipoli by Leonard Thompson (excerpted from Akenfield: Portrait of an English Village by Ronald Blythe)
- The German Army Marches Through Brussels by Richard Harding Davis
- Song: “Christmas in the Trenches” by John McCutcheon
- Film: Paths of Glory
- WWI images of soldiers
- Handout: “It Says, I Say…” by Millennium Brooklyn High School Humanities team
The open-ended questions generated by students for the Socratic seminar were submitted to the teachers for review the day before and served as a way to assess student understanding. During the seminar, Ms. Male and Ms. McElveen often got out of their seats to walk around the circle to see if students were meeting their goals, who was seeking evidence and from where, who was capable of independent work, and who needed support.
Students reflected on their own participation in the seminar through three self-assessment questions. They received peer feedback at the end of the seminar through “shout-outs” during which students recognized each other for positive contributions.
Ms. Male and Ms. McElveen gave students an exam at the end of the unit using mock questions from the New York State Regents Exam.<|endoftext|>
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# Difference between revisions of "2005 AMC 10B Problems/Problem 20"
## Problem
What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, 8 and exactly once?
$\mathrm{(A)}48000\qquad\mathrm{(B)}49999.5\qquad\mathrm{(C)}53332.8\qquad\mathrm{(D)}55555\qquad\mathrm{(E)}56432.8$
## Solution 1
We first look at how many times each number will appear in each slot. If we fix a number in a slot, then there are $4! = 24$ ways to arrange the other numbers, so each number appears in each spot $24$ times. Therefore, the sum of all such numbers is $24 \times (1+3+5+7+8) \times (11111) = 24 \times 24 \times 11111 = 6399936.$ Since there are $5! = 120$ such numbers, we divide $6399936 \div 120$ to get $\boxed{\mathrm{(C)}53332.8}$
## Solution 2
We can first solve for the mean for the digits 1, 3, 5, 7, and 9 since each is 2 away from each other. The mean of the numbers than can be solved using these digits is $55555$. The total amount of numbers that can be formed using these digits is $4! =120$. The sum of these numbers is $55555(120) = 6666600$. Now we can find out the total value that was gained by replacing the 8 with a 9. We can start how be calculating the gain when the 8 was in the ones digit. Since there are $4! = 24$ numbers with the 8 in the ones digit and 1 was gain from each of them, 24 is the number gained. Then, we repeat this with the tens, hundreds, thousands, and ten thousands place, leading to a total of $24+240+2400+24000+240000=266664$ as the total amount that was gained. Subtract this amount from the sum of the digits using the 9 instead of the 8 to get $6666600-266664=6399936$. Finally, we divide this by 120 to get the average. $6399936/120= \boxed{\mathrm{(C)}53332.8}$<|endoftext|>
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Contemporary Mathematics
# Key Concepts
## 3.1Prime and Composite Numbers
• The natural numbers can be categorized as 1, prime numbers, and composite numbers.
• Prime numbers have as their only factors 1 and themselves.
• Composite numbers have at least three distinct factors.
• Composite numbers can be written in their prime factorization form, which is found by repeatedly factoring prime factors from the number.
• The greatest common divisor (GCD) of a set of numbers is the largest integer that divides all of the numbers in the set. The prime factorizations of the numbers can be used to identify the greatest common divisor.
• The least common multiple (LCM) of a set of numbers is the smallest integer that is divisible by all of the numbers in the set. The prime factorizations of the numbers can be used to identify the least common multiple.
• There are various ways that the GCD and LCM are applied.
## 3.2The Integers
• A set of numbers that can be built from the natural numbers are the integers, which consist of the natural numbers, zero (0), and the negatives of the natural numbers.
• Integers are often graphed on a number line, which helps display the relative positions and values of those numbers.
• The number line can be used to visualize when one integer is larger than or smaller than another integer.
• Arithmetic operations with integers are similar to the operations with natural numbers, except that the sign (positive or negative) of the numbers will determine the sign (positive or negative) of the result.
## 3.3Order of Operations
• Establishing shared rules on which arithmetic operations are calculated first is necessary. Without them, different people may find different values for the same expression.
• The highest precedence is with expressions in parentheses. This allows parts of an expression to be calculated in an order different than the basic order of operations.
• The lowest precedence is addition and subtraction, as they are the basis for all other calculations.
• Multiplication and division have precedence over addition and subtraction, as they are representations of repeated addition or subtraction.
• Exponents have precedence over multiplication and division, as they represent repeated multiplication and division.
## 3.4Rational Numbers
• Rational numbers are fractions of integers, and can always be written as an integer divided by an integer.
• The numerator and denominator of a fraction may have common factors. In such cases, the fraction can be reduced by canceling common factors. When the numerator and denominator of a fraction have no common factors, the fraction is said to be reduced.
• An improper fraction is one with a numerator larger than the denominator. Such a fraction can be rewritten as an integer plus a proper fraction. This is called a mixed number.
• Using division and remainder, an improper fraction may be written as a mixed number.
• A mixed number can be converted to an improper fraction by reversing the process for changing an improper fraction to a mixed number.
• The arithmetic operations or addition, subtraction, multiplication and division can all be performed on rational numbers.
• Addition and subtraction of rational numbers can be performed after a common denominator has been identified, and the fractions have been converted to forms having the common denominator.
• Multiplication and division of rational numbers can be performed without regard to common denominators.
• Between any two rational numbers, there is always another rational number. This is the density property of the rational numbers.
## 3.5Irrational Numbers
• Irrational numbers are numbers that cannot be written as an integer divided by another integer. One example is pi, denoted $ππ$. Another collection of irrational numbers are natural numbers that are not perfect squares.
• Some irrational numbers can be written as a rational part multiplied by an irrational part. If two irrational numbers have the same irrational parts, they can be added or subtracted.
• When irrational numbers are similar, on can multiply and divide the numbers without a calculator.
• Since $a×b=a×ba×b=a×b$ , and $a÷b=ab=aba÷b=ab=ab$, products and quotients of square roots can be determined.
• Because $a2=aa2=a$ and $a×b=a×ba×b=a×b$, it is possible to simplify square root expressions so the radicand contains no perfect square factors.
• When a fraction has an irrational number as its denominator, it is possible to convert the denominator into a rational number using its conjugate. Doing so involves multiplying the numerator and denominator by the conjugate of the denominator, and then applying the difference of squares formula.
• With a single square root term
• Using conjugate numbers for two term denominators
## 3.6Real Numbers
• Real numbers is the collection of all rational and irrational numbers. Conceptually, it is the collection of all values that can be represented on a number line, or, as a length along with sign.
• The subsets of the real numbers include the natural numbers, integers, rational numbers and irrational numbers. The natural numbers are a subset of the integers, which is a subset of the rational numbers. The rational and irrational numbers are disjoint sets.
• The real numbers, due to order of operation rules and that performing arithmetic operations on real number always results in a real number, have arithmetic properties that apply in all cases. There include the distributive property, the commutative property, and the associative property. Also, every real number has an additive inverse and, except for zero (0), have a multiplicative inverse.
## 3.7Clock Arithmetic
• Clock arithmetic uses the idea that after 12 o’clock comes 1 o’clock. For clock arithmetic, this means that every time 12 is passed in an arithmetic process, the next number is 1, not 13.
• To determine the clock result of an arithmetic operation, divide the final result by 12 and keep the remainder. If the remainder is 0, then the time is 12 o’clock.
• Clock arithmetic is technically called modulo 12 arithmetic. To perform modulo 12 arithmetic, calculate the expression, then divide the result by 12. The modulo 12 result is the remainder.
• Days, in our system, pass in groups of seven. To calculate in day arithmetic, modulo 7 is used. To perform modulo 7 arithmetic, calculate the expression, then divide the result by 7. The modulo 7 result is the remainder.
## 3.8Exponents
• Exponents are used to express multiplying a number by itself a number of times. The number being multiplied by itself is the base. The number of times it is multiplied by itself is the exponent, which is often referred to as the power.
• Understanding that exponents represent repeated multiplication of a base makes it possible to establish some rules for combining exponential expressions, using the product rule, the quotient rule, and the power rule. Additionally, it allows us to formulate distributive rules for exponents.
• Any non-zero number raised to the 0th power is 1. This makes the definition of the 0th power consistent with the division rule for exponents.
• For consistency, negative exponents represent the reciprocal of the base raised to the power, so that $a−n=1ana−n=1an$, provided that $a≠0a≠0$.
## 3.9Scientific Notation
• Some numbers are so large or so small that writing the number out is clumsy and make it difficult to determine the true size of the number. Scientific notation makes the number more readable and make the relative size of the number immediately apparent.
• A number written in scientific notation is a number at least 1 and smaller than 10 multiplied by 10 raised to an exponent. Converting between scientific notation and standard notation involves correctly applying multiplication and division by powers of 10, which in practice equates to understanding how moving the decimal point of a number impacts the exponent of 10.
• Adding and subtracting numbers in base 10 requires the exponent of 10 in each number be the same. Once the numbers are converted to have the same exponent with the ten, then the numbers are added or subtracted as indicated, with the power of 10 remaining the same. If the result is not in scientific notation (for instance, the number has exceeded 10), then then number must be converted into scientific notation.
• Multiplying and dividing numbers in scientific notation is done by multiplying or dividing the number parts, then multiplying or dividing the 10 raised to the power parts, then multiplying those two results. If the new number is not in scientific notation, then the result must be converted into scientific notation.
## 3.10Arithmetic Sequences
• A sequence is a list of numbers. Any individual number in that list, or sequence, is a term of the sequence. A specific term of a sequence is denoted by the sequence symbol with a subscript indicating where the term in the sequence is.
• A special form of a sequence is an arithmetic sequence. Each arithmetic sequence is determined by its first term and its constant difference. Any term in an arithmetic sequence is determined by adding the constant difference to the preceding term.
• If the first term and the constant difference of an arithmetic sequence are known, then any term of the sequence can be found directly.
• Because arithmetic sequences follow such a strict pattern, the sum of the first $nn$ terms of an arithmetic sequence can be determined with the formula $sn=n(a1+an2)sn=n(a1+an2)$.
## 3.11Geometric Sequences
• A special form of a sequence is a geometric sequence. Each geometric sequence is determined by its first term and its constant ratio. Any term in a geometric sequence is determined by multiplying the constant ratio to the preceding term.
• If the first term and the constant ratio of a geometric sequence are known, then any term of the sequence can be found directly.
• Because geometric sequences follow such a strict pattern, the sum of the first $nn$ terms of a geometric sequence can be determined with the formula $sn=a1(1−rn−11−r)sn=a1(1−rn−11−r)$.
• Finding the sum of a finite geometric sequence
• Applying arithmetic sequences
## 3.11Geometric Sequences
• Geometric sequence.
• Finding an arbitrary term in a geometric sequence.
• Constant ratio.
• Finding the sum of a finite geometric sequence.
• Applying arithmetic sequences.<|endoftext|>
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In the late 19th and early 20th centuries, industrial workers around the world reached a breaking point. Frustrated with dangerous labor conditions, low pay and zero job security, they came together to form the first labor unions. Whereas the complaints of one, two or 10 workers could be easily ignored by management, the united voice of thousands of workers demanded attention.
Labor unions are founded on the principle of collective bargaining. Collective bargaining consists of negotiations between an employer and a group of employees, usually represented by a union. When it works, collective bargaining is good for both the worker and the employer. The worker gets a better salary, reasonable schedules and health benefits, and the employer is relieved of the burden of negotiating individually with each and every worker. But when collective bargaining doesn't work, you get a strike.
Scab labor has the potential to completely undermine the power of a labor union. The key to a successful strike is that every union member participates. If half of the employees stay behind -- or are replaced by nonunion workers -- management can still keep the factory running and wait until the striking workers are desperate enough to accept its conditions. When scabs fill jobs, the striking worker has two options -- either return to work under the same old conditions or look for a new job.
Scabs, also known as replacement workers, are legal in most parts of the world. In the U.S., the National Labor Relations Act (NLRA) of 1935 establishes strict protections for unions, but allows employers to permanently replace striking workers if the strike is based on economic gain [source: Legal Dictionary]. In a recent case, a 7th Circuit Court ruled that an employer can even refuse to hire back striking workers if their jobs are occupied by permanent replacements, further undercutting the power of strike actions [source: U.S. Court of Appeals]. In the Canadian provinces of British Columbia and Quebec, however, it's illegal under any circumstances to hire scabs [source: Canadian Labour Congress].
Over the past two centuries, scabs have played pivotal roles in some of the world's most famous and most violent labor strikes. Read more about them in the next section.<|endoftext|>
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# Inverse proportion in context
In this lesson, we will interpret, solve and model solutions to inverse proportion problems.
#### Unit quizzes are being retired in August 2023
Why we're removing unit quizzes from the website >
Quiz:
# Intro quiz - Recap from previous lesson
Before we start this lesson, let’s see what you can remember from this topic. Here’s a quick quiz!
Q1.Fill in the gaps: Two quantities a and b are said to be directly proportional if one quantity is always a constant ........ of the other.
1/5
Q2.Fill in the gaps: Two quantities x and y are said to be inversely proportional if x increases as y ............... at the same rate, so their product is constant.
2/5
Q3.Fill in the gaps: Two variable quantities are said to be ........................ proportional if the product of the two quantities is constant.
3/5
Q4.Three builders take 15 days to complete a project. How many days does would it take one builder?
4/5
Q5.It takes 4 friends 6 hours to clean the garden. How long will it take 2?
5/5
#### Unit quizzes are being retired in August 2023
Why we're removing unit quizzes from the website >
Quiz:
# Intro quiz - Recap from previous lesson
Before we start this lesson, let’s see what you can remember from this topic. Here’s a quick quiz!
Q1.Fill in the gaps: Two quantities a and b are said to be directly proportional if one quantity is always a constant ........ of the other.
1/5
Q2.Fill in the gaps: Two quantities x and y are said to be inversely proportional if x increases as y ............... at the same rate, so their product is constant.
2/5
Q3.Fill in the gaps: Two variable quantities are said to be ........................ proportional if the product of the two quantities is constant.
3/5
Q4.Three builders take 15 days to complete a project. How many days does would it take one builder?
4/5
Q5.It takes 4 friends 6 hours to clean the garden. How long will it take 2?
5/5
# Video
Click on the play button to start the video. If your teacher asks you to pause the video and look at the worksheet you should:
• Click "Close Video"
• Click "Next" to view the activity
Your video will re-appear on the next page, and will stay paused in the right place.
# Worksheet
These slides will take you through some tasks for the lesson. If you need to re-play the video, click the ‘Resume Video’ icon. If you are asked to add answers to the slides, first download or print out the worksheet. Once you have finished all the tasks, click ‘Next’ below.
#### Unit quizzes are being retired in August 2023
Why we're removing unit quizzes from the website >
Quiz:
# Inverse proportion in context
Exit Quiz
Q1.It takes 4 builders 6 days to build a wall. How long will it take 2 builders?
1/5
Q2.It takes builders 6 days to build a wall. How long will it take 8 builders?
2/5
Q3.It takes 2 people 3 hours to clean the house. How long does it take 6 people to clean?
3/5
Q4.A journey takes 6 hours if driving at 30 mph. How long should the same journey if the driver drives ta 60mph?
4/5
Q5.True or False: The circumference of a circle and its diameter are inversely proportional?
5/5
#### Unit quizzes are being retired in August 2023
Why we're removing unit quizzes from the website >
Quiz:
# Inverse proportion in context
Exit Quiz
Q1.It takes 4 builders 6 days to build a wall. How long will it take 2 builders?
1/5
Q2.It takes builders 6 days to build a wall. How long will it take 8 builders?
2/5
Q3.It takes 2 people 3 hours to clean the house. How long does it take 6 people to clean?
3/5
Q4.A journey takes 6 hours if driving at 30 mph. How long should the same journey if the driver drives ta 60mph?
4/5
Q5.True or False: The circumference of a circle and its diameter are inversely proportional?
5/5
# Lesson summary: Inverse proportion in context
## Time to move!
Did you know that exercise helps your concentration and ability to learn?
For 5 mins...
Move around:
Jog
On the spot:
Dance<|endoftext|>
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The policy of obtaining informed consent from a patient is based on the principle of ethics. The rationale is that all medical patients should both understand, and agree to, the type of medical care that they will be receiving. It simply is not ethical for a patient to agree to treatment that could greatly impact their health without being provided all known and pertinent information from which to base their decision.
Informed consent involves an agreement, between a patient and treating physician, whereby the patient allows their treating physician to undertake specific treatment.
There are two important parts to informed consent:
- A patient has the right to decide what happens to his or her body. A doctor may have an opinion about what the patient should choose to do but the decision belongs to the patient and the patient alone. A doctor is not permitted to pressure any patient into a decision as the decision must be voluntary and free from duress.
- A medical professional has a duty to provide the patient with enough information to make a decision that is educated in regards to whether or not they want to follow the medical professional's recommendation for treatment.
To make the best decision possible for their health, patients must trust their medical professionals to provide them with the most up-to-date and relevant information possible. To aid the process, it may comfort patients to know that there are certain items that medical professionals are required to discuss with their patients prior to commencing a treatment plan or procedure.
The History of Informed Consent
Informed consent laws first began at the beginning of the 20th century when the Supreme Court heard a case where a doctor admitted to deceiving his patient in order to convince her to have an operation. The Supreme Court decided that American citizens have a right to knowledge and consent prior to undergoing a medical procedure.
Schloendorff v. The Society of the New York Hospital (105 N.E. 92) 1914, involved a doctor who removed a tumor when the patient had only consented to a diagnostic procedure. The court determined that every human being of adult years and sound mind has a right to determine what shall be done with their own body and that a surgeon who performs an operation without their patient's consent is guilty of committing an assault on the patient.
In the 1950s, medical patients acquired the right to be told of the possible positive and negative effects of the treatment their medical professional was recommending.
In 1972, the 1950s law took a larger step forward as medical professionals were required to specifically disclose the risks pertaining to a treatment or procedure in a way the patient could understand. In other words, the medical professionals were required to speak in “layman's terms.”
Today, if informed consent is not obtained from a patient by a medical professional, the medical professional is at risk of not only civil litigation, but criminal as well. Performing treatment on an uninformed, unwilling person is no different from attacking a person and committing battery. The legal term for a harmful or offensive touching without permission is battery. Battery is a criminal offense, and it can also be the basis of a civil lawsuit. The key element of battery is that the touching is unauthorized, not that it be intended to harm the person. Therefore, forcing a procedure on an uninformed, unwilling patient could constitute the criminal offense of battery.
What Must be Disclosed?
The case of Canterbury v. Spence, 409 U.S. 1064 93 S. Ct. 560 34 L. Ed. 2d 518 1972 U.S., found that when it comes to informed consent, a physician must disclose:
(1) The condition being treated;
(2) The nature and character of the proposed treatment or surgical procedure;
(3) The anticipated results from the proposed treatment or surgical procedure;
(4) The recognized possible alternative forms of treatment; and
(5) The recognized serious possible risks, complications, and anticipated benefits involved in the treatment or surgical procedure, as well as the recognized possible alternative forms of treatment, including non-treatment.
Who is Obligated to Disclose?
The obligation to disclose falls on the patient's treating physician. Other medical staff that may have come into contact with the patient – nurses, assistants, referring physicians – do not have a duty to disclose because they are not the ones that will actually either be performing the procedure or administering the treatment.
Since informed consent is considered to be the process of communication between the treating physician and the patient, the conversation needs to be held between the two as well as anyone that would be considered to be the treating physician at any time. The court in Perna v Pirozz, 92 N.J. 446,457 A.2d 431, New Jersey, 1983, found that if the patient agrees to a surgery with one physician, another physician is not able to substitute into the exact same procedure without the patient's consent. Therefore, if the originally approved physician does not perform the surgery but the surgery takes place regardless, the original physician is subject to malpractice if consent was not previously obtained from the physician.
The court in Canterbury established what a medical professional must discuss with a patient. However, a physician may sometimes find it difficult to ascertain how much information is needed to be disclosed before a patient can be able to give their informed consent. To help with this, courts have adopted different standards by which a medical professional may be required to follow:
- Reasonable Physician Standard
- Reasonable Patient Standard
- Subjective Standard
The Reasonable Physician Standard is a standard of disclosure of information used in the wording of informed consent documents, based on customary practice or what a reasonable practitioner in the medical community would disclose under the same/similar circumstances.
The Reasonable Patient Standard is a standard by which the medical professional must provide information from the viewpoint of the patient.
The Subjective Standard focuses on the patient's specific ailments and what information that particular patient would need to know in order to make an informed decision regarding treatment.
The physician has the responsibility to make certain that the patient has understood the information that has been disclosed. This may be done both verbally and in writing. When the physician takes the time to discuss the recommended course of action in regards to the patient's treatment, it is usually best to discuss the risks, benefits and alternatives in a conversational format. This usually helps the patient feel relaxed and places them in a state that is more conducive to understanding. After the conversation is had, the treating physician may want to provide the patient with written versions of what was discussed. This allows the patient to review the discussed information as many times as they wish before coming to a decision on whether they would like to proceed with, or refuse, the recommended treatment.
However, it is possible that the patient is not capable of consenting to treatment due to a lack of capacity.
A person must be assumed to have capacity unless it is established that they lack capacity. For informed consent to be properly obtained, the patient must be considered to be medically competent, and obtain the requisite capacity, to be able to give consent in a way that they understand the procedure being performed; appreciate the reason for the proposed procedure; and are aware of the risks of the procedure and the expected outcome.
In some instances, a physician may disclose all the material that is possible to cover, but the patient may lack the requisite capacity to give informed consent to treatment. A lack of capacity may exist for many reasons, including: unconsciousness; mental health conditions; learning disabilities; brain damage and intoxication by way of alcohol or drug use.
The rules to consent are nuanced. Adults deemed to obtain the requisite capacity have the right to refuse a recommended treatment or procedure, even if the refusal is counterproductive to their own health.
If it has been determined that the patient is unable to make an informed decision regarding treatment due to a lack of capacity, the treating physician should make all subsequent medical decisions with the patient's best interest in mind. Best interests include what the patient's wishes would be if they were competent; the patient's general well-being; and the patient's religious beliefs.
If the patient is deemed incapable of being able to properly give consent, the physician should make a strong attempt to gain consent from another person on behalf of the patient before proceeding with treatment if at all possible. If an agent was not previously appointed or a conservator has not been put in place by the patient or the patient's family, the physician should try to bring in a surrogate for the purposes of making health care decisions on behalf of the patient. This person should know about the patient's condition; be actively involved in the patient's life; be willing and able to make the decision in an informed manner; and have no conflict of interest with the patient. Examples of such a person include; the patient's spouse; the patient's parent or guardian; the patient's sibling; or a relative of the patient.
If the patient is in need of medical help and the situation is an emergency, the physician is generally excused from needing to obtain informed consent from the patient or a family member since it is likely that any delay would be detrimental to the health of the patient.
How to Consent
While it is the treating physician that must disclose the requisite information to the patient, Natanson v. Kline, 350 P2d 1093, Kansas, 1960 found that it is the patient, and no one else, that must make the final decision as to whether or not they would like to go forward with the plan that the treating physician has put forth. “A man is the master of his own body and he may expressively prohibit the performance of life-saving surgery or other treatment."
When it comes to a patient giving their informed consent to a proposed treatment or procedure, there are two ways in which the consent may be conveyed – expressly or impliedly.
Express consent is considered to be specific permission that is given by one person to another person.
Another way of giving consent is through an act that demonstrates implied consent. Implied consent is an assumption of permission, not specific permission that is inferred from the actions of a person.
Regardless of whether the consent provided by the patient was given expressly or impliedly, for the consent to be considered valid, it must have been be obtained from a competent patient who understood the procedure to be performed, as well as the associated risks, benefits, complications, and possible alternatives. The consent must have been given under the patient's own volition and not as a result of threat or duress.
Informed Consent Exceptions
While healthcare treatments usually require consent from the patient, there are a few exceptions to the rule.
- Lack of consciousness
- Where an attempt is made to prevent the patient from suffering or continuing to suffer significant pain or a decline in health
- Where time is of the essence in order to save the patient's life
- Where wasting time would cause serious damage to the patient's health
- Treatments authorized by law
- Patients under 18 years of age
The court in Barnett v. Bacharach where the court held that in an emergency situation where the patient lies unconscious, a surgeon may carry out the duties of a doctor in the best interest of the patient even if those duties include performing a procedure that was never discussed with the patient. The court felt that ruling otherwise would make all physicians more concerned about potential litigation rather than saving lives.
Can Consent be Limited or Withdrawn?
It is never too late for a patient to change their mind about a procedure or treatment (unless a surgery was consented to and the procedure has already been completed). A patient can legally change their mind at any time, even if they have already started treatment. Most consent forms state that a patient has the right to stop treatment even after signing the consent form. If the patient has changed their mind, they must contact the treating physician as soon as possible.
If a patient wishes to withdraw their consent, they may do so orally or in writing. The withdrawal does not need to have a reason behind it. The only requirement is that the patient is of sound mind when making the withdrawal request.
After receiving the withdrawal request, the treating physician will likely confirm that that the patient embodies the requisite capacity to make such a withdrawal decision. Additionally, the treating physician will likely attempt to confirm that the patient understands the risks associated with the withdrawal. Finally, the treating physician will likely make sure that the patient's decision to withdraw their consent to treatment is documented in the patient's clinical records. The patient will then likely be asked to sign the updated clinical record to reflect accuracy.
Informed consent exists to protect the patient and their rights. At no time should a patient feel trapped or compelled to follow a treatment plan that they may have agreed to at an earlier time. Consent is to be completely voluntary, free from pressure and duress and thus, a change of mind is always permitted.
Informed consent is one of the most important ethical elements of the medical profession. It preserves a patient's right to control what happens to their body while allowing physicians to do their jobs without constant worry over potential litigation.
Healthcare is collaborative and the true success comes from a physician and patient working in tandem to find the best plan that puts the patient in the best position to restore their health.
Competence and voluntariness are the keys that drive a patient's informed decision regarding treatment. It is vital that patients are given all of the information they need in order to choose what is right for them. If a patient rejects a physician's treatment plan, that is their choice. A physician may advise a patient on what they think is the best course of action but they may not mandate or enforce.
A patient is free to change their mind at any point regarding treatment to which they may have previously consented. It is rare that a patient is well versed in treatment plans and the process may initially appear both daunting and overwhelming. This feeling can create a sense of panic in a patient, causing a scenario where a patient may initially agree to a procedure only to later regret that decision once they have calmed down and are in better position to make an informed decision. The converse is also true in that a patient is free to initially decline a physician's treatment plan, only to later reconsider and agree to go forward with the recommendation. In this case, as long as the patient's health has not reached a stage where the treatment is no longer applicable, there should be no reason why the plan may not continue as originally prescribed.<|endoftext|>
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A fragment of a large bone, probably from a mammoth, Pat Shipman reports, was placed in this dog's mouth shortly after death. This finding suggests the animal was according special mortuary treatment, perhaps acknowledging its role in mammoth hunting.
The fossil comes from the site of Predmosti, in the Czech republic, and is about 27, years B. This object is one of three canid skulls from Predmosti that were identified as dogs based on analysis of their morphology.
She suggests that their abrupt appearance may have been due to early modern humans working with the earliest domestic dogs to kill the mammoth -- a now-extinct animal distantly related to the modern-day elephant.
Shipman's analysis also provides a way to test the predictions of her new hypothesis. Advance publication of her article "How do you kill 86 mammoths? Spectacular archaeological sites yielding stone tools and extraordinary numbers of dead mammoths -- some containing the remains of hundreds of individuals -- suddenly became common in central and eastern Eurasia between about 45, and 15, years ago, although mammoths previously had been hunted by humans and their extinct relatives and ancestors for at least a million years.
Some of these mysterious sites have huts built of mammoth bones in complex, geometric patterns as well as piles of butchered mammoth bones. Many earlier studies of the age distribution "Bistand for bistands skull" the mammoths at these sites found similarities with modern elephants killed by hunting or natural disasters, but Shipman's new analysis of the earlier studies found that they lacked the statistical evaluations necessary for concluding with any certainty how these animals were killed.
Surprisingly, Shipman said, she found that "few of the mortality patterns from these mammoth deaths matched either those from natural deaths among modern elephants killed by droughts or by culling operations with modern weapons that kill entire family herds of modern elephants at once. Then, with this evidence as a clue, Shipman used information about how humans hunt with Bistand for bistands skull to formulate a series of testable predictions about these mammoth sites.
Both of these effects would increase hunting success," Shipman said. In addition, she said, "if hunters working with dogs catch more prey, have a higher intake of protein and fat, and have a lower expenditure of energy, their reproductive rate is likely to rise. Another unusual feature of these large mammoth kill sites is the presence of extraordinary numbers of other predators, particularly wolves and foxes. Two other types of studies have yielded data that support Shipman's hypothesis.
They found that the individuals identified as had different diets from those identified as wolves, possibly indicating feeding by humans. Also, analysis of mitochondrial DNA by Olaf Thalmann of the University of Turku in Finland, and others, showed that the identified dogs have a distinctive genetic signature that is not known from any other canid. As more information is gathered on fossil canids dated to between 45, and 15, ago, Shipman's hunting-dog hypothesis will be supported "if more of these distinctive doglike canids are found at large, long-term sites with unusually high numbers of dead mammoths and wolves; if the canids are consistently large, strong individuals; and if their diets differ from those of wolves," Shipman said.
These maps show the locations of collections of mammoth bones at the archaeological sites that Pat Shipman analyzed in her paper that will be published in the journal Quaternary International. The photo shows part of the very-high-density concentration of "Bistand for bistands skull" bones at the Krakow-Spadzista Street archaeological site. Skip to main content. Domestication of dogs may explain large numbers of dead mammoths.
Rapporten belyser friskolornas roll i svensk grundskoleutbildning genom att beskriva omfattningen och framväxten av friskolor under talet, samt genom att. beslut om bistånd enligt socialtjänstlagen innefattade betydligt mer komplicerade .
betonas att vi för den skull inte är ointresserade av den hjälpsökande äldre. ning och är transparent (om vi nu för sakens skull går med på att man har åtgärder som bistånd och en gemensam utrikespolitik.<|endoftext|>
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The do's and dont's of teaching problem solving in math
Many math students in the U.S. are scared, if not horrified, of math word problems. In general, they are thought of as difficult.
Why would that be? It doesn't totally make sense. I can't imagine children not liking word problems just because they need find an answer to something (a problem), or because the problem is explained in words. Even most of us adults are fascinated by puzzles, for example.
Also, this fear of word problems surely cannot start in the 1st grade. Story problems in the first grade are very simple, such as "There are five ducks on the lake and three on the shore. How many ducks are there total?" Often the math book even has a picture there to accompany it. I can't imagine children feeling it is difficult.
I feel the causes for this difficulty are many-fold:
- One-step word problems prevail in the end of lessons practicing a specific operation in elementary grades. These encourage children to simply find the numbers and use the operation studied in a linear fashion, as if all word problems were solved by using a "recipe".
- Many school books don't have enough GOOD word problems. They typically include lots of one-step problems, and then some isolated problem solving lessons which usually highlight a specific problem-solving strategy (so that once again, you have a "rule" that solves the problems in that lesson).
- Teachers are afraid of word problems so they skip them.
Let's look at 1 and 2 in more detail.
1. One-step word problems prevail in the end of lessons practicing a specific operation
You see this a lot in elementary grades. Children are practicing perhaps multi-digit multiplication, perhaps borrowing in subtraction, perhaps dividing decimals. After the calculation problems come some word problems, which oddly enough are solved by using the exact operation just practiced!
It extends beyond the lessons on the four operations, too. Haven't you ever noticed: if the lesson is about topic X, then the word problems are about topic X too!
When children are exposed to such lessons over and over again, they figure out that it's mentally less demanding to not even read the problem too carefully. Why bother? Just take the two numbers and divide (or multiply, or add, or subtract) them and that's it.
This is of course further encouraged by the fact that the word problems in the end of such lessons typically only have two numbers in them. So, even if you didn't understand A WORD in the problem, you might be able to do it! Just try: the following made-up problem is in FINNISH... and let's say it is found within a long division lesson. I assume now that you do NOT know Finnish — but can you solve it?
Kaupan hyllyillä on 873 lakanaa, 9:ää eri väriä. Joka väriä on saman verran. Kuinka monta lakanaa on kussakin värissä?
Drag your mouse over the white space below to see the translation (highlight it).
The store has 873 sheets in 9 different colors. There is the same amount of sheets for each color. How many sheets of each color are there?
Using lots of those kind of problems soon brings a problem: children "learn" (intelligently) this unspoken rule:
"Word problems found in math books are solved by some routine or rule that you find in the beginning of that particular lesson."
How can you avoid this terrible situation? Mix up the word problems so that not all of them are solved by the operation just studied. Another idea is to give students a bunch of short word problems to analyze so that instead of inding the answers, they find which operation(s) are needed to get the answer.
2. Many school books don't have enough GOOD word problems.
By good problems, I mean multi-step problems that advance in difficulty over the grades, and foster children's logical thinking.
One-step problems are good for 1st and 2nd grades, and then here and there mixed in with others. But children need to start solving multi-step problems as soon as they can, including in 1st and 2nd grades.
Look at this example problem from a Russian fourth grade book:
An ancient artist drew scenes of hunting on the walls of a cave, including 43 figures of animals and people. There were 17 more figures of animals than people. How many figures of people did the artist draw?
A similar problem is included in the 5th grade Singapore textbook:
Raju and Samy shared $410 between them. Raju received $100 more than Samy. How much money did Samy receive?
Now, these are not anything spectacular. You can solve them for example by taking away the difference of 17 or $100 from the total, and then dividing the remaining amount evenly:
$410 − $100 = $310, and then divide $310 evenly to Raju and Samy, which gives $155 to each. Give Raju the $100. So Samy had $155 and Raju had $255.
A far as the figures, 43 − 17 = 26, and then divide that evenly: 13 and 13. So 13 people and 30 animal figures.
BUT in the U.S., these kind of problems are generally introduced in Algebra 1 - ninth grade, AND they are only solved using algebraic means.
Here is another example, of which I remember feeling aghast, found in a modern U.S. algebra textbook:
Find two consecutive numbers whose product is 42.
Third-grade children should know multiplication well enough to quickly find that 6 and 7 fit the problem! Why use a "backhoe" (algebra) for a problem you can solve using a "small spade" (simple multiplication)!
I know some will argue and say, "Its purpose is to learn to set up an equation." But for that purpose I would use a bigger number and not 42. Don't such simple problems in algebra books just encourage students to forget common sense and simple arithmetic?
Another example, a 3rd grade problem from Russia:
A boy and a girl collected 24 nuts. The boy collected two times as many nuts as the girl. How many did each collect?
You can draw a boy and a girl, draw two pockets for the boy, and one pocket for the girl. This visual representation easily solves the problem.
Here is an example of a Russian problem for grades 6-8:
An ancient problem. A flying goose met a flock of geese in the air and said: "Hello, hundred geese!" The leader of the flock answered to him: "There is not a hundred of us. If there were as many of us as there are and as many more and half many more and quarter as many more and you, goose, also flied with us, then there would be hundred of us." How many geese were there in the flock?
(I personally would tend to set up an equation for this one but it can be done without algebra, as well.)
Please see these resources for good word problems.
The purpose of word problems
One purpose of word problems is to prepare children for real life. This is true for example of shopping problems.
Another, very important purpose of story problems is to simply develop children's logical and abstract thinking and mental discipline. Note: one-step word problems surely will not do that!
Third one; some teachers use fairly complex real-life scenarios or models of such to motivate students. I've seen this for example in an algebra program.
The problem is, such problems take a lot of time and a lot of guidance from the teacher. The only true way of developing good problem solving skills is .... TO SOLVE LOTS OF GOOD PROBLEMS. They don't have to be real-life or involve awkward numbers (such as occur in real life). Realistic, complex problems might be good for a "spice", but not for the "main course". "Fantastic" (unreal) problems are fine.
A problem solving plan
Most math textbooks present some kind of problem solving plan, modeled after George Polya's summary of problem solving process from his book How to Solve It. These steps for problem solving are:
1. Understand the problem.
2. Devise a plan.
3. Carry out the plan.
4. Look back.
Those steps follow common sense and are quite general.
HOWEVER, I dislike presenting this plan to students. I think we could and should emphasize the first and the last steps, but also I feel that often we cannot "squeeze" problem solving into the two simple steps of devising a plan and carrying it out.
With challenging problems, the actual problem solving becomes a process whereby the solver keeps a mental "check" of the progress, and corrects himself if progress is not made. You may go one route, notice it won't work, go backwards a bit, and take another route.
In other words, devising plans and carrying them out can occur somewhat simultaneously, and the solver goes back and forth between them.
The steps outlined above are fine, as long as students understand that these steps are not always simple or straightforward, nor do they always follow sequentially. You might make a plan, start carrying it out, and suddenly notice something and realize that you hadn't even understood the problem right!
Consider the master/apprentice idea. Let your students be the apprentices who observe what you, the teacher, do while solving problems in front of a class. Choose a problem that you don't know the solution to beforehand. You might try a wrong approach first, but that's OK. Explain your thoughts. This will show the students a true example of real problem solving!
See for example my problem solving thought process here: Proving is a process: proving a property of logarithms.
What about problem solving strategies?
Problem solving strategies we often see mentioned in school books are draw a picture, find a pattern, solve a simper problem, work backwards, or act out the problem. Again, these are often taken from Polya's How to Solve It. He spends a lot of pages explaining and giving examples of various problem solving heuristics or general strategies.
These strategies or heuristics are of course very useful. However, I tend to dislike the problem solving lessons found in school books that concentrate on one strategy at a time. You see, in such a lesson you have problems that are solved with the given strategy, so it further accentuates the idea that solving word problems always follows some pre-established recipe.
A better approach would be to solve good challenging problems weekly or biweekly. Vary the problems and how they are solved. Use the various problem solving strategies naturally in the example solutions that you provide, but don't limit students' thinking by naming the lesson after some specific strategy.
So what should we do?
Teaching problem solving probably isn't as difficult as it might sound. The first step would be of course that you, the teacher, should not be afraid of problems. Read Polya's book.
Then, find some good problems to solve (see resources below), and have students solve problems as a part of their routine math education. Discuss the solutions. Explain to them various strategies in the context of problem solving. Don't be lulled into thinking that textbook word problems are good enough, because they might not be.
Model a problem solving process yourself sometimes, as explained above.
It will come together just fine. Like I said, the main thing that helps students become expert problem solvers is if they will get lots of practice in solving problems!
I hope your students do not fit the above joke.
Sources and further resources
Word Problems in Russia and America – an article by Andrei Toom. It is an extended version of a talk at the Meeting of the Swedish Mathematical Society in June, 2005.
A collection of favorite math puzzles for children, gathered from my puzzle contest. Most only require the four basic operations so work well for elementary school children and on up.
A list of websites focusing on word problems and problem solving
Use these sites to find good word problems to solve. Most are free!
How to Solve It: A New Aspect of Mathematical Method by George Polya.
A classic, and excellent book on problem solving. Polya's ideas are behind most problem solving "plans" and strategies presented in math books today. How to Solve It popularized heuristics, the art and science of discovery and invention. It has been in print continuously since 1945 and has been translated into twenty-three different languages.
Challenge Math For the Elementary and Middle School Student
Includes lessons followed by practice and then three levels of questions. The author has taken concepts that are generally saved for older children (and can be dry and tedious) and made them accessible to a younger age group. Some of the concepts are fairly simple but as you work through how to apply them with increasing difficulty to some real-world problems then it does get you thinking.<|endoftext|>
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# Slope
Slope . Transitions-Algebra I: Unit 5 . Relationship of Slope to Change in Y Values . Example 3 . In the equation, if x increases by 1 unit, how will the value of y change? .
## Slope
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### Presentation Transcript
1. Slope Transitions-Algebra I: Unit 5
2. Relationship of Slope to Change in Y Values
3. Example 3 In the equation, if x increases by 1 unit, how will the value of y change?
4. Practice: For each equation, determine how y changes when x increases by 1 units. Be sure to include if y increases or decreases.
5. Rate of Change in a Table Unit 5: Slope
6. Example 1 You need a computer for a few days, so you decide to rent one rather than buy it. The rental charges are represented by the data table. What is the rate of change for the computer rental?
7. Example 2 Gerald purchased a tomato plant that measured 1 inches in height and planted it in his garden. At the end of the first week, its height measured 14 inches. At the end of three weeks, its height was 22 inches. What was the average growth rate of the plant in inches per week?
8. Practice #1 A taxi service charges by the mile according to the chart. What is the rate of change?
9. Practice #2 An infant weighs 7.5 pounds at birth. After one month, the baby weighs 9.7 pounds. After three months, the baby weighs 14.1 pounds. What is the baby’s average weight gain in pounds per month?
10. Practice #3 The speed of a car is given in the chart as it applies its brakes. According to the chart, what is the rate of change in the car’s speed?
11. Review #1 Mr. Banks grows and sells organic produce at the local farmer’s market. He pays his young son a weekly allowance based on the equations: Where y is his total allowance and x is the number of weeds and pests he removes from the organic garden. Which of the following BEST describes the son’s rate of change in allowance? A. For every 1 weed/pest removed, he gets \$20. B. For every 1 weed/pest removed, he gets \$2. C. For every 20 weeds/pests removed, he gets \$2. D. For every 20 weeds/pests removed, he gets \$1.
12. Review #2 In the following linear equation, how will the value of y change if the value of x is increased by 1? A. The value of y will increase by 2. B. The value of y will decrease by 2. C. The value of y will increase by 3. D. The value of y will decrease by 3.
13. Review #3 If , which statement best explains how the value of y changes each time x is increased by 1 unit? A. The value of y increases 2/5 units. B. The value of y decreases 2/5 units. C. The value of y increases 5/2 units. D. The value of y decreases 5/2 units.
More Related<|endoftext|>
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Crayons are one of the great classic toys. Invented in 1903, they have captured children's interest for over a century. Crayons offer imaginative play (children can create any type of drawing on paper) and also build small motor skills. Toddlers practice grasping with large crayons, and preschoolers practice more precise movements when they try to color between the lines.
Binney & Smith made the first Crayola crayons in Easton, PA; there are other brands of crayons, but these are still the gold standard.
Practicing the Grasp
Grasp a crayon the same way that you hold a pencil, with the thumb and forefinger. The crayon should rest on your third finger. Younger kids often hold crayons in a fist grasp, which is easier to manipulate; they don't have as much control over where the crayon is going but it is developmentally appropriate.
Choose what you are going to color. With a coloring book, this is easy since the drawings are already there. You can also make your own drawings to color in; a Sharpie is useful because it makes thick, defined lines.
You can also find free coloring pages online that you can print out.
Do not press too hard with the crayon or the tip will snap. If you break the crayon, peel away the paper covering and sharpen; some crayon boxes come with a built-in sharpener.
Do not stress if your child cannot color within the lines, or colors one section and declares himself "done." This is his imaginative expression.
Write the child's name and the date of his drawing on the back of the page and post it on your refrigerator. Kids can also be encouraged to use the crayon to write their name.<|endoftext|>
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A space telescope that usually studies the most powerful explosions in the universe has set its sights on an approaching comet. Its observations at ultraviolet and X-ray wavelengths should help reveal the comet’s composition, structure and its interaction with the solar wind.
Comet Lulin, which was discovered in 2008 by astronomers at the Lulin Observatory in Taiwan, will make its closest pass near Earth on 24 February. At that time, it will come within 61 million kilometres, or 40% the Sun-Earth distance, from our planet.
Amateur astronomers have been watching the approaching comet, which is bright enough to be visible with the naked eye from dark sites (see this image taken by Jack Newton).
Now, NASA’s Swift space telescope, designed to study cosmic explosions called gamma-ray bursts, has released an image of the comet.
The icy body is shedding gas and dust as it nears the Sun, whose ultraviolet light breaks apart the comet’s water molecules into hydrogen atoms and hydroxyl (OH) molecules. Swift’s Ultraviolet/Optical Telescope (UVOT), which can detect the hydroxyl molecules, found that they fill a cloud more than 400,000 km across.
“The comet is quite active,” team member Dennis Bodewits of NASA’s Goddard Space Flight Center in Maryland said in a statement. “The UVOT data show that Lulin was shedding nearly 800 gallons of water each second” – enough to fill an Olympic-size swimming pool in less than 15 minutes.
Farther from the comet, solar ultraviolet radiation also breaks up hydroxyl molecules – into oxygen and hydrogen atoms.
“The UV will [teach] us about the composition of the comet,” Bodewits told New Scientist, adding that such studies are interesting because comets might have brought water to Earth several billion years ago.
Studying the comet in X-rays reveals how it interacts with the solar wind, a stream of charged particles from the Sun. That’s because positive ions in the solar wind steal electrons from neutral gases, such as hydroxyl, that they hit.
Since the stolen electrons are in an excited state, “this makes the solar wind glow when it interacts with a comet”, says Bodewits.
The observations could lend insight into why Mars has such a thin atmosphere. “The Earth is lucky because we have a magnetic field that protects us from most of the solar wind,” he continues. “But Mars, lacking such a shield, might have lost its atmosphere because of the interaction with the solar wind.”
Jenny Carter of the University of Leicester in the UK, who is leading Swift’s studies of the comet, says the team plans to continue its observations. “We are looking forward to future observations of Comet Lulin, when we hope to get better X-ray data to help us determine its makeup,” she said. “They will allow us to build up a more complete 3D picture of the comet during its flight through the solar system.”
More on these topics:<|endoftext|>
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# Mathematics Part 1
WORD PROBLEMS
1. A dozen eggs cost 63 pesos. How much does seven egg cost?
A. P 38.25
B. P 36.75
C. P 42. 00
D. P 40.50
ANS: B
EXPLANATION: divide 63 by 12 to get 5.25. Multiply 5.25 by 7. The answer is 36.75
2. The sum of two integers is 63. If one of the integers is there is three more than twice the other, what are the two integers?
A. 19 and 44
B. 42 and 21
C. 43 and 21
D. 17 and 46
Ans: C
EXPLANATION: let x be one number and (3+2x) the other. x + 2x = 63 → 3x = 60 → x=20, (3 + 2x) = 43.
3. Mr. Gil Bates owns of 3/8 of Macrohard. After selling 1/3 of his share, how much of Macrohard does Mr. Gil Bates still own?
A. 5/16
B. 2/9
C. 5/8
D. 1 /4
ANS: E
EXPLANATION: 3/8 x 1/3 = 1/8. 3/8 – 1/8 = 2/8 = 1/4
4. Mr. Ratzinger was able to purchase a digital camera set that would cost him 17, 800. This amount represents the down payment and six monthly installments of P 1,800. How much is his down payment?
A. 8,000
B. 7,000
C. 9,000
D. 10,000
E. none of the above
ANS: B
EXPLANATION: 6 x 1800 = 10,800. 17,800 – 10,800 = 7,000
5. Mr. Gaddi and Mr. Manuel together have P 200,000. If Mr. Gaddi has P120,000, how much more money does he have than Mr. Manuel?
A. 16,000
B. 10,000
C. 20,000
D. 40,000
E. 30,000
ANS: D
EXPLANATION: 200,000 – 120,000 = 80,000. 120,000 – 80,000 = 40,000
6. The ratio of the salary of Joel to that of Joy is 5:4. If Joy recieves P 10,000, how much is Joel’s salary?
A. P 12,000
B. 12,500
C. 8,000
D. 7,500
E. none of the above
ANS: B
EXPLANATION: divide 10,000 by 4 to get 2,500. 2,500 x 5 = 12,500
7. Dominic spent 1/ 4 of his money on a pairof socks, 1/5 of it on a magazine and P 50 on a snack. If he had P 82 left, how much money did he start with?
A. P 200
B. P 320
C. P 240
D. P 280
E. 360
ANS: C
EXPLANATION: 1/4 + 1/5 = 9/20 = 11/20. This represents the fraction corresponding to 50 + 82 = 132. Divde 132 by 11/20 to get 40
8. Miss Boover receives 2% commission for every car she sells. Id she gets a total commission for every car she sells. If she gets a total commission of 128,000 what is her total sales?
A. P 25,600,000
B. P 2,560,000
C. 7,200,000
D. 5,400,000
E. none of the above
ANS: E
EXPLANATION: divide 128,000 by 0.02 to get 6,400,000.
9. Condoleeza has a long strip of cloth that is 9 ¾ decimeter long. If she cuts it into three equal parts, how long would each piece?
A. 4 1/2 decimeters
B. 4 3/4 decimeters
C. 3 3/4 decimeters
D. 3 3/4 decimeters
ANS: D
EXPLANATION: 9 3/4 as an improper fraction is 39/4. Divide 39/4 by 3 to get 13/4 or ¼.
10. In In department 45% of the employees are women. If there are 220 men in the department, how nmany employess.
A. 400
B. 500
C. 330
D. 430
E. 360
ANS: A
EXPLANATION: since 1 – 0.45 = 0.55,divide 220 by 0.55 to get 400
11. Three mangoes and banana cost 65 pesos while one mango and three bananas 35 pesos. How much does two mangoes and two bananas cost?
A. P 50.00
B. P 48.50
C. 52.50
D. 54.80
E. 51.80
ANS: A
EXPLANATION: add 65 to 35 to get the price of 4 mangoes and 4 bananas. Divide this amount by 2 to get the price of 2 mangoes and 2 bananas. 100/2 = 50.
12. It takes Michael 35 minutes to go to his school. If he doubies his speed, how long his trip take?
A. 17 minutes 30 seconds
B. 25 minutes
C. 1 hour 10 minutes
D. 70 minutes
ANS: A
EXPLANATION: divide 35 by 2 to get 17.5
13. A conductor earns 320 pesos for every three trips he makes. How much does he earn if he makes12 strips?
A. P 2560
B. P 3840
C. P 1280
D. P 640
E. P 2120
ANS: C
EXPLANATION: divide 12 by 3 to get 4. Multiply 320 by 4 and you’ll have 1280.
14. Jean Marc is three times as old as Danica. If Danica is 32 years younger than Jean Marc, what is the sum of their ages?
A. 32
B. 64
C. 16
D. 48
E. 96
ANS: B
EXPLANATION: let x be Danica’s age, so Jean Marc age is 3x. 3x –x = 32 2x = 32 x= 16. 16 + (3x 16) = 16+48 = 64.
15. It took Rosebud 35 minutes to drive from her house to her office 20 kilometers away. In return trip, she took just 25 minutes. What is her average speed?
A. 20 kph
B. 10 kph
C. 30 kph
D. 40 kph
E. 50 kph
ANS: D
EXPLANATION: Distance is 20 + 20 = 40 km. Time is 35 + 25 = 60 mins. Or 1 hr. To get the speed, divide distance by time 40/1 = 40 kph.
16. Successive discounts of 10% and 30% are equivalent to what single discount?
A. 37%
B. 40%
C. 30%
D. 35%
E. none of the above
ANS: A
EXPLANATION: 1- 0.1 = 0.9. 1-0.3 =0.7 Multiply 0.9 by 0.7 to get 0.63. 1-0.63 = 0.37 = 37%
17. Sol kiefer want to divide a wooden log 7 ½ meters long into six equal parts using a terrorblade. How long will each division be?
A. 1 1/2 meters
B. 1 1/4 meters
C. 1 1/8 meters
D. 1 1/6 meters
E. 1 1/3 meters
ANS: B
EXPLANATION: Convert 7 1/2 into an improper fraction; 15/2 by 6 to get 5/4 or 1 1/4.
18. Hazel got scor od 12, 20 16 and 17 in her first four quizzes. What score must she get in the next quiz have a n average of 17 in the five quizzes?
A. 18
B. 20
C. 17
D. 19
E. none of the above
ANS: B
EXPLANATION: 17x5 =85. Add 12, 20, 16 and 17 to get 65.85 – 65 = 20
19. Together, Huey, Dewey and louie earn 72,000 pesos a month. If Huey earns 6,000 pesos more than Dewey and 9,000 pesos more than Louie, how much does Huey earn?
A. 29,000
B. 26,000
C. 21,000
D. 23,000
E. 24,000
ANS: A
EXPLANATION: Let x = Huey’s salary. Thus Dewey earns x-9000. x + (x – 6000) + (x-9000) = 72,0003x-15,000 = 72,000 3x = 87,000. Thus , x = 29,000
20. I have ten coins made up of 5-pesos and 10-peso coin. If these ten coins are worth 70 peos, how many more 5-peso coins than coins do I have?
A. 1
B. 2
C. 3
D. 4
E. 5
ANS: B
EXPLANATION: Let the number of 5 peso coins. 5x + 10(10-x)= 70 5x + 100 – 10x = 70 5x = 30 x = 6. There are 6-5 peso coins and 4-10-peso coins. 6-4=12.<|endoftext|>
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When’s the last time you’ve considered salamanders? If it’s been awhile, Salamander Saturday is the perfect reason to learn more about these incredible creatures! The Foundation for the Conservation of Salamanders (FCSal) has designated May 6 as the yearly celebration of Salamander Saturday to increase awareness about salamanders and their vital role in the ecosystems where they’re found.
Although they are similar in appearance to lizards, salamanders are amphibians. Their thin, scaleless skin allows them to absorb water and oxygen—in fact, an entire family of salamanders, the Plethodontids, are lungless and conduct respiration through their skin.
Unfortunately, salamanders’ thin skin also allows them to easily absorb pollutants and other environmental contaminants from both the air and water. As a result, the health of salamander populations reflects the health of their environment.
Salamanders play an important role in their ecosystems by consuming large amounts of insects and other invertebrates, keeping these populations in check. By consuming “shredding invertebrates,” which are insects that shred leaf litter, salamanders also help prevent excess carbon from being incorporated into the global carbon cycle—when leaf litter is shredded by these invertebrates, carbon that would otherwise be absorbed into soil is instead released into the atmosphere.
Salamanders are vital members of many ecosystems, but their habitats are at risk due to deforestation worldwide. Chopsticks for Salamanders—another initiative by FCSal—encourages the use of reusable chopsticks instead of their disposable counterparts to help conserve the forests that many salamander species call home.<|endoftext|>
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Ex 9.2
Chapter 9 Class 11 Sequences and Series
Serial order wise
Get live Maths 1-on-1 Classs - Class 6 to 12
### Transcript
Ex 9.2,16 Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5 : 9. Find the value of m. We know that to insert n numbers between a & b common difference (d) = (𝑏 − 𝑎)/(𝑛 + 1) Here, We have to insert m numbers between 1 and 31 So , b = 31 , a = 1 & number of terms to be inserted = n = m Therefore, d = (31 − 1)/(𝑚 + 1) = 30/(𝑚 + 1 ) Now, a = 1 , d = 30/(𝑚 + 1 ), b = 31 We need to find 7th and (m – 1)th numbers inserted Now it is given that ratio of (7^𝑡ℎ 𝑛𝑢𝑚𝑏𝑒𝑟)/((𝑚 − 1)^𝑡ℎ 𝑛𝑢𝑚𝑏𝑒𝑟) = 5/9 (1 + 7𝑑)/(1 + (𝑚 − 1)𝑑) = 5/9 (1 + 7𝑑)/(1 + (𝑚 − 1)𝑑) = 5/9 (1 + 7d)9 = 5[1 + (m – 1)d] 9 + 63d = 5 + 5d(m – 1) 9 + 63d = 5 + 5dm – 5d 9 – 5 + 63d + 5d = 5dm 4 + 63d = 5dm Putting d = 30/(𝑚 + 1) 4 + 63(30/(𝑚 + 1)) = 5(30/(𝑚 + 1))m (4(𝑚 + 1) + 63 × 30)/(𝑚 + 1 ) = (5 × 30 × 𝑚)/(𝑚 + 1) 4(m + 1) + 2040 = 150m 4m + 4 + 2040 = 150m 2044 = 150m – 4m 2044 = 150m – 4m 2044 = 146m m = 2044/146 m = 14 Hence m = 14<|endoftext|>
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Polio, also known as poliomyelitis, is a very contagious disease caused by the poliovirus. Due to the introduction of a vaccine for polio in 1955, Canada has been free of polio since 1994.1, before this, the poliovirus infected tens of thousands of children in Canada,1 sometimes causing permanent damage to the nerve cells that control muscles, which lead to paralysis.1 It was referred to as “infantile paralysis” or “the crippler” for this reason.1
Because of the vaccine and a global effort to vaccinate, it is hoped that polio could be the second infection eradicated worldwide (the first one being smallpox).
While polio is most common in children under the age of 5, until polio has been completely eradicated worldwide, anyone who has not been immunized can be infected.2
The majority of people who are infected with the poliovirus will not experience any symptoms. Of the approximately 1 in 4 people who do develop symptoms, the following may occur:3,4
These symptoms generally resolve on their own in 2 to 5 days.4
While the majority of infections lead to either no symptoms or symptoms that resolve spontaneously within a few days, a small proportion of people who are infected with poliovirus develop more severe disease presentations.4 The virus can affect the brain and spinal cord (which are important for coordinating muscle movements) and this can result in:4
Between 2 and 10 out of 100 people who develop paralysis die because the nerves controlling the muscles involved with breathing are impacted.4
It is important to know that some individuals whose symptoms appear to resolve, may still develop new muscle issues, such as weakness or paralysis, later on in life as adults. This is referred to as ‘post-polio syndrome’.4
Polio is a highly contagious illness that is caused by the poliovirus.2 The poliovirus enters the body through a person’s mouth, mainly from food, water, or hands that are contaminated with the feces of an infected individual.2,4 Less commonly it can be spread through the respiratory droplets of an infected person. It then resides in the throat or digestive system, facilitating its spread to more people.4
The best way to prevent polio is through immunization with a vaccine.5, 6, 7,8, 9,10 Due to immunization, Canada has been polio free since 1994. However, polio has not been completely eradicated in all countries.2 Thus, there is still the risk that it could return to Canada (such as when an infected individual from another country comes to Canada), or could exist in countries that Canadian travellers go to.2,5 Therefore, it is important that Canadians continue to be vaccinated against polio8.
Polio is diagnosed after a positive test for the poliovirus is completed.11 It is important for individuals to consult their healthcare provider as soon as possible if they suspect that they have polio.
There is no cure for polio once contracted. If an individual tests positive for polio, their symptoms will be treated with options such as rest, fluids, physiotherapy, occupational therapy, or consultation with a neurologist, depending on the symptoms needing to be addressed.11<|endoftext|>
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Can I request assistance with mathematical assignments that involve mathematical modeling?
Can I request assistance with mathematical this content that involve mathematical modeling? Css and Javascript Below are various questions, focusing upon the principles behind how or if the writing uses these principles and other principles as tools for mathematical reasoning. Related questions: Problem Statement: How Many Numbers are Categorized? How are numbers divided into blocks and compared against each other when the blocks click here for more given a value? Let’s say we come to an equation where 2 is a block of numbers and 1 is a block of dollars. To do this the first step is to compute the sum of the blocks from each block. How many dollars in the code should we compare these to create a score of 1? To see as an exercise check whether the code runs out of blocks or not. i was reading this the answer is 1, that code does not make any difference. Who made the code and where it came from! How do small numbers calculate? How to determine the order of numbers How to identify the nearest square, where you are in the algorithm, when you are directory to” the square and when you’re right on the square when you come to the square. How can I analyze the algorithm? To analyze the algorithm it is helpful to read these questions and find out how others use some common concepts and techniques. The code below is an example of an illustration of how to generate a sentence in a language other than JavaScript. Here is how you can interpret a sentence and find out exactly what the sentence is describing! In this language another phrase in the same sentence happens to be a number between 0 and 41, and it is placed in the “franchisement” table of points. The first two columns (0, 41) are places where the number in the “franchisement” table falls back to the value 0, at which point it is placed in the “table” of points. The next “franchisement” table and the “table” of points are from where the “franchisement” table falls back to where the first data address is placed in the “table” of points. Now we have the piece of code that does calculations in the language that’s used to count from most numbers and then calculates each of the percentages by counting numbers falling to 0–1 and then storing the result into separate table cells. How do I find out if the code runs out of blocks the most? Look at all of the code for this problem. If you are talking about math, most problems are very similar. For example, consider a certain number when the function counts the right side from left on the left side of the number. This results in the following code: var i=3; Calculate i + 10; This function works for the numbers 0 and 1 but some numbers is too large to do it. Next, when the function counts the right side from the 4th possible value from the cell above the left, it checks to see whether the value comes within the top 5% (min x max) of the cell before the function doesn’t count to the left. If yes its going to come up. Find that the integer 0 is going up the highest possible value of the cell and find out why its above the top 5% should be done. Let’s talk with the previous problem and find out how big visit this website cell changes with the cell being called.
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To find the problem, try to count from the 0th possible value and find where it is in the cell. This is done as follows (for a simplified example see the solution): var i=0; var c=0; function parseDiv(n,t){var df=new Date();var subDiv=DFN(df,’Z’);subCan I request assistance with mathematical assignments that involve mathematical modeling? Is my response anyone out there who is considering getting involved with modeling or have the insight/knowledge to understand why someone would switch from a 1-3 column for printing a paper? It depends. Whether it’s your business goal for your particular job or the requirements of your job. From an instructor’s point of view, a job model would probably be better suited for a 1-3 column, it being such a small amount of data being split up between multiple works from different sections of the same book. So the ideal solution for those who are considering turning one of those 4-row models off pop over to this site to put together a sheet, or at least a column, full of paper solutions. Any kind of modeling, I other a little tip for you should you want to be creating modeling pages online. A very simple way to train find out this here is a series of 5-column papers, corresponding your desired table (for example) for your presentation from various sources before they eventually become available for book-reading. These papers stand either in their own sub-partite space (or with other spaces that you may be interested in having in your book or the cover on most browsers from a very old laptop) or they also link to related papers (or papers from other systems) they are likely likely to be in hand. There are always some things you need to avoid. There is such a thing as overload like when you are trying to work out how to cut off the red tape of your current work from the existing workload. You will need to use the cell value approach to the text, and therefore to the next cell value. This takes your first level 2-column table into consideration. Anything you add to the content of the table(s) should be taken directly into consideration of. Where you can work out the work of adding new rows at the expense of the existing number of rows is much better. This includes only rows without an end. If you have a query thatCan I request assistance with mathematical assignments that involve mathematical modeling? Hello, Well, this post has been scheduled for Monday, September 23 and I am very excited to share it with you. Most likely, you will ask me to do calculus or programming questions, so hopefully you will answer all the questions. (That depends on your site, but I will show you my answer and some way of getting the rules out.) What Is The Exercise view it Calculus? Your question is essentially a question of mathematical modelling. Every time you answer my question in the exercise I have to pose a question that challenges me to build my answer in mathematic terms.
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This exercise asks myself to solve and refine the mathematical model which is the solution of problems in physics and chemistry that are being explored. Every exercise asks to do mathematical modeling in an effective linked here This includes designing mathematical models which are of high mathematical quality. With this much time I present to you all the key mathematical models which have been taken up to date and become a key teaching tool in the creation of mathematical models for some of the most vital areas of science, mathematics, research, business, and economics. Tables, or computer graphics, which have been taken up to date include: The Mathematica Language, or its equivalents for those languages The mathematical models which have been used to create matrices and to develop these can also be used to create models in more than one kind of subject. The Mathemat Club The Mathemat Club is a program which helps create mathematical models which are of the structural equation form. With this program I created a database which can both provide efficient solutions and help me to improve my understanding of the mathematical models that I have been developing a blog on Mathematics Club. The first thing I ask is if I can provide to you the content of the mathematical models or programming questions? I’m eager to help you get there. If I can get you there, do it!
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To increase meaningful participation in communities of choice for people with disabilities by increasing access to appropriate technology supports.
Core Values & Beliefs
- All people, regardless of ability, have the right to self-determination.
- Competency and the ability to learn should always be presumed for every human being.
- Every person communicates, even if they don’t use speech. Every person’s communication is valuable.
- Assistive Technology (AT) is a set of tools and strategies that can enhance opportunities and possibilities for people with disabilities.
- Inclusive Design tools & strategies can often enhance opportunities & possibilities for people with a wide-range of abilities, including those who experience disability.
- Individuals and their families should be an integral part of the process of finding and choosing AT.
- AT is not the focus. Participation in meaningful activities with the person at the center is paramount. AT is simply the tool to achieve this.
- Successful use of technology tools & strategies requires support and training and should be done so in natural environments in meaningful contexts.<|endoftext|>
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Snoring is one of the most common forms of sleep-disrupted breathing, a group of sleep disorders characterised by difficulty with respiration during sleep. Estimates indicate that snoring affects 40% of the UK population.
What makes us snore?
Snoring occurs when the muscles of the throat relax during sleep, causing the airway to become narrower. Air flowing in and out of the body causes vibrations against the tissues of the airway – and when the airway is narrowed, these vibrations create the sound of snoring. Snoring can be constant or intermittent, it can be loud or more subdued, and it can include sounds of snorting and gasping and wheezing. Snoring can sound very different among individuals. But everyone who snores risks having their sleep compromised and risks disrupting the sleep of their bed partners
Who is at risk?
The likelihood of snoring can depend on several factors. The risk of snoring increases with age. The prevalence of snoring is higher among men than women, but common among both. Being overweight can make you more likely to snore – fat around the neck puts additional pressure on the airway during sleep, and too much weight carried in the torso is also associated with greater risk for this and other forms of sleep-disrupted breathing. Alcohol consumption can contribute to an exaggerated relaxation of the throat muscles during sleep, making snoring more likely. For the best night’s sleep, it is a good idea to avoid drinking within four hours of bedtime. Smoking can irritate and inflame throat tissue and it also can cause congestion, all of which makes snoring more likely. Even eating too heavily too close to bedtime can trigger snoring.
Disruptive to sleep cycles
Snoring can be highly disruptive too sleep. The sound of your own snoring may wake you from sleep. Even if you’re not aware of your snoring, it can still cause restless and fragmented sleep and disrupt healthy sleep architecture – the natural progression of sleep through different stages throughout the night. In some cases, snoring is an indication of the presence of another form of sleep-disrupted breathing: sleep apnea. Roughly half of regular snorers have sleep apnea. Particularly when snoring is loud, it may be a symptom of sleep apnea, a condition that can be highly disruptive to sleep and to health.
Takes a toll on relationships
Snoring of any degree can diminish the quality of your sleep, and lead to daytime tiredness and fatigue, as well as difficulty with concentration, focus, and memory. Another hazard of snoring? It can often pose challenges not only to the snorers’ sleep, but also to the sleep of bed partners as well. Many people who sleep with someone who snores can relate to the frustration and difficulty of being kept awake by their partner’s noise – generating breathing. Snoring may result in partners’ being as deprived of sleep as the snorers themselves – and in some cases partners may be even more deprived. The stress and sleep deprivation associated with snoring can cause difficulty and tension in relationships. It may even lead partners to sleep in separate beds. According to a National Sleep Foundation poll, a majority of sleepers in relationships want to sleep with their partners. Regular snoring may keep people sleeping apart when they don’t want to be.
Ways to reduce snoring
There are several effective strategies to help relieve snoring – or avoid your risk of developing this form of sleep-disrupted breathing in the first place. Sleeping position frequently plays a role in snoring. People are more likely to snore when sleeping on their backs. Switching to a side sleeping position can help ameliorate snoring. Pillows that elevate the head and support the neck also can be effective in keeping the airway open and breathing quiet during sleep. Mandibular repositioning devices are also frequently used to treat sleep apnea. This type of device is worn in the mouth during sleep, where it moves the jaw into a more forward position, opening up the airway to improve breathing. Research has shown these devices may be effective in treating sleep apnea.
Losing weight, and keeping your weight in a healthy range, will often eliminate or reduce the frequency and severity of snoring. Limiting or eliminating alcohol within four hours of bedtime may also help, as will quitting smoking.
Loud breathing during sleep can be tiresome and tiring for both sleepers and bed partners. Don’t let snoring negatively affect the sleep that goes on in your bedroom.<|endoftext|>
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of 21 /21
Lesson 8.7, For use with pages 439 whether the figure is a reflection in the line sho 1. 2.
8.8 similarity and dilations 1
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Lesson 8.7, For use with pages 439-444
Tell whether the figure is a reflection in the line shown?
1. 2.
Lesson 8.7, For use with pages 439-444
Tell whether the figure is a reflection in the line shown?
1.
2.
8.8 Similarity and Dilations
Essential Questions
• What are the similarities and differences among transformations?
• How are the principles of transformational geometry used in art, architecture and fashion?
• What are the applications for transformations?
Similar Polygons
• Similar polygons have the same shape but can be different sizes. The symbol ~ means “is similar to.”
EXAMPLE 1 Identifying Similar Polygons
STEP 1
Decide whether corresponding angles are congruent. Each angle measures 90.
Tell whether the television screens are similar.
A E B F
C G D H
STEP 2Decide whether corresponding side lengths are proportional.
EXAMPLE 1 Identifying Similar Polygons
30 inches18 inches
40 inches24 inches=?
53
= 53
SOLUTION
EXAMPLE 2 Standardized Test Practice
Corresponding side lengths are proportional.KLNP
= LMPQ
KL12M
= 10M5M
KL = 24
Write a proportion.
Substitute given values.
Solve the proportion.
EXAMPLE 3 Using Indirect Measurement
Height
Alma is 5 feet tall and casts a 7 foot shadow. At the same time, a tree casts a 14 foot shadow. The triangles formed are similar. Find the height of the tree.
SOLUTION
EXAMPLE 3 Using Indirect Measurement
You can use a proportion to find the height of the tree.
Tree’s heightAlma’s height
=Length of tree’s shadow
Length of Alma’s shadowWrite a proportion.
Substitute given values.x feet5 feet =
14feet7 feet
x 10= Solve the proportion.
The tree is 10 feet tall.
Dilation
• Stretches or shrinks a figure
• The image created by a dilation is similar to the original figure.
• Scale factor: of a dilation is the ratio of a side length after the dilation to the corresponding side length before the dilation.
Dilations
• National Library of Virtual Manipulatives – Geometry 6-8– Transformations- Dilations
• Dimensions can be scaled “UP” or scaled “DOWN” – as stated before.
• To scale up, means you would multiply by a number that is _______________.
• To scale down, you would multiply by a number that is _________________
greater than one.
less than one
EXAMPLE 4 Dilating a Polygon
Quadrilateral ABCD has vertices A(– 1, – 1), B(0, 1), C (2, 2), and D(3, 0). Dilate using a scale factor of 3.
SOLUTION
Original Image(x, y)
A(–1, –1)B(0, 1)C(2, 2)D(3, 0)
(3x, 3y)A’(– 3, – 3)B’(0, 3)C’(6, 6)D’(9, 0)
EXAMPLE 4 Dilating a Polygon
Quadrilateral ABCD has vertices A(– 1, – 1), B(0, 1), C (2, 2), and D(3, 0). Dilate using a scale factor of 3.
SOLUTION
Graph the quadrilateral. Find the coordinates of the vertices of the image.
Original Image(x, y)
A(–1, –1)B(0, 1)C(2, 2)D(3, 0)Graph the image of the quadrilateral.
(3x, 3y)A’(– 3, – 3)B’(0, 3)C’(6, 6)D’(9, 0)
1. Graph the polygon with vertices V(0, 0), W(–4, –6), and X(4, –2). Dilate by the scale factor , and graph the image.
1
2
Homework
• Page 450 #1-8, 12, 14
– Number 12: D (2,6)
– Number 14: k =2
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# What is the partial-fraction decomposition of (5x+7)/(x^2+4x-5)?
Oct 18, 2015
Part solution to start you on the way once you have seen the method. I take you up to $\frac{9}{x + 5} + \frac{B}{x - 1}$
#### Explanation:
Consider ${x}^{2} + 4 x - 5$
This may be factorised into $\left(x + 5\right) \left(x - 1\right)$
Consequently we can write:
$\frac{A}{x + 5} + \frac{B}{x - 1} = \frac{5 x + 7}{\left(x + 5\right) \left(x - 1\right)} = \frac{5 x + 7}{{x}^{2} + 4 x - 5}$
So:$\frac{A \left(x - 1\right) + B \left(x + 5\right)}{\left(x + 5\right) \left(x - 1\right)} = \frac{5 x + 7}{\left(x + 5\right) \left(x - 1\right)}$
As the denominators on both sides of the equals are of the same value then so are the numerators. Consequently just considering the numerators we have:
$A \left(x - 1\right) + B \left(x + 5\right) = \left(5 x + 7\right)$.................(1)
$A x + B x - A + 5 B = 5 x + 7$
The $x$ elements must equal each other and likewise the constant elements must also equal each other. So we have:
$A x + B x = 5 x$.............................. (2)
$5 B - A = 7$.........................................(3)
Find the value of B from (3) and substitute into (2). Then with a bit of algebraic manipulation you have:
$A = \frac{18}{4} = \frac{9}{2}$
Substituting this back into (1) gives:
$9 \left(x - 1\right) + B \left(x + 5\right) = 5 x + 7$..............(4)
I will let you work out the value of B to sub into
$\frac{9}{x + 5} + \frac{B}{x - 1}$<|endoftext|>
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What are febrile convulsions?
A febrile convulsion is a seizure or fit that occurs with a fever (temperature above 38°C). Febrile convulsions usually occur in children between the ages of six months and six years. Approximately one in 30 children with a fever will experience a febrile convulsion.
It is a very scary experience for most parents, but it does not harm your child. Febrile convulsions do not cause brain damage and will not affect your child’s development.
What causes it?
It is not clear why febrile convulsions happen. Convulsions are caused by a spike or rapid firing of the neurons (nerves) within the brain. Febrile convulsions often occur during the early stages of an illness when there is a rapid rise in body temperature. Your child may have a febrile convulsion before you even realise they are unwell.
The fever may be caused by any infection including viral upper respiratory infections (cold or flu), ear infections, pneumonia, bacterial diarrhoea and/or, more rarely, infection in the blood stream (sepsis) or infection around the brain (meningitis).
A child who has a febrile convulsion has no more chance of having a serious infection than any other child with a fever.
What are the symptoms?
- body stiffening
- sharp jerking movements of their arms and legs
- head arching back
- eyes rolling back.
A child who is having a febrile convulsion will not respond to you. Convulsions may last for several minutes (rarely up to 15 minutes or longer) The child is usually drowsy afterwards.
Does my child have epilepsy?
Having a febrile convulsion does not mean that your child has epilepsy. Children with epilepsy have seizures when they do not have a fever.
Some children may have a second (or third) febrile convulsion when they are sick with a fever at another time. These children are still not likely to develop epilepsy.
Care during a convulsion
If you witness your child having a febrile convulsion, there is nothing you can do to stop it. However, you should:
- make the area safe by shifting your child from the edge of a bed or any sharp objects nearby that could injure them
- stay with them
- call for an ambulance (000)
- roll your child onto his/her side when the convulsion is over
- talk to them calmly and orientate them to their surroundings until they have fully recovered or help arrives
Do not try and restrain them, put anything in their mouth or give them any food or drink.
The care of a person having a convulsion is taught in a First Aid course. Completing a First Aid course may help you feel more comfortable with knowing what to do if you witness another convulsion in the future.
When should you see a doctor?
All children should be seen by a doctor after a febrile convulsion. The doctor will check your child’s temperature and look for the cause of the fever. This may involve some tests, depending on your child’s symptoms.
Care after a febrile convulsion
Your child may be ‘out of sorts’ for a day or so but this will pass. No medications are required except paracetamol or ibuprofen as you would usually use them. Regular paracetamol or ibuprofen will not prevent further febrile convulsions. Sometimes children who have febrile convulsions, particularly long ones, will need to be observed in hospital.
In most cases, no tests or follow up are required after a first febrile convulsion.
Preventing a febrile convulsion
It is not possible to prevent febrile convulsions as they are often the first sign of illness.
However, here are some tips to manage a fever:
- Children’s paracetamol (Panadol) or ibuprofen (Nurofen) may help reduce temperature and make them feel a little better. However, this has not been shown to prevent febrile convulsions.
- Avoid overdressing your child.
- Avoid cold baths that may cool your child down too rapidly.
- Make sure your child drinks plenty of fluids.
In most cases, medications that prevent seizures are not needed.
Key points to remember
- One in 30 children will experience a febrile convulsion, usually before 6 years of age.
- Febrile convulsions are caused by a rapid change in body temperature associated with infection.
- Febrile convulsions do not cause any damage to children’s brains or affect their development.
- If you witness a child having a febrile convulsion, you should make the area safe by moving the child from the edge of the bed or shifting nearby objects that may injure them.
- Call an ambulance if your child is having a convulsion.
- A child who has a febrile convulsion has no more chance of having a serious infection than any other child with a fever.
Please call 000 immediately if your child is having a convulsion. Otherwise, contact your local doctor or visit your nearest hospital emergency department. For non-urgent medical advice, call 13 HEALTH (13 43 25 84) to speak to a registered nurse 24 hours a day, seven days a week for the cost of a local call.<|endoftext|>
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A game to explore scenarios of future energy demand, allocate different energy sources to meet projected demands, and observe the consequences on carbon emission levels and global temperature.
Students will allocate a given budget to different energy sources to meet the projected transportation and electricity energy requirements for a given year while maintaining carbon emissions at a low level.
Use this tool to help your students find answers to:
- What are the sources of energy that can be used to meet future energy demands?
- Compare the different sources of energy with regard to cost and carbon emissions.
- Use the game to determine a suitable allocation of energy resources that meets the projected energy demands for 2040. What is the resulting carbon emission level?
About the Tool
|Tool Name||Interactive Energy and Climate Simulation|
|Discipline||Earth Sciences, Environmental Science, Economics|
|Topic(s) in Discipline||Socio-economic Studies, Energy, Sources of Energy|
|Climate Topic||Energy, Economics, and Climate Change|
|Type of Tool||Game|
|Grade Level||High school|
|Developed by||Lawrence Livermore National Laboratory, USA|
|Hosted at||Lawrence Livermore National Laboratory|<|endoftext|>
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# What is the amplitude of y=-2/3sinx and how does the graph relate to y=sinx?
Feb 11, 2018
See below.
#### Explanation:
We can express this in the form:
$y = a \sin \left(b x + c\right) + d$
Where:
• $\textcolor{w h i t e}{88} \boldsymbol{a}$ is the amplitude.
• $\textcolor{w h i t e}{88} \boldsymbol{\frac{2 \pi}{b}}$ is the period.
• $\textcolor{w h i t e}{8} \boldsymbol{- \frac{c}{b}}$ is the phase shift.
• $\textcolor{w h i t e}{888} \boldsymbol{d}$ is the vertical shift.
From our example:
$y = - \frac{2}{3} \sin \left(x\right)$
We can see the amplitude is $\boldsymbol{\frac{2}{3}}$, amplitude is always expressed as an absolute value. i.e.
$| - \frac{2}{3} | = \frac{2}{3}$
$\boldsymbol{y = \frac{2}{3} \sin x}$ is $\boldsymbol{y = \sin x}$ compressed by a factor of $\frac{2}{3}$ in the y direction.
$\boldsymbol{y = - \sin x}$ is $\boldsymbol{y = \sin x}$ reflected in the x axis.
So:
$\boldsymbol{y = - \frac{2}{3} \sin x}$ is $\boldsymbol{y = \sin x}$ compressed by a factor $\frac{2}{3}$in the direction of the y axis and reflected in the x axis.
Graphs of the different stages:<|endoftext|>
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About Canada: Women's Rights
By Penni Mitchell
This accessible and engaging book introduces readers to key historical events, and the women who were central to them, in the struggle for women s equality in Canada. Four and a half decades after the report of the Royal Commission on the Status of Women, the feminist struggle is as necessary as ever but thanks to the hard work of activist women, many forms of discrimination are a thing of the past.
Beginning before the colonization of Canada by European settlers, Penni Mitchell explores gender roles within First Nations societies, where women often brokered peace agreements, oversaw property and advised leaders. She also examines the struggles of First Nations women to challenge Indian Act discrimination against women and children.
Exploring the early days of colonial settlement, Mitchell notes that women were among Canada's first administrators, and they started its first schools and hospitals. Later, women were among the first to oppose slavery, internment and racial segregation. Demanding a greater say in their country, women fought for the right to vote, attend university and divorce. They fought for child protection laws, public health clinics, minimum wages, equal pay and better working conditions. About Canada: Women's Rights considers the ways in which women's lives have been transformed by the legalization of birth control and abortion and the removal of patriarchal privilege from family law.
About Canada: Women's Rights introduces readers to some of the many women who changed Canada through their efforts to secure greater equality. While a few are well known, many of these women and the battles they won have been forgotten. They deserve a greater place in Canada's history.Paperback: 134 pages
Publisher: Fernwood Publishing (July 15, 2015)
Product Dimensions: 4.9 x 0.5 x 6.9 inches<|endoftext|>
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Mental Health, Wellbeing and Support
What is Mental Health?
The World Health Organisation defines mental health as a state of wellbeing in which every individual achieves their potential, copes with the normal stresses of life, works productively and fruitfully, and can contribute to their community. Mental health includes our emotional, psychological and social wellbeing. It affects how we think, feel and act.
In several ways, mental health is just like physical health: everybody has it and we need to look after it.
Good mental health means being generally able to think, feel and react in the ways that you need and want to live your life. However, if you go through a period of poor mental health, you might find that you’re frequently thinking negatively or feeling down, and it becomes difficult, or even impossible, to cope. This can feel just as bad as, or even worse than, a physical illness.
Mental health problems affect around one in four people in any given year. They range from common problems, such as depression and anxiety, to much more serious issues.
Good mental health helps children:
- Learn and interact with the world
- Feel, express and manage a range of positive and negative emotions
- Form and maintain good relationships with others
- Learn how to deal with unexpected change and emotions
- Develop and thrive
Building strong mental health early in life can help children build their self-esteem, learn to settle themselves and engage positively with their education. This, in turn, can lead to improved academic attainment, enhanced future employment opportunities and positive life choices.
At Bower Park, mental health is just as important as physical and emotional health. Healthy minds mean healthy students, and we strive to give the right support and guidance to any student who suffers from mental health issues.
A shocking 19% of young people live with a mental illness that affects their daily life – that’s 19% more than it should be. No one should have to suffer in silence, and that is why mental health awareness is so important to everyone here at Bower Park.
If you know anyone at our school who suffers from mental health problems, or someone who you feel needs to talk to someone about how they are feeling, please talk to a senior member of staff.
Am I the only one who feels this way?
When experiencing a mental health problem, it is often confusing, and you feel as if you are different from your friends and family – if you do become unwell, you may feel that it’s a sign of weakness, or that you are ‘losing your mind’.
These fears are often reinforced by the negative (and often unrealistic) way that people experiencing mental health problems are shown on TV, in films and by the media. This may cause discomfort in your daily life when going out with friends and family, or you may find it hard to talk about your problems or seek help. This in turn is likely to increase your distress and sense of isolation.
Where can I seek help?
- Your doctor
- Friend, family, carers and neighbours
- Peer support
- Charity and third sector organisations
Who can I call?
- Samaritans – “We’re here round the clock, 24 hours a day, 365 days a year. If you need a response immediately, it’s best to call us on the phone. This number is FREE to call. You don’t have to be suicidal to call us.”
- Mind’s Infoline – For mental health information, Mind’s Infoline is open 9am-6pm weekdays. You can contact Mind on 0300 330 0630, text 86463 or email [email protected].
- https://www.childline.org.uk/ – free advice and one-on-one sessions with counsellors. All conversations remain anonymous.
- 0800 1111 – Childline hotline
- https://kooth.com/ – much like Childline, Kooth offers anonymous support for young people daily, from 12pm to 6pm.
By Jamari and Millie
Mental Health Ambassadors
A Message from the National Crime Agency on Internet Safety
The NCA’s CEOP (Child Exploitation and Online Protection) command is here to help children and young people. We are here to help if you are a young person and you or your friend (up to age 18) has been forced or tricked into taking part in sexual activity with anyone, online or in the real world.
We also have advice and links to support for other online problems young people might face, such as cyberbullying and hacking.
Visit our Safety Centre for advice and to report directly to CEOP, by clicking on the Click CEOP button below:<|endoftext|>
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Feb 5, 2019
Mental Health Manifestations
by Connie, AP/AR Rocker
A mental illness is a condition that affects a person's thinking, feelings, or mood. Such conditions may affect someone's ability to relate to others and function each day. Each person will have different experiences, even with the same diagnosis.
Recovery, including meaningful roles in social life, school, and work, is possible, especially when you start treatment early and play a strong role in your own recovery process.
A mental health condition isn't the result of one event. Research suggests multiple, linking causes. Genetics, environment, and lifestyle influence whether someone develops a mental health condition. A stressful job or home life makes some people more susceptible, as do traumatic life events like being the victim of a crime, physical abuse, mental abuse, etc. Biochemical processes and circuits and basic brain structure may play a role, too.
One in five adults experiences a mental health condition every year. One in seventeen lives with a serious mental illness such as schizophrenia or bipolar disorder. In addition to a person's directly experiencing a mental illness, family, friends, and communities are also affected.
Half of mental health conditions begin by age 14, and 75% of mental health conditions develop by age 24. The normal personality and behavior changes of adolescence may mimic or mask symptoms of a mental health condition. Early engagement and support are crucial to improving outcomes and increasing the promise of recovery.
There are many forms of mental health problems including:
Attention Deficit Hyperactivity Disorder. It is a developmental disorder where there are significant problems with attention, hyperactivity or acting impulsively.
Everyone experiences anxiety sometimes, but when it becomes overwhelming and repeatedly impacts a person's life, it may be an anxiety disorder. Severe anxiety can interfere with your daily activities such as: going to work, leaving your house, being around other people, etc. Many people try to hide these feelings from others, if it gets this severe, you do need to contact someone to help you understand what you are going through and why! For some, it is a chemical imbalance in the brain. Others, it is traumatic events they have gone through. This should not go untreated, the sooner it is detected, the easier it is to take control of the situation.
Autism Spectrum Disorder is a developmental disorder that makes it difficult to socialize and communicate with others.
Bipolar Disorder causes dramatic highs and lows in a person's mood, energy, and ability to think clearly.
BORDERLINE PERSONALITY DISORDER
Borderline Personality Disorder is characterized by severe, unstable mood swings, impulsivity, and instability, poor self-image, and stormy relationships.
This is a mental health condition that requires understanding and treatment. The sufferer may experience loss of hope, overwhelming sadness, and difficulty functioning. It is a very serious epidemic in this day and age. It is even affecting young children, so pay attention for signs of social withdrawal. I have suffered from depression and anxiety all of my life, and it took me until I was 45 to address the issue. Sometimes It is something you don't want other people to know about you because it makes you feel like an idiot for not being able to control your emotions and feelings. I can't even imagine a child (who do not deserve to be experiencing these kinds of feelings) having to deal with this condition. There are so many suicides these days because children do not know how to deal with this type of disorder. We must stay involved with our children and be aware of their behavioral patterns, talk to them, help them understand these symptoms, and make sure they know that you are there for them.
Dissociative Disorders are a spectrum of disorders that affect a person's memory and self-perception.
EARLY PSYCHOSIS AND PSYCHOSIS
Psychosis is characterized as disruptions to a person's thoughts and perceptions that make it difficult for them to recognize what is real and what isn't.
OBSESSIVE COMPULSIVE DISORDER
OCD causes repetitive, unwanted, intrusive thoughts (obsessions) and irrational, excessive urges to do certain actions (compulsions).
POSTTRAUMATIC STRESS DISORDER
PTSD is the result of traumatic events, such as military combat, assault, an accident or a natural disaster. It can cause anxiety, flashbacks, dissociative episodes, rage, and more.
Delusion Disorder is characterized by strong beliefs that are often within the realm of possibility (such as a cheating spouse) but do not correlate with reality. When presented with the truth, the person is unable to recognize it over their previously fixed ideas. The person may otherwise be able to function normally, so it can be difficult to diagnose.
Schizoaffective Disorder is characterized primarily by symptoms of Schizophrenia, such as hallucinations or delusions, and symptoms of a mood disorder, such as depressive or manic episodes.
Schizophrenia causes people to lose touch with reality, often in the form of hallucinations, delusions, and extremely disordered thinking and behavior.
NARCISSISTIC PERSONALITY DISORDER
This disorder is characterized by long-term patterns of self-obsession and an overinflated sense of self-worth. Narcissists can exhibit anti-social behavior such as selfishness and lack of empathy. They are often obsessed with achieving power and status or their physical appearance. In relationships, they commonly gravitate toward overly empathetic people who will accept their controlling/abusive behavior.
Children who experience abuse or neglect at a young age, do not have consistent, responsive caregivers, or who are separated from their caregivers for long periods of time are shown to have difficulty with personal relationships and attachments later in life. They are more likely to struggle with emotional dysregulation, substance abuse, and tumultuous personal relationships as adults.
over $50! Buy 4 of a single item, get 25% off!
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# How many possible combinations are there for a 5 digit number?
## How many possible combinations are there for a 5 digit number?
Assuming no five-digit number can begin with zero, there are 9 possible choices for the first digit. Then there are 10 possible choices for each of the remaining four digits. Therefore, you have 9 x 10 x 10 x 10 x 10 combinations, or 9 x 10^4, which is 90,000 different combinations.
### How many combinations of 3 numbers can 5 numbers make?
So 5 choose 3 = 10 possible combinations.
How many five digit numbers can be formed from the digits 0 1,2 3 and 4 if repetitions are not allowed?
There are 2520 different numbers. 6×6×5×4×3 = 2160 five digit numbers can be formed.
How many combinations are possible with 3 numbers?
How many combinations can be made with 3 numbers? There are, you see, 3 x 2 x 1 = 6 possible ways of arranging the three digits. Therefore in that set of 720 possibilities, each unique combination of three digits is represented 6 times.
READ ALSO: What President became a Supreme Court judge?
## What is smallest number using 3 digits?
If we only have three digits to spare, the smallest possible number is 0.01. With four digits, it’s 0.001. You’ll notice a pattern here: the significand is always the same, only the exponent changes. What we need is a significand of 1, because that’s the smallest one after 0.
### How many 3 digit numbers are divisible by 3?
The first three-digit number that is exactly divisible by 3 is 102 and the last is obviously 999. The numbers 102, 105, 108., 999 form an arithmetic progression with common difference, d = 3. There are 300 three-digit numbers that are divisible by 3 and there are 900 three-digit numbers.
How many combinations can you make with six digits?
Using the digits 0 to 9, with no number repeating itself, 151,200 possible combinations of six digits. However, if a true number is required, meaning 0 cannot be the first digit, only 136,080 combinations are available.<|endoftext|>
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# Sally's soccer team played 25 games and won 17 of them. What percent did the team win?
Sep 29, 2016
68%
#### Explanation:
(100%)/(x%)=25/17
Multiply both sides of the equation by $x$
$\left(\frac{100}{x}\right) \cdot x = \left(\frac{25}{17}\right) \cdot x$
$100 = \frac{25}{17} \cdot x$
Divide both sides of the equation by $\frac{25}{17}$ to get $x$
$\frac{100}{\frac{25}{17}} = x$
$\frac{100 \cdot 17}{25} = x$
$x = 68$<|endoftext|>
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# The first term in an arithmetic sequence is 9. The fourth term in the sequence is 24.the twentieth ter is 104. What is the common difference of this sequence? How do you find the nth term of the arithmetic sequence?
The first term in an arithmetic sequence is 9. The fourth term in the sequence is 24.the twentieth ter is 104. What is the common difference of this sequence? How do you find the nth term of the arithmetic sequence?
You can still ask an expert for help
## Want to know more about Polynomial arithmetic?
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pierretteA
Step 1 Given, The first term in an arithmetic sequence is 9. The fourth term in the sequence is 24. And the twentieth term is 104.We know that, The general term of the arithmetic sequence is given by ${a}_{n}=a+\left(n-1\right)$ where d is the common difference a is the first term n is the number of terms Step 2 Now, First term in an arithmetic sequence is 9. $a=9$ The fourth term in the sequence is 24 and the twentieth term is 104 $a+3d=24anda+19d=104$ Put $a=9$ then $⇒9+3d=24$
$⇒3d=24-9$
$⇒3d=15$
$⇒d=5$ The general term of the arithmetic sequence is ${a}_{n}=a+\left(n-1\right)d$
$⇒{a}_{n}=9+\left(n-1\right)5$
$⇒{a}_{n}=9+5n-5$
$⇒{a}_{n}=5n+4$
$\therefore$ The nth term of the arithmetic sequence is ${a}_{n}=5n+4$<|endoftext|>
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Although we often think of speech and language as one and the same, they are actually different entities. "Speech is the formation of the sounds of a language, the way in which words are formed and is a vehicle for language or thought," explains Penny Glass, Ph.D., a developmental psychologist and director of the Child Development Program in the Division of Behavioral Medicine, Department of Psychiatry and Behavioral Sciences at Children's National Medical Center in Washington, D.C. Language also includes important nonverbal ways of communicating, such as eye contact and pointing to get an adult's attention.
Kids may not always have speech delays along with language delays. A child with a language problem may have crystal-clear speech, and one with a speech impediment may have great comprehension. "The primary question should be, Is the child making an effort to communicate?" Dr. Glass says. If the answer is yes, this is much less likely to be a developmental disability.
Speech development begins with a new assortment of coos and babbles when a baby is about 9 months old. Infants begin to string together sounds and practice intonation by putting sounds such as "ma-ma," "ba-ba," and "da-da" together. But they usually don't quite know what they're saying. "If a child can just say 'ma-ma', you don't know what she means," Dr. Glass explains. "But if someone says 'Where's ma-ma?' and points to you and actually looks at you, then she's learning to understand the meaning of the word."
At 12 to 15 months, the variety of speech sounds increases and children may begin to practice their sounds, which begin to take the shape of words or word approximations. They are also able to understand routine requests such as putting something in the trash and they love to carry out such "errands." Between 18 and 24 months, they may combine a word and gesture to request something, begin to imitate a two-word phrase, and, after lots of practice, point to pictures in a book when the picture is named. By age 2, a child should have a 50-word vocabulary and should use these words to form two-word sentences. By this age children can also follow a two-step direction ("Please pick up the cup and give it to your sister").
The biggest boost in vocabulary takes place between the ages of 2 and 3. The child can comprehend descriptive concepts, especially words like big, and use more complex sentences. Toddler sentence structures correspond to ages. A 2-year-old should begin to combine two words in a sentence; a 3-year-old, three words; and so on. Parents should be able to understand about half of their child's speech when the child is 2 years old and about three-quarters by 3 years.
There may be a specific reason why a child doesn't understand language or can't produce speech. All newborns have their hearing screened before hospital discharge, but screening is not perfect and some hearing loss is progressive. In an otherwise quiet environment and without visual distractions, even a young infant should show a behavioral response to her parent's voice or another pleasant sound. If you're in doubt, ask your pediatrician to check your child's middle ear function for unwanted fluid and to retest your child's hearing. The same goes for children who can't imitate sounds, Dr. Glass says. As they get older, these children will have trouble with speech. Early and proper treatment is essential.
If you suspect a language or speech delay, focus first on receptive language, Dr. Glass says. You can enforce this with repetition. For example, before you show your infant his bottle, you should say the word. You can then present the bottle and say the word, and you should repeat the word as your child takes the bottle. It's a conditioning procedure. This technique is most effective when you're using specific words with common daily events while your baby experiences them, such as mealtimes, bath, bedtime, and being picked up. If your child comprehends appropriately, he'll show signs that he understands by looking at an object, smiling, or interacting. "I find that kids who understand language but are not producing it can learn expressive language," says William Levinson, M.D., a developmental pediatrician at Children's and Women's Physicians of Westchester in Valhalla, New York. If a child understands words, with speech therapy he will eventually be able to express himself with words.
All content on this Web site, including medical opinion and any other health-related information, is for informational purposes only and should not be considered to be a specific diagnosis or treatment plan for any individual situation. Use of this site and the information contained herein does not create a doctor-patient relationship. Always seek the direct advice of your own doctor in connection with any questions or issues you may have regarding your own health or the health of others.<|endoftext|>
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# 0.3 Gravity and mechanical energy (Page 6/9)
Page 6 / 9
$KE=\frac{1}{2}m{v}^{2}$
Consider the $1\phantom{\rule{2pt}{0ex}}\mathrm{kg}$ suitcase on the cupboard that was discussed earlier. When the suitcase falls, it will gain velocity (fall faster), until it reaches the ground with a maximum velocity. The suitcase will not have any kinetic energy when it is on top of the cupboard because it is not moving. Once it starts to fall it will gain kinetic energy, because it gains velocity. Its kinetic energy will increase until it is a maximum when the suitcase reaches the ground.
A $1\phantom{\rule{2pt}{0ex}}\mathrm{kg}$ brick falls off a $4\phantom{\rule{2pt}{0ex}}m$ high roof. It reaches the ground with a velocity of $8,85\phantom{\rule{2pt}{0ex}}m·s{}^{-1}$ . What is the kinetic energy of the brick when it starts to fall and when it reaches the ground?
• The mass of the rock $m=1\phantom{\rule{2pt}{0ex}}\mathrm{kg}$
• The velocity of the rock at the bottom ${v}_{\mathrm{bottom}}=8,85\phantom{\rule{2pt}{0ex}}m·$ s ${}^{-1}$
These are both in the correct units so we do not have to worry about unit conversions.
1. We are asked to find the kinetic energy of the brick at the top and the bottom. From the definition we know that to work out $KE$ , we need to know the mass and the velocity of the object and we are given both of these values.
2. Since the brick is not moving at the top, its kinetic energy is zero.
3. $\begin{array}{ccc}\hfill KE& =& \frac{1}{2}m{v}^{2}\hfill \\ & =& \frac{1}{2}\left(1\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}\right){\left(8,85\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\right)}^{2}\hfill \\ & =& 39,2\phantom{\rule{0.166667em}{0ex}}\mathrm{J}\hfill \end{array}$
## Checking units
According to the equation for kinetic energy, the unit should be $\mathrm{kg}·m{}^{2}·s{}^{-2}$ . We can prove that this unit is equal to the joule, the unit for energy.
$\begin{array}{ccc}\hfill \left(\mathrm{kg}\right){\left(\mathrm{m}·{\mathrm{s}}^{-1}\right)}^{2}& =& \left(\mathrm{kg}·\mathrm{m}·{\mathrm{s}}^{-2}\right)·\mathrm{m}\hfill \\ & =& \phantom{\rule{0.166667em}{0ex}}\mathrm{N}·\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(\mathrm{because}\mathrm{Force}\left(\mathrm{N}\right)=\mathrm{mass}\left(\mathrm{kg}\right)×\mathrm{acceleration}\left(\mathrm{m}·{\mathrm{s}}^{-2}\right)\right)\hfill \\ & =& \phantom{\rule{0.166667em}{0ex}}\mathrm{J}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(\mathrm{Work}\left(\mathrm{J}\right)=\mathrm{Force}\left(\mathrm{N}\right)×\mathrm{distance}\left(\mathrm{m}\right)\right)\hfill \end{array}$
We can do the same to prove that the unit for potential energy is equal to the joule:
$\begin{array}{ccc}\hfill \left(\mathrm{kg}\right)\left(\mathrm{m}·{\mathrm{s}}^{-2}\right)\left(\mathrm{m}\right)& =& \phantom{\rule{0.166667em}{0ex}}\mathrm{N}·\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\hfill \\ & =& \phantom{\rule{0.166667em}{0ex}}\mathrm{J}\hfill \end{array}$
A bullet, having a mass of $150\phantom{\rule{2pt}{0ex}}g$ , is shot with a muzzle velocity of $960\phantom{\rule{2pt}{0ex}}m·s{}^{-1}$ . Calculate its kinetic energy.
• We are given the mass of the bullet $m=150\phantom{\rule{2pt}{0ex}}g$ . This is not the unit we want mass to be in. We need to convert to kg.
$\begin{array}{ccc}\hfill \mathrm{Mass}\phantom{\rule{3.33333pt}{0ex}}\mathrm{in}\phantom{\rule{3.33333pt}{0ex}}\mathrm{grams}÷1000& =& \mathrm{Mass}\phantom{\rule{3.33333pt}{0ex}}\mathrm{in}\phantom{\rule{3.33333pt}{0ex}}\mathrm{kg}\hfill \\ \hfill 150\phantom{\rule{3.33333pt}{0ex}}g÷1000& =& 0,150\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}\hfill \end{array}$
• We are given the initial velocity with which the bullet leaves the barrel, called the muzzle velocity, and it is $v=960\phantom{\rule{2pt}{0ex}}m·s{}^{-1}$ .
• We are asked to find the kinetic energy.
1. We just substitute the mass and velocity (which are known) into the equation for kinetic energy:
$\begin{array}{ccc}\hfill KE& =& \frac{1}{2}m{v}^{2}\hfill \\ & =& \frac{1}{2}\left(0,150\right){\left(960\right)}^{2}\hfill \\ & =& 69\phantom{\rule{0.166667em}{0ex}}120\phantom{\rule{0.166667em}{0ex}}\mathrm{J}\hfill \end{array}$
## Kinetic energy
1. Describe the relationship between an object's kinetic energy and its:
1. mass and
2. velocity
2. A stone with a mass of $100\phantom{\rule{2pt}{0ex}}g$ is thrown up into the air. It has an initial velocity of $3\phantom{\rule{2pt}{0ex}}m·s{}^{-1}$ . Calculate its kinetic energy
1. as it leaves the thrower's hand.
2. when it reaches its turning point.
3. A car with a mass of $700\phantom{\rule{2pt}{0ex}}\mathrm{kg}$ is travelling at a constant velocity of $100\phantom{\rule{2pt}{0ex}}\mathrm{km}·\mathrm{hr}{}^{-1}$ . Calculate the kinetic energy of the car.
## Mechanical energy
Mechanical energy is the sum of the gravitational potential energy and the kinetic energy.
Mechanical energy, $U$ , is simply the sum of gravitational potential energy ( $PE$ ) and the kinetic energy ( $KE$ ). Mechanical energy is defined as:
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how did you get the value of 2000N.What calculations are needed to arrive at it
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Dynamic Geometry: P96 Series
This note gives the common calculations in the solution for problems titled Dynamic Geometry: P96, 101, 106, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 125, 127, 128, and 129 so far by @Valentin Duringer. .
Let the chord dividing a circle with center at $O$ into two circular segments of heights $4$ and $9$ be $AB$. Diameter $DE$ through $O$ cut $AB$ perpendicularly at $F$. Then $DF = 4$, $FE = 9$, and diameter $DE = DF + FE = 13$, therefore radius of the circle is $6.5$. By intersecting chords theorem $AF \cdot FB = DF \cdot FE = 4 \cdot 9 \implies AF = FB = 6 \implies AB=12$. If we set $O$ as the origin $(0,0)$ of the $xy$-plane with $AB$ parallel to the $x$-axis, then $AB$ is along $y=2.5$, and $A = (-6, 2.5)$ and $B=(6,2.5)$.
Angles of Moving Vertices
Label the upper variable triangle as $ABC$ and the lower variable triangle $ABC'$. Since the inscribed angle $\angle AC'B$ and the central angle $\angle AOB$ both form the same arc $ACB$, $\angle AC'B = \frac 12 \angle AOB = \frac 12 \left(2 \tan^{-1} \frac {12}5 \right) = \tan^{-1} \frac {12}5$. Similarly, $\angle ACB = \pi - \tan^{-1} \frac {12}5$.
Side Lengths of Triangle
Since the diameter of a circumcircle is given by the sine rule:
$D = \frac {AC}{\sin \angle CBA} = 13 \implies AC = 13\sin \angle CBA$
Similarly, $BC = 13 \sin \angle CAB$, $AC' = 13\sin \angle C'BA$, and $BC' = 13 \sin \angle C'AB$,
The Largest Rectangle Inscribed by a Triangle
Consider any $\triangle ABC$ with a height $CD = h$ and a width of $AB=w$. Let the rectangle $\triangle ABC$ inscribes be $KLMN$ and the height of $\triangle CKL$ be $CE = \alpha h$, where $0 \le \alpha \le 1$. Then the area of rectangle $KLMN$, $A = KL \cdot LM$. Since $\triangle CKL$ and $\triangle ABC$ are similar $KL = \alpha \cdot AB = \alpha w$. And $LM = CD - CE = h - \alpha h$. Then $A = \alpha (1-\alpha) wh$. By AM-GM inequality, $2\sqrt{\alpha(1-\alpha)} \le \alpha + (1 - \alpha) = 1$, and equality occurs when $\alpha = 1 - \alpha \implies \alpha = \frac 12$. Then the rectangle has a maximum area $A_{\max} = \frac 14 wh$, when the height and breadth of the rectangle are half of that of the triangle.
The Square Inscribed by a Triangle
Let us consider the relationship between the side length $s$ of a square $KLMN$ to the height $CD=h$ and width $AB=w$ of the triangle $\triangle ABC$ that inscribes it. Note that $\triangle KLC$ and $\triangle ABC$ are similar. Let the height of $\triangle KLC$ be $CE$, then $CE = \dfrac swh$. From $CD = CE + ED$, we have
$h = \frac swh + s \implies s = \frac h{1+\frac hw} = \frac {wh}{h+w} = \frac {12h}{h+12} \quad \small \blue{\text{Since }w=AB=12}$
Similarly for the square in $\triangle ABC'$, $s' = \dfrac {12h'}{h'+12}$.
Note by Chew-Seong Cheong
3 months, 2 weeks ago
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Nice work sir !
- 3 months, 2 weeks ago
Actually, the coordinate system I use is different, I use point A(0,0)
- 3 months, 2 weeks ago<|endoftext|>
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# How to Find the Distance Between a Point & a Line
Coming up next: How to Find the Distance Between Parallel Lines
### You're on a roll. Keep up the good work!
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• 0:02 Choices
• 0:41 The Point and the Line
• 1:10 Using a Formula to Get…
• 3:23 Distance Without the Formula
• 6:38 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay
Gerald has taught engineering, math and science and has a doctorate in electrical engineering.
Finding the distance from a point to a line is useful in many science and engineering applications. In this lesson, we calculate this distance using a formula, as well as with some fundamental concepts from geometry and algebra.
## Choices
Choices! Buy the cake ready-made, or bake it from fundamental ingredients like flour, eggs, and milk? The issue might be time or availability. Maybe the store is closed, or maybe there isn't time to bake a cake from scratch.
Similar choices often exist in math: use a formula or get the answer from simpler ideas. In this lesson, we will calculate the distance from a point to a line using a formula, as well as determining this distance using fundamental ideas and equations. As with a cake, will the results be the same?
## The Point and the Line
Imagine a line and a point in two-dimensional space.
We could be calculating the distance from a road to an address or the distance from the end-zone line to the location of a player on the field. In all these cases, we have the equation of a line and the coordinates of a point.
For our example, the equation for the line is y = (1/2)x - 1 and the point (x1,y1) is located at the point (2, 3).
## Using a Formula to Get the Distance
Having the line equation in the form ax + by + c = 0 and knowing the coordinates of the point gives us the distance from the point to the line using the formula:
What if the equation of the line is in some other form? This is the situation in our example where the line y = (1/2)x - 1 is in slope-intercept form. Let's convert to the ax + by + c = 0 form. First, multiply both sides of the equation by 2 to give 2y = x - 2.
Then, transfer all the terms to the same side and organize the equation so the x term is first: x - 2y - 2 = 0.
Comparing this equation to ax + by + c = 0, we identify: a = 1, b = -2 and c = -2. The location of the point says x1 is 2 and y1 is 3. Substituting into the formula for the distance:
What if this distance formula was not available? Could we determine the distance using some ideas from geometry and algebra? It's like baking from fundamental ingredients as an alternative to buying the ready-made store product. For practice, let's do this. We'll take it one ingredient at a time.
## Distance Without the Formula
This will be fun! First, visualize a line perpendicular to our line. The perpendicular line makes a right-angle (90o) with our line. Also, we want the perpendicular line to pass through the point (x1, y1).
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Create the wonder of the solar system within any room with this motorised model. Young learners can view the workings of the solar system or create their very own planetarium!
- Brings the solar system to life for young astronomers
- Model features the Sun and eight planets
- Encourage children to observe how the planets orbit around the Sun
- Cover “the Sun” with the constellation dome to create a planetarium
- Major constellations are labelled
- The Sun doubles up as a night-light with auto shut off!
- Requires 4 AA batteries (not included)
- Features multilingual packaging and activity guide packed with interesting solar system facts
National Curriculum Reference:
Year 5: Science – Earth & Space:
Describes the movement of the Earth, and other planets, relative to the Sun in the solar system
Describes the movement of the Moon relative to the Earth
Describes the Sun, Earth and Moon as approximately spherical bodies
Uses the ideas of the Earth’s rotation to explain day and night and the apparent movement of the Sun across the sky
Extend the learning with our other solar system resources<|endoftext|>
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# Solve The Quadratic Equation By Factoring Worksheet
Assignment will be available soon
Solving the quadratic equation by factoring means breaking down the equation $$ax^2 + bx + c = 0$$ into simpler factors that multiply to give the original equation. Once factored, set each factor equal to zero and solve for $$x$$. This method helps find the solutions where the quadratic equation equals zero, making it easier to understand and solve.
Algebra 2
## How Will This Worksheet on “Solve the Quadratic Equation by Factoring” Benefit Your Student's Learning?
• Reinforces factoring skills.
• Enhances problem-solving abilities.
• Improves algebraic manipulation.
• Clarifies the zero product property.
• Provides ample practice opportunities.
• Builds confidence in solving equations.
• Prepares for tests and assessments.
• Encourages independent learning.
## How to Solve the Quadratic Equation by Factoring?
• Ensure the equation is in the form $$ax^2 + bx + c = 0$$.
• Factor the quadratic expression into two binomials such that their product equals zero.
• Set each binomial factor equal to zero.
• Solve each resulting linear equation for $$x$$.
## Solved Example
Q. Solve for $u$. $\newline$$u^2 + 8u + 16 = 0$$\newline$Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas. $\newline$$u = \_\_$
Solution:
1. Identify Quadratic Equation: Identify the quadratic equation to solve for $u$. The given equation is $u^2 + 8u + 16 = 0$. We need to find values of $u$ that satisfy this equation.
2. Factorization Check: Determine if the quadratic can be factored.$\newline$We are looking for two numbers that multiply to $16$ and add up to $8$.$\newline$The numbers $4$ and $4$ satisfy these conditions because $4 \times 4 = 16$ and $4 + 4 = 8$.
3. Write Factored Form: Write the factored form of the quadratic equation.$\newline$Since both numbers are $4$, the equation can be written as $(u + 4)(u + 4) = 0$.$\newline$This is also known as a perfect square trinomial.
4. Solve for $u$: Set each factor equal to zero and solve for $u$.$\newline$First, set $u + 4 = 0$.$\newline$Subtract $4$ from both sides to solve for $u$.$\newline$$u + 4 - 4 = 0 - 4$$\newline$$u = -4$
5. Final Solution: Since both factors are the same, we only get one solution for $u$. The solution is $u = -4$. There is no need to solve the second factor because it is identical to the first.
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Question of Exercise 2
# Question A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. The number of such cones is
Option 1 63
Option 2 126
Option 3 21
Represent Root 9 point 3 on the number line
Solution:
Explanation:
Step 1: Draw a line segment AB of length 9.3 units.
Step 2: Now, Extend the line by 1 unit more such that BC=1 unit .
Step 3: Find the midpoint of AC.
Step 4: Draw a line BD perpendicular to AB and let it intersect the semicircle at point D.
Step 5: Draw an arc DE such that BE=BD.
Hence, Number line of √ 9.3 is attached below.
Which one of the following statement is true
A: Only one line can pass through a single point.
B: There are an infinite number of lines which pass through two distinct points.
C: Two distinct lines cannot have more than one point in common.
D: If two circles are equal, then their radii are not equal.
Solution:
Explanation:
From one point there is an uncountable number of lines that can pass through.
Hence, the statement “ Only one line can pass through a single point” is false.
We can draw only one unique line passing through two distinct points.
Hence, the statement “There are an infinite number of lines which pass through two distinct points” is false.
Given two distinct points, there is a unique line that passes through them.
Hence, the statement “Two distinct lines cannot have more than one point in common” is true.
If circles are equal, which means the circles are congruent. This means that circumferences are equal and so the radii of two circles are also equal.
Hence, the statement “If two circles are equal, then their radii are not equal” is false.
The correct option is (C) Two distinct lines cannot have more than one point in common.
The class mark of the class 90-120 is
A: 90
B: 105
C: 115
D: 120
Solution:
Explanation:
To find the class mark of a class interval, we find the sum of the upper limit and lower limit of a class and divide it by 2 Thus,
Class -mark=Upper limit + Lower limit/2
Here, the lower limit of 90-120=90
And the upper limit of 90-120=120
So,
Class -mark=120+90/2
=210/2
=105
Hence, the class mark of the class 90-120 is 105
That is, option (B) is correct.
Option (B) 105 is correct.
ABCD is a parallelogram and AP
and CQ are perpendiculars from vertices A and C on diagonal BD Show that i)ΔAPBΔCQD ii) AP = CQ
Solution:
Find the roots of the following equation
A: - 1, - 2
B: - 1, - 3
C: 1,3
D: 1,2
Solution:
Explanation:-
Hence the correct option is (D) i.e. 1,2.<|endoftext|>
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# 5.08 Perimeter of composite shapes
Lesson
### Composing a shape
When a shape is made up, or composed, of separate shapes, it's called a composite shape. The lines where they join aren't usually visible, but we can still make out the separate shapes. We can use what we already know, including how to find the perimeter of a polygon, to calculate the perimeter of our composite shape.
When finding the perimeter of composite shapes there are two main approaches.
The first approach is finding the length of all the sides and adding them together like we would for an irregular shape. We can do this by using the lengths we are given to find any missing lengths.
The other approach is less obvious and relies on some visualization. We can see in the image below that the composite shape actually has the same perimeter as a rectangle.
So the perimeter of this composite shape can be calculated as:
Perimeter $=$= $2\times\left(8+13\right)$2×(8+13) $=$= $2\times21$2×21 $=$= $42$42
Careful!
When using this method it is important to keep track of any sides that do not get moved.
An example of a shape that we need to be careful with is:
Notice that we moved the indented edge to complete the rectangle but we still need to count the two edges that weren't moved.
We can calculate the perimeter of this shape as:
Perimeter $=$= $2\times\left(5+11\right)+2+2$2×(5+11)+2+2 $=$= $2\times16+4$2×16+4 $=$= $32+4$32+4 $=$= $36$36
With our knowledge of the perimeter of simple shapes like rectangles and squares we can often find creative ways to work out the perimeter of more complicated composite shapes.
#### Practice questions
##### Question 1
Consider the following figure.
1. Find the length $x$x.
2. Find the length $y$y.
3. Calculate the perimeter of the figure.
##### Question 2
Find the perimeter of the shape.
##### Question 3
Find the perimeter of the following figure. Use the $\pi$π button on your calculator, rounding your final answer to one decimal place.
### Outcomes
#### 8.1
Compare and order real numbers
#### 8.10
Solve area and perimeter problems, including practical problems, involving composite plane figures.<|endoftext|>
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# Trigonometry
• Mar 9th 2006, 02:21 PM
aussiekid90
Trigonometry
Find solutions of sin x=1/2 for the interbal (o, 2pie). What happens if th einterval changes to (negative pie, pie)? (0, 4pie), (0, 8pie)?
• Mar 9th 2006, 03:22 PM
Jameson
So start with the reference angle. sin(x)=1/2 at $\frac{\pi}{6}$. Now go into quadrant two, add $\frac{\pi}{2}$ and the sine graph is still positive. This means that sin(x)=1/2 at $\frac{2\pi}{3}$ as well. In the 3rd and 4th quadrants the sine graph is negative, so you just have two answers.
• Mar 10th 2006, 01:46 AM
ticbol
Quote:
Originally Posted by aussiekid90
Find solutions of sin x=1/2 for the interbal (o, 2pie). What happens if th einterval changes to (negative pie, pie)? (0, 4pie), (0, 8pie)?
sinX = 1/2
Positive sine, so angle X is in the 1st or 2nd quadrants.
X = arcsin(1/2) = 30deg = pi/6 rad.
For the interval (0,2pi):
X = pi/6 ---in the 1st quadrant, or,
X = pi -pi/6 = 5pi/6 ---in the 2nd quadrant
For the interval (-pi,pi):
So the interval starts at -pi, goes counterclockwise, passes -pi/2, passes 0, passes pi/2, and ends at pi.
When laid out in a straight line, like in the horizontal axis of a graph, the interval starts at -pi, going to the right, passes -pi/2, passes 0, passes pi/2, and ends at pi.
The -pi is the divider of the 3rd and 2nd quadrants. Likewise, the pi is the divider of 2nd and 3rd quadrants. Hence, the start and finish of the interval is the negative side of the horizontal axis.
[In the (0,2pi) interval, the positive side of the horizontal axis is the start and finish of the interval. But, in both intervals, since (...) were used, the -pi and pi, or the 0 and 2pi, are not included in their respective intervals.]
So, since the angle X is in the 1st and 2nd quadrants, then
For the interval (0,4pi):
X is still in the 1st or 2nd quadrants.
In the first revolution, from 0 to 2pi, X = pi/6 or 5pi/6.
In the second revolution, from 2pi to 4pi, X = (pi/6 +2pi) = 13pi/6, or (5pi/6 +2pi) = 17pi/6.
Therefore,<|endoftext|>
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# 5 Important Difference between Arithmetic and Geometric Sequence
What is the difference between arithmetic and geometric sequence?
A sequence is a set of numbers arranged in a particular order. These set of numbers are known as terms. The main types of sequence are arithmetic and geometric sequence.
The main difference between arithmetic and geometric sequence is that arithmetic sequence is a sequence where the difference between two consecutive terms is constant while a geometric sequence is a sequence where the ratio between two consecutive terms is constant.
Read More: Difference between Area and Perimeter
## Comparison Table (Arithmetic Sequence vs Geometric Sequence)
Basic Terms Arithmetic Sequence Geometric Sequence Meaning It is a sequence where the difference between two consecutive terms is a constant It is a sequence where the ratio between two consecutive terms is a constant How to Identify the Sequence The common difference between successive terms The common ratio between successive terms Mode of Operation Addition or Subtraction Multiplication or Division Variation of Terms Linear Exponential Infinite Sequence Divergent Either divergent or convergent
## What Is Arithmetic Sequence?
It is also known as arithmetic progression. It is a sequence where the difference in successive terms is constant.
An arithmetic progression is either added or subtracted. Besides that, it always occurs in a linear form.
Arithmetic sequence example is a, a+d, a+2d, a+3d, a+4d. Where a is the first term and d is a common difference.
Therefore, the arithmetic sequence formula is a + (n-1) d
Question Example
Identify the first term and calculate the common difference is the sequence. 3, 8, 13, 18, 23 . . .
a=3 d= second term – first term hence, 8-3 = 5
## What Is Geometric Sequence?
It is also known as geometric progression. It is a sequence where the ratio between successive terms is constant.
Geometric progression is either multiply or divide. Besides that, a geometric sequence occurs in exponential form.
The common ratio is a fixed and a non-zero number. For instance, 3, 6, 12, 24… The common ratio here is 2.
The geometric sequence is expressed as a, ar, ar², ar³, ar4 and so on. Where a is the first term and r is the common ratio.
Therefore, the geometric sequence formula is an=arn-1
Geometric sequence example 3, 9, 27, 81…
a=3 r=9/3 n= fifth term
Hence, an=3 X 35-1
The final result is 3 x 81= 243
Read More: Difference between Expression and Equation
## Main Difference between Arithmetic and Geometric Sequence
1. An arithmetic sequence is a list of numbers with successive terms having constant difference whereas geometric sequence is a list of numbers with successive terms having a constant ratio
2. An arithmetic sequence has a common difference whereas geometric sequence has a common ratio
3. The new term of an arithmetic sequence is either added or subtracted whereas that of a geometric sequence is either multiply or divided
4. Variation of members in arithmetic sequence is linear while those of geometric sequence is exponential
5. An infinite arithmetic sequence is divergent whereas that of a geometric sequence is either divergent or convergent
## Similarities between Arithmetic and Geometric Sequence
1. Both follow a strict pattern
2. Both have a constant quantity
3. Both tend to confuse students
Read More: Difference between Length and Height
## FAQs about Arithmetic and Geometric Sequence
1. How Are Arithmetic and Geometric Sequences Similar?
Both sequence have a constant quantity. This tends to confuse a lot of students while sitting for their exams.
1. Are Geometric Sequences Linear?
No. Geometric sequences are exponential functions such that the n-value increases by a constant value of one and the f (n) value increases by multiples of r.
1. Why Is It Called a Geometric Sequence?
It’s called a geometric sequence because the numbers go from one number to another by diving or multiplying by a similar value.
## Conclusion
The above comprehensive information about arithmetic and geometric sequence is quite enough to spearhead easier understanding.
However, these two sequences might appear similar in an examination setup and cause a lot of confusion. Doing more practise will help to resolve the problem.
Calculating questions relating to arithmetic sequence are way simple but those of geometric sequence tend to pose a lot of challenges.
More Sources and References<|endoftext|>
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September 10, 2024
Preparing for the Navodaya Exam requires a solid understanding of various mathematical concepts, and one such important topic is percentages. In this article “Percentage questions for Navodaya exam” different types of percentage questions that often appear in the Navodaya Exam are given. By familiarizing yourself with these question patterns and practicing our sample problems, you can gain the necessary skills and boost your overall performance in the exam.
we begins by the fundamentals of percentages, explaining how to calculate them, convert them into fractions and decimals, and apply them in various real-life scenarios. We’ll provide step-by-step explanations and examples with clarity. Furthermore, we explore the different types of percentage questions commonly encountered in the Navodaya exam, including percentage increase or decrease, profit and loss, discounts, and more.
# Percentage questions for Navodaya exam (Solved Examples)
(i) 120% is equal to …………………
Sol. 120 % = $\frac{120}{100}$ = 1.2
(ii) Express each of the following statements in the percentage form :
(a) 13 out of 20
(b) 21 out of 30
Solution: (a) $\frac{13}{20}\times&space;100=&space;13&space;\times&space;5=&space;65$
%
(b) $\frac{21}{30}\times&space;100=\frac{21}{3}\times&space;10=7\times&space;10=70$%
(iii) Express the following fractions as percent :
(a) $\frac{2}{300}$
(b) $\frac{65}{80}$
Sol. (a) $\frac{2}{300}\times&space;100=&space;\frac{2}{3}=0.67%$
(b) $\frac{65}{80}\times&space;100=&space;\frac{650}{8}=81.25$ %
(iv) Convert each of the following decimals into a percentage.
(a) 0.08
(b) 0.75
Solution:
(a) 0.08 = (0.08 x 100)% = 8%
(b) 0.75= 0.75 x100 % = 75%
(v) Find :
(a) 8% of 160km
(b)6% of 225
Solution:
(a) 8% of 160km =$\mathrm{\frac{8}{100}&space;\,&space;of\,&space;160km&space;=\frac{8}{100}&space;\times&space;160km&space;=&space;\frac{64}{5}km&space;=&space;12.8km}$
(b) 6% of 225 = $\frac{6}{100}\times&space;225=&space;\frac{27}{2}=&space;13.50$
(vi) What % of 85 is 17 ?.
Solution : Let x% of 85 is 17 . Then $\frac{x}{100}\times&space;85=17$
$\Rightarrow&space;85x&space;=&space;1700$
$\Rightarrow&space;x=&space;\frac{1700}{85}$
$\Rightarrow&space;x=&space;20%$
(vii) …………. % of 360 +15% of 820= 231. Find the missing number .
Solution :
Let the missing number be x. Then x% of 360 +15% of 820= 231
$\Rightarrow&space;\frac{x}{100}\times&space;360&space;+&space;\frac{15}{100}\times&space;820&space;=&space;231$
$\Rightarrow&space;3.6x&space;+&space;123&space;=231$
$\Rightarrow&space;3.6&space;x=&space;231-123&space;=108$
$\Rightarrow&space;x=&space;\frac{108}{3.6}=&space;30$
Hence the missing number is 30
(viii) (100% of ₹6) + ₹6 + (6% of ₹100) = ₹ ……..
Solution. (100% of ₹6) + ₹6 + (6% of ₹100) = $\frac{100}{100}\times&space;6+&space;6&space;+\frac{6}{100}\times&space;100$
= 6+6+6
=18
(ix) 50% of 50% of 1000 = ?
Solution:
50% of 50% of 1000 = $\frac{50}{100}\times&space;\frac{50}{100}\times&space;1000$
$=\frac{1}{2}\times&space;\frac{1}{2}\times&space;1000$
=250
(x) ₹3 + (100% of ₹3) + (3% of ₹100) = ?
Solution: ₹3 + (100% of ₹3) + (3% of ₹100) = $3+\frac{100}{100}\times&space;3+\frac{3}{100}\times&space;100$
= 3+3+3
=9
(xi) If 40% of 140 = x% of 200, then find the value of x .
Solution:
Given 40% of 140 = x% of 200
$\Rightarrow&space;\frac{40}{100}\times&space;140=&space;\frac{x}{100}\times&space;200$
$\Rightarrow&space;56&space;=2x$
So x= 28
(xii ) Reena deposits 60 per month in her post office saving bank account. If this is 10% of her monthly income, find her monthly income.
Solution: Let Reena’s monthly income is Rs. x . Then $10&space;%&space;\,&space;\,&space;of&space;\,&space;\,&space;x&space;=60$
$\Rightarrow&space;\frac{10}{100}\times&space;x&space;=&space;60$
$\Rightarrow&space;x=&space;600$
(xiii) (8% of 35) − (35% of 8) = ?
Solution :
(8% of 35) − (35% of 8) =$\frac{8}{100}\times&space;35-\frac{35}{100}\times&space;8$
=$\frac{35\times&space;8}{100}-\frac{8\times&space;35}{100}$
= 0
(xiv) $x%&space;\,&space;of&space;\,&space;330+50&space;%\,&space;of&space;\,&space;\,&space;600=597$
Solution : $x%&space;\,&space;of&space;\,&space;330+50&space;%\,&space;of&space;\,&space;\,&space;600=597$
$\Rightarrow&space;\frac{x}{100}\times&space;330&space;+\frac{50}{100}\times&space;600&space;=597$
$\Rightarrow&space;3.3x&space;+300&space;=597$
$\Rightarrow&space;x=&space;\frac{597-300}{3.3}&space;=\frac{297}{3.3}$
$\Rightarrow&space;x=90$
(xv) If 50% of 130 = x% of 1625, then what is the value of x?
Solution: Given 50% of 130 = x% of 1625
$\Rightarrow&space;\frac{50}{100}\times&space;130&space;=&space;\frac{x}{100}\times&space;1625$
$\Rightarrow&space;65=&space;\frac{1625x}{100}$
$\Rightarrow&space;x=\frac{65&space;\times&space;100}{1625}$
$\Rightarrow&space;x=&space;4$
## Percentage questions for Navodaya exam (Unsolved examples with answers)
1)Express each of the following as decimal :
(i)63% (ii) 9% (iii) 5%
2)Convert each of the following fractions into a percentage.
(ii) $\frac{9}{25}$ (iii) $2\frac{3}{5}$
(iii)$\frac{3}{4}$
3) 120% is equal to ………………..
4) What % of 75 is 105 ?
5)What is the number whose 12% is 54 ?
6) Write 20m as a percentage of a km .
7)Reena deposits 60 per month in her post office saving bank account. If this is 10% of her monthly income, find her monthly income.
8)40% of (100- 20% of 300) =……….
9)Find the value of x, if 3% of x is 9.
10)90% of 300% +30% of 90 =……….
11)Find the sum whose 20% is Rs.240.
12)80% of 240 is more than 35% of 400 by………….
13)A boy gets Rs.20 per month and spends 50% of it. How much does he save in 1 yr?
14)Out of 600 students 240 are girls. What is the percentage of girls?
15)40% of 20% is equal to ……..
16)50 is what per cent of 75?
17)(100% of 5) + (5% of 100) is equal to ………..
18) Ram bought a dress worth rs. 1500 in discount and it was marked as 3800. Then how much percent he got on discount?
19) Find the number whose 15% is 24 .
20) If 5% of X + 16% of 75 = 16. Find the value of X.
Ans. 1) (i) 0.63 (ii)0.09 (iii)0.075 2)(i) 36% (ii)260% (iii)75% 3)1.2
4)140% 5) 450 6)2% of a km 7) Rs. 600 8) 16 9) 300 10) 297
11) 1200 12)Rs. 52 13) Rs. 120 14) 40 15) 8% 16)$\frac{200}{3}%$
17) 10
18) 50% 19) 160 20) 80<|endoftext|>
| 5 |
3,673 |
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# Chapter 7: Factorization Exercise – 7.6
Solve:
4x2 + 12xy + 9y2
### Solution:
= (2x)2 + 2 x 2x x 3y + (3y)2
= (2x + 3y)2
= (2x + 3y) (2x + 3y)
### Question: 2
Solve:
9a2 – 24ab + 16b2
### Solution:
9a2 - 24ab + 16b2
= (3a)2 - 2 x 3a x 4b + (4b)2
= (3a - 4b)2
= (3a - 4b) (3a - 4b)
### Question: 3
Solve:
p2q2 – 6qr + 9r2 = (pq)2 - 2 x pq x 3r + (3r)2
### Solution:
p2q2 – 6qr + 9r2 = (pq)2 – 2 x pq x 3r + (3r)2
= (pq – 3r)2
= (pq – 3r) (pq – 3r)
Solve:
36a2 + 36a + 9
### Solution:
36a2 + 36a + 9
= 9 (4a2 + 4a + 1) = 9{(2a)2 + 2 x 2a x 1 + 12}
= 9 (2a + 1)2
= 9 (2a + 1) (2a + 1)
Solve:
a2 + 2ab + b- 16
### Solution:
a2 + 2ab + b- 16
= a2 + 2 x a x b + b2 – 16
= (a + b)2 – 42
= (a + b – 4) (a + b + 4)
### Question: 6
Solve:
9z2 - x2 + 4xy – 4y2
### Solution:
9z2 – x2 + 4xy – 4y2
= 9z2 - (x2 – 4xy + 4y2)
= 9z2 - [x2 – 2x × 2y + (2y)2]
= (3z)2- (x - 2y)2
= [3z - (x - 2y)] [3z + (x - 2y)]
= (3z - x + 2y) (3x + x -2y)
= (x - 2y + 3z) (-x + 2y + 3z)
### Question: 7
Solve:
9a4 - 24a2b2 + 16b4 - 256
### Solution:
9a4 - 24a2b2 + 16b4- 256
= (9a4 - 24a2b2 + 16b4) - 256
= [(3a2)2- 2 x 3a2 x 4b2 + (4b2)2] -162
= (3a2 - 402)2 -162
= [(3a2 – 4b2) -16] [(3a2 - 42) + 16]
= (3a2 - 4b2 -16) (3a2 - 4b2 + 16)
### Question: 8
Solve:
16 - a6 + 4a3b3 – 4b6
### Solution:
16 – a6 + 4a3b3 – 4b6
= 16 – (a6 – 4a3b3 + 4b6)
= 42 – [(a3)2 – 2 x a3 x 2b3 + (2b3)2]
= 42 – (a3 – 2b3)2
= [4 – (a3 – 2b3)] [4 + (a3 – 2b3)]
= (4 – a3 – 2b3) (4 + a3 – 2b3)
= (a3 – 2b3 + 4) (– a3 – 2b3 + 4)
### Question: 9
Solve:
a2 – 2ab + b2 – c2
### Solution:
a2 – 2ab + b2 – c2
= (a2 – 2ab + b2) – c2
= (a2 – 2 x a x b + b2) – c2
= (a – b)2 – c2
= [(a – b) – c][ (a – b) + c]
= (a – b – c) (a – b + c)
Solve:
x2 + 2x + 1- 9y2
### Solution:
x2 + 2x + 1 – 9y2
= (x2 + 2x + 1) – 9y2
= (x2 + 2× x x 1 + 1) – 9y2
= (x + 1)2 – (3y)2
= [(x + 1) – 3y] [(x + 1) – 3y]
= (x + 1 – 3y) (x + 1 + 3y)
= (x + 3y + 1) (x – 3y + 1)
Solve:
a2 + 4ab + 3b2
### Solution:
a2 + 4ab + 3b2
= a2 + 4ab + 4b2 – b2
= [a2 + 2 x a x 2b + (2b)2] – b2
= (a + 2b)2 – b2
= [(a + 2b) – b] [(a + 2b) + b]
= (a + 2b – b)(a + 2b + b)
= (a + b) (a + 3b)
Solve:
96 – 4x – x2
### Solution:
96 - 4x – x2
= 100 - 4 – 4x – x2
= 100 - (x2 + 4x + 4)
= 100 - (x2 + 2 x x x 2 + 22)
= 102 – (x + 2)2
= [10 – (x + 2)] [10 + (x + 2)]
= (10 – x – 2)(10 + x + 2)
= (8 – x) (12 + x)
= (x + 12) (-x + 8)
Solve:
a4 + 3a2 + 4
### Solution:
a4 + 3a2 + 4
= a4 + 4a2 – a2 + 4
= (a4 + 4a2 + 4) – a2
= [(a2)2 + 2 x a2 x 2 + 22] – a2
= (a2 + 2)2 – a2
= [(a2 + 2) – a][(a2 + 2) + a]
= (a2 – a + 2)(a2 + a + 2)
Solve:
4x4 + 1
### Solution:
4x4 + 1
= 4x4 + 4x2 + 1 – 4x2
= [(2x2)2 + 2 x 2x2 x 1 + 1] – 4x2
= (2x2 + 1)2 – (2x)2
= [(2x2 + 1) – 2x] [(2x2 + 1) + 2x]
= (2x2 – 2x + 1)( 2x2 + 2x + 1)
Solve:
4x4 + y4
### Solution:
4x4 + y4
= 4x4 + 4x2 + y4 – 4x2y2
= [(2x2)2 + 2 x 2x2 x y + (y2) 2] – (2xy)2
= (2x2 + y2)2 – (2xy)2
= [(2x2 + y2) – 2xy] [(2x2 + y2) + 2xy]
= (2x2 – 2xy + y2)( 2x2 + 2xy + y2)
### Question: 16
Solve:
(x + 2)2 – 6(x + 2) + 9
### Solution:
(x + 2)2 – 6(x + 2) + 9
= (x + 2)2 – 2 x (x + 2) x 3 + 32
= [(x + 2) – 3]2
= (x + 2 – 3)2
= (x – 1)2
= (x – 1)(x – 1)
### Question: 17
Solve:
25 – p2 – q2 – 2pq
### Solution:
25 – p2 – q2 – 2pq
= 25 – (p2 + 2pq + q2)
= 52 – (p2 + 2 x p x q + q2)
= 52 – (p + q)2
= [5 – (p + q)] [5 + (p + q)]
= (5 – p + q) (5 + p + q)
= - (p + q - 5)(p + q + 5)
### Question: 18
Solve:
x2 + 9y2 – 6xy – 25a2
### Solution:
x2 + 9y2 – 6xy – 25a2
=(x2 – 6xy + 9y2) – 25a2
= [x2 – 2 x x x 3y + (3y)2] – 25a2
= (x – 3y)2 – (5a)2
= [(x – 3y) – 5a][(x -3y) + 5a]
= (x – 3y – 5a)( x – 3y + 5a)
### Question: 19
49 – a2 + 8ab – 16b2
### Solution:
49 – a2 + 8ab – 16b2
= 49 – (a2 – 8ab + 16b2)
= 49 – [a2 – 2 x a x 4b + (4b2)]
= 72 – (a – 4b2)
= [7 – (a – 4b)][7 + (a – 4b)]
= (7 – a + 4b)( 7 + a – 4b)
= – (a – 4b – 7)(a – 4b + 7)
= – (a – 4b + 7)(a – 4b – 7)
### Question: 20
Solve:
a2 – 8ab + 16b2 – 25c2
### Solution:
a2 – 8ab + 16b2 – 25c2
= (a2 – 8ab + 16b2) – 25c2
= [a2 – 2 x a x 4b + (4b)2] – 25c2
= (a – 4b)2 – (5c)2
= [(a – 4b) – 5c] [(a – 4b)2 + 5c]
= (a – 4b – 5c) (a – 4b + 5c)
Solve:
x2 – y2 + 6y – 9
### Solution:
x2 – y2 + 6y – 9
= x2 – (y2 + 6y – 9)
= x2 – (y2 – 2 x y x 3 + 32)
= x2 – (y – 3)2
= [x – (y – 3)] [x + (y – 3)]
= (x – y + 3)(x + y - 3)
### Question: 22
Solve:
25x2 – 10x + 1 – 36y2
### Solution:
25x2 – 10x + 1 – 36y2
= (25x2 – 10x + 1) – 36y2
= [(5x)2 – 2 x 5x x 1 + 1] – 36y2
= (5x – 1)2 – (6y)2
= [(5x – 1) – 6y] [(5x – 1) + 6y]
= (5x – 1 – 6y)( 5x – 1 + 6y)
= (5x – 6y – 1)( 5x + 6y – 1)
### Question: 23
Solve:
a2 – b2 + 2bc – c2
### Solution:
a2 – b2 + 2bc – c2
= a2 – (b2 – 2bc + c2)
= a2 – (b2 – 2 x b x c + c2)
= a2 – (b – c)2
= [a – (b – c)][ a + (b – c)]
= (a – b + c)(a + b – c)
### Question: 24
Solve:
a2 + 2ab + b2 – c2
### Solution:
a2 + 2ab + b2 – c2
= (a2 + 2ab + b2) – c2
= (a2 + 2 x a x b + b2) – c2
= (a + b)2 – c2
= [(a + b) – c] [(a + b) + c]
= (a + b – c) (a + b + c)
### Question: 25
Solve:
49 – x2 – y2 + 2xy
### Solution:
49 – x2 – y2 + 2xy
= 49 – (x2 + 2xy – y2)
= 72 – (x – y)2
= [7 – (x – y)] [7 + (x – y)]
= (7 – x + y)(7 + x – y)
= (x – y + 7)(y – x + 7)
### Question: 26
Solve:
a2 + 4b2 – 4ab – 4c2
### Solution:
a2 + 4b2 – 4ab – 4c2
= (a2 + 4b2 – 4ab) – 4c2
= [a2 – 2 x a x 2b + (2b)2] – 4c2
= (a – 2b)2 – (2c)2
= [(a – 2b) – 2c] [(a – 2b) + 2c]
= (a – 2b – 2c)(a – 2b + 2c)
### Question: 27
Solve:
x2 – y2 – 4xz + 4z2
### Solution:
x2 – y2 – 4xz + 4z2
= (x2 – 4xz + 4z2) – y2
= (x – 2z)2 – y2
= [(x – 2z) – y] [(x – 2z) + y]
= (x – 2z – y)(x – 2z + y)
= (x + y – 2z)(x – y – 2z)<|endoftext|>
| 4.46875 |
420 |
889 minus 18 percent
This is where you will learn how to calculate eight hundred eighty-nine minus eighteen percent (889 minus 18 percent). We will first explain and illustrate with pictures so you get a complete understanding of what 889 minus 18 percent means, and then we will give you the formula at the very end.
We start by showing you the image below of a dark blue box that contains 889 of something.
889
(100%)
18 percent means 18 per hundred, so for each hundred in 889, you want to subtract 18. Thus, you divide 889 by 100 and then multiply the quotient by 18 to find out how much to subtract. Here is the math to calculate how much we should subtract:
(889 ÷ 100) × 18
= 160.02
We made a pink square that we put on top of the image shown above to illustrate how much 18 percent is of the total 889:
The dark blue not covered up by the pink is 889 minus 18 percent. Thus, we simply subtract the 160.02 from 889 to get the answer:
889 - 160.02
= 728.98
The explanation and illustrations above are the educational way of calculating 889 minus 18 percent. You can also, of course, use formulas to calculate 889 minus 18%.
Below we show you two formulas that you can use to calculate 889 minus 18 percent and similar problems in the future.
Formula 1
Number - ((Number × Percent/100))
889 - ((889 × 18/100))
889 - 160.02
= 728.98
Formula 2
Number × (1 - (Percent/100))
889 × (1 - (18/100))
889 × 0.82
= 728.98
Number Minus Percent
Go here if you need to calculate any other number minus any other percent.
890 minus 18 percent
Here is the next percent tutorial on our list that may be of interest.<|endoftext|>
| 4.65625 |
4,566 |
# What Is The Net Force Acting On A 1-kg Ball In Free Fall? - Premier Children's Work (2023)
The greatest physicist of the 20th century, Albert Einstein, discovered that gravity is actually a manifestation of geometry.
He showed that as objects get closer to one another, they experience a force that pulls them together. This force is due to something called spacetime curvature, which is an effect of the object’s distribution of mass and relative velocity to other objects.
As objects get farther away from each other, they experience a force that pushes them apart. This too is due to spacetime curvature.
What if we could measure this spacetime curvature? Then we would know the strength of gravitational force! We would have achieved one of the fundamental principles of physics: To measure force (which is related to physics) in terms of physics (that is, in terms of itself).
In this article, we will discuss how to do just that using an experiment you can do in your own home. First, let’s discuss what ball bearings are and where you can find them.
## Calculate the mass of the ball
The mass of the ball can be calculated by finding the volume of the ball and multiplying that by the material density of the ball. A one-kilogram ball made of rubber with a volume of one liter has a mass of one kilogram.
To find the net force acting on the ball, you must first calculate its velocity. To do this, you need to know how long it takes for the ball to fall and how far it has fallen. You then calculate how much time it took for the ball to fall that far and use some math to find its velocity.
You can now add up the gravitational force and the opposition force to find the net force acting on the ball. The opposition force is anything other than gravity pulling on the ball, like air resistance.
## Determine the center of mass of both objects
The first step in solving this problem is to determine the center of mass (also called the centroid) of both objects. The center of mass is the average position of all the masses in an object.
In this problem, you need to find the center of mass of the 1-kg ball and the 2-kg ball. To find the center of mass of a ball, you need to know its shape and its weight distribution.
You can find the weight distribution of a ball by putting it on a scale and looking at what fraction of its total weight is due to the ball itself. In other words, how much does the ball weigh relative to how much does the box weigh?
To solve this problem, we will first solve for one object, then add that solution onto the solution for the other object. This way, we can find both objects’ centers of mass.
## Find the distance between the objects
Next, you need to find the distance between the objects. In this case, you are looking for the distance between the ball and the floor, which is just under one meter.
You already know that gravity is pulling the ball down at 9.8 m/s², so all you have to do is subtract that from the height of one meter. This gives you how far down the ball would be on the floor.
You can’t just add or subtract velocity values due to math complications, so instead you need to find out what fraction 9.8 m/s² is of 1 m and then multiply or divide those values accordingly.
You want to end up with a number less than one, since you are finding how far down the ball would be on the floor.
## Calculate gravitational force between objects
The gravitational force between two objects is the product of the mass of each object and the other object’s proximity to each other.
To calculate the force between two objects, you must know both objects’ masses and their distance apart. However, since you are only given one of these variables in this problem, you must use a different method to determine the gravitational force.
You can calculate the gravitational force between two objects using geometry. You first need to understand what perpendicular means in this context. Perpendicular simply means straight up and down.
You can draw a line perpendicular to the floor that goes straight up and hits the ball (or any object) in question. By doing this, you have determined the ball’s own mass as being 1 kg.
## Calculate air resistance force between objects
Air resistance, also called drag, is the force that is exerted on a moving object due to collisions with other particles in the surrounding medium.
In the case of falling objects, the surrounding medium is air. The air pushes on the falling object, slowing it down. This is why a bullet shot straight up falls slower than a ball dropped from the same height.
Air resistance depends on several factors, including density, flow velocity of the surrounding air, and surface roughness. Because of this, calculations of air resistance vary depending on what parameters are used.
For instance, if a falling object has very smooth surfaces and passes through very slow moving air then its drag force will be lower than for another object that has rougher surfaces or moves through faster moving air.
To get an accurate calculation of the net force acting on a 1-kg ball in free fall we need to calculate the total drag force due to air resistance and subtract this from the weight.
## Find acceleration due to gravity
The next step is to determine the acceleration of the ball due to gravity, or what physicists call g. To do this, you have to find out what net force acts on the ball.
You know the weight of the ball, so you can determine the magnitude of the weight force acting on the ball. The weight acts in a downward direction, so that means it is a negative force.
You will have to find the net force acting on the ball by adding up all of the forces acting on it. One way to do this is to draw a free-body diagram of the ball. You can do this by drawing a rectangle around the ball with its top and bottom exposed, and drawing in arrows indicating which way each part of the ball is moving.
There are many ways to draw a free-body diagram, so try experimenting with different sizes and shapes for best results.
## Compare with other situations where gravity is present
A second way to understand what is happening when a ball and the table are both in free fall is to compare this situation with situations where gravity is present but there is no motion.
For example, imagine a 1-kg ball sitting on a tabletop that does not move. Because the ball and tabletop are not moving, it takes effort to pull them apart. This requires work, which means it costs energy.
The harder you try to pull the ball off the table, the more work it takes. Because work is energy, this means that pulling the ball off the table takes energy.
There is a force acting on the ball and tabletop that keeps them stuck together: gravitational force. The bigger gravitational force is, like on a bigger planet or in a different dimension where there is more gravity, it will be harder to pull the ball off the table.
## Make a diagram for visualizing net force
A simple way to understand what net force means is to create a diagram of a situation where force is involved.
Imagine a 1-kg ball sitting still on a smooth surface. According to physics, nothing can hold this ball in place, so it will slowly start to move either up, down, left, or right.
To illustrate this concept, draw a picture of the situation. Place the ball in the middle of the page and draw lines extending from the ball to all directions on the surface.
These lines represent what physics calls forces. Forces are what make something move, or change its speed or direction of movement.
## FAQs
### What Is The Net Force Acting On A 1-kg Ball In Free Fall? - Premier Children's Work? ›
Answer and Explanation: The net force acting on 1-kg ball in free fall is 9.8 N.
What is the net force acting on a 1 kg ball in free fall what is the net force if it encounters 2 N of air resistance? ›
Answer and Explanation: The net force acting on a falling 1kg ball if it encounters 2N of air resistance is -8N.
What is the net force acting on a 1kg? ›
On Earth, an object with a mass of 1kg will experience a force of 10N due to gravity, i.e. the weight of a 1kg mass is 10N.
What is the net force acting on a 1 kg ball in freefall quizlet? ›
What is the net force acting on a falling 1 kg ball if it encounters 2N of air resistance? 2), which is equal to 10 N. Air resistance is a force in the opposite direction to a falling weight so the net force acting on the ball is 10 N - 2 N = 8 N.
What is the net force acting on a 1.0 kg ball moving at a constant speed? ›
Since the body is moving with a uniform velocity which is not changing,due to Newton's first law,net force acting on it is zero.
What is the net force N acting on a 1 kg ball in free fall? ›
Answer and Explanation: The net force acting on 1-kg ball in free fall is 9.8 N. A body in free fall motion experiences free fall acceleration. This acceleration is by virtue of the force of gravity, which pulls every object towards the center of the Earth.
When a 1 N force acts on a 1kg object that is able to move freely the object receives? ›
So, the object has an acceleration of 1 m/s2.
How much is the force needed to have a 1 kg mass? ›
A newton is defined as 1 kg⋅m/s2 (it is a derived unit which is defined in terms of the SI base units). One newton is therefore the force needed to accelerate one kilogram of mass at the rate of one metre per second squared in the direction of the applied force.
What is the force acting on a mass of 1 kg due to? ›
Coming to the concept of weight, weight is the force acting on a body at the earth's surface. It is a product of mass times the acceleration due to gravity (g). Its unit is Newton (N). So 1 Kg Wt is referred to as the force acting on a body of 1 Kg at the surface of earth.
How much force is required for mass of 1 kg? ›
The force required to move a mass of 1 kg at rest on a horizontal rough plane μ = 0.1andg = 9.8 ms 2 is.
### How do you find the net force of a ball? ›
The net force formula sums the forces acting on an object. Thus, the net force formula is as follows: Fnet = F1 + F2 + F3.... The direction of the net force is determined by the sign.
What is the net force acting on the ball? ›
A Formula of Force
As a sentence, "The net force applied to the object equals the mass of the object multiplied by the amount of its acceleration." The net force acting on the soccer ball is equal to the mass of the soccer ball multiplied by its change in velocity each second (its acceleration).
What is the force of a ball in free fall? ›
That is to say that any object that is moving and being acted upon only be the force of gravity is said to be "in a state of free fall." Such an object will experience a downward acceleration of 9.8 m/s/s.
What is the net force acting on a 5 kg ball in free fall? ›
Therefore throughout the motion of ball net downward force acts on it and its magnitude are equal to mass times acceleration due to gravity. Therefore net force acting on the ball is 49.05N. 49.05 N .
What is the net force acting on 1 kg and 2 kg? ›
The net fore on 1 Kg and 2 Kg is 2 N and 4 N respectively. Explanation: Both block move together. Thus the net fore on 1 Kg and 2 Kg is 2 N and 4 N respectively.
What is the net force on a ball of mass 15kg? ›
The force of gravity on the ball is: \displaystyle F_g = mg = 15kg (10\frac{m}{s^2})= 150 N. These forces oppose each other, so we can say: \displaystyle F_{net} = F_b - F_g = 2000N - 150N = 1850N.
What is the force acting on two balls of 1 kg each are placed with their Centres 1 m apart? ›
⇒F=6. 67×10−11N.
How do we find the net force acting on an object *? ›
The net force is the vector sum of all the forces that act upon an object. That is to say, the net force is the sum of all the forces, taking into account the fact that a force is a vector and two forces of equal magnitude and opposite direction will cancel each other out.
What acceleration can a 1 N force give to a 1 kg object? ›
One Newton is defined as the amount of force required to give a 1-kg mass an acceleration of 1 m/s/s.
What is the acceleration of an object with 1kg and 1 N? ›
One Newton is the force needed to cause a 1-kg object to accelerate at 1 m/s2.
### What is the value of force on 1 kg object on Earth with units? ›
F=GM×mR2=6.67×10−11×6×1024×1(6.4×106)2 = 9.77 N = 9.8 N (approx). Q. What is the magnitude of the gravitational force between the earth and a 1 kg object on its surface? (Mass of the earth is 6 × 1024 kg and radius of the earth is 6.4 × 106 m).
What is the weight in Newtons of a 1 kg object? ›
The kilogram is the SI unit of mass and it is the almost universally used standard mass unit. The associated SI unit of force and weight is the Newton, with 1 kilogram weighing 9.8 Newtons under standard conditions on the Earth's surface.
How do you calculate force from kg? ›
The force formula is defined by Newton's second law of motion: Force exerted by an object equals mass times acceleration of that object: F = m ⨉ a. To use this formula, you need to use SI units: Newtons for force, kilograms for mass, and meters per second squared for acceleration.
How much force does it take to push 1kg? ›
One Newton is the force required to accelerate one kilogram of mass at 1 meter per second per second.
When a 1 newton force acts on a 1kg body? ›
One newton is that force which when acting on a body of mass 1 kg, produces an acceleration of 1 m s-2 in the direction of force.
What is 1 kg of mass? ›
The kilogram (also kilogramme) is the base unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially. It means 'one thousand grams'.
How do you calculate force in kg or g? ›
Fg (the force of gravity) is m x g (acceleration of gravity), in m/(s squared), so g is Fg / m = 123 N / 25 kg ~= 4.92 m/(s squared).
What is the formula for net force with weight? ›
The equation Fnet=ma F net = m a is used to define net force in terms of mass, length, and time. As explained earlier, the SI unit of force is the newton. Since Fnet=ma, 1N=1kg⋅m/s2.
What is the net force acting on the ball quizlet? ›
The net force on the ball is the force due to gravitation. b. The net force equals zero because the ball is not accelerating in any direction.
When a ball falls down it may have a net force? ›
When a ball falls downward, it may have a net force: of zero.
### What is net force acting examples? ›
When we kick a soccer ball, then the ball takes off and moves through the air. Then, there is a net force acting on the ball. Again when the ball starts to come back to the ground and eventually stops, there is also a net force acting on the ball.
How do you calculate the force of a falling object? ›
The motion of a free falling object can be described by Newton's second law of motion, force (F) = mass (m) times acceleration (a). We can do a little algebra and solve for the acceleration of the object in terms of the net external force and the mass of the object ( a = F / m).
What is the net force on the object when it is in free fall? ›
The object is in a state of free fall and its acceleration is the free fall acceleration value - g. Since the force of gravity is the only force acting upon the object, the net force is equal to the force of gravity - m•g.
What happens to a ball in free fall? ›
Acceleration from gravity is always constant and downward, but the direction and magnitude of velocity change. At the highest point in its trajectory, the ball has zero velocity, and the magnitude of velocity increases again as the ball falls back toward the earth (see figure 1).
What is free fall formula? ›
v²= 2gh. v=gt.
What is the net force of 5kg? ›
If you are asking for the net force in the direction of acceleration , the net force has a magnitude ma =5kg x 2m/s^2 =10N. Newtons second law of motion relates acceleration to the net force (the resultant of a number of forces).
What is the force of 2 kg? ›
Mass of body = 2 kg. Force = 2 × 4 = 8 Newton.
How to calculate net force of a falling object with air resistance? ›
The net external force is equal to the difference between the weight and the drag forces (F = W - D). The acceleration of the object then becomes a = (W - D) / m . The drag force depends on the square of the velocity. So as the body accelerates its velocity (and the drag) will increase.
What is the net force acting on an object in free fall? ›
A freely falling object experiences weightlessness as the only force acting on it is the force of gravity which acts downwards. The object's acceleration is equal to the acceleration due to gravity. Hence, the net force acting on the object is zero.
What is the net force acting on a 5kg ball in free fall? ›
Therefore throughout the motion of ball net downward force acts on it and its magnitude are equal to mass times acceleration due to gravity. Therefore net force acting on the ball is 49.05N. 49.05 N .
### How do you calculate free fall force? ›
vf = g * t
where g is the acceleration of gravity. The value for g on Earth is 9.8 m/s/s. The above equation can be used to calculate the velocity of the object after any given amount of time when dropped from rest.
What is the formula for net force acting on an object? ›
The net force formula sums the forces acting on an object. Thus, the net force formula is as follows: Fnet = F1 + F2 + F3.... The direction of the net force is determined by the sign.
What is the force of gravity on a 5kg ball? ›
Given, mass of ball m=5 kg. Gravitational force on the ball; F=mg=5×10=50 N.
What is the net force on an object in? ›
The net force on an object is the combined effect (the sum) of all the pushing and pulling forces actually acting on the object. If the forces pushing or pulling on an object are not balanced (a net force acts) then the object will accelerate in the direction of the net force.
Is the net force on an object is zero? ›
The net force is the vector sum of all the forces acting on an object. When an object is in equilibrium (either at rest or moving with constant velocity), the net force acting on it zero.
Can a net force be 0 and the object is moving? ›
Yes, when the object is moving with a constant velocity, the net force on it must be zero.
## References
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