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Obesity has historically been associated with wealth and indulgence, and poverty with starvation. However, now in the 21th Century people who live in poverty are more likely to be obese, especially in the developed world.
Obesity is a global crisis with 39% of adults being overweight or obese in 2016. Poverty plays a role in this as evidence has shown that poorer people are more likely to be food insecure and that households that are food insecure have the highest BMI and obesity prevalence. Food security is when a person always has availability and adequate access to sufficient, safe, nutritious food to maintain a healthy and active lifestyle. The two major ways that poverty lead to food insecurity are lack of availability and lack of access to healthy food.
Food deserts – lack of availability
A ‘food desert’ describes an area that lacks access to fresh food. These food deserts are associated with lower quality diets, higher obesity rates and are usually in deprived areas. A contributing factor to these food deserts is that there are fewer supermarkets in deprived areas. This may be due to some characteristics of deprived areas deterring brands from building there.
Examples of deterring characteristics:
– Increased levels of violence
– Inadequate transport infrastructure
– Zoning impediments
Poorer people are less likely to have a large supermarket in their local area and are less likely to be able to travel to one as owning a car or using public transport is expensive. This means that poorer people are reliant on smaller, more expensive shops. The reason that shopping at smaller shops costs more money is because they tend not to carry own-brand foods, which are cheaper, and they have fewer deals. This is an example of a poverty premium – where poorer consumers have to pay more for products than richer consumers.
Expense of healthy food – lack of access
Evidence has shown that healthier food is more expensive, and it has been getting more expensive over time. A study in 2012 demonstrated this – it found that per 1,000 calories unhealthy food cost just £2.50, but healthy food cost £7.50. This seems shocking, so for a practical example, let’s consider snacks.
An example of a healthy snack is an apple – a serving being 1 apple.
An example of an unhealthy snack is a bourbon biscuit – a serving being 2 biscuits.
As clearly shown, bourbons, even though they are the unhealthier option, are much cheaper and provide a higher number of calories. This is often the case with food, so people on a tighter budget are forced to buy more unhealthy food in order to get the number of calories they need in a day.
Solutions sometimes don’t help
Some solutions to help people that are food insecure also provide unhealthy options. For example, The Trussell Trust, which is the main food bank in the UK, include on their website a list of what is usually found in a food parcel.
– Tinned tomatoes/ pasta sauce
– Tinned meat
– Tinned vegetables
– Tinned fruit
– Lentils, beans and pulses
– UHT milk
– Fruit juice
A lot of the foods provided are processed carbohydrates or long-life food. The food tends not to be fresh as they want the food to last as long as possible, but this means that the food tends to be less healthy and more likely to feed into the problem of obesity.
Problems caused by obesity
Obesity leads to a variety of problems and reduces life expectancy by an average of 3-10 years. It is estimated that obesity contributes to at least 1 out of 13 deaths in Europe. Here are some examples of the problems obesity can lead to:
Day to day problems – breathlessness, snoring, excessive sweating, joint and back pain
Psychological – low confidence and self-esteem, feeling isolated, depression, bullying
Serious health issues – type 2 diabetes, high blood pressure, cancer, asthma, reduced fertility, high cholesterol
Poverty is not the only cause for obesity. People make their own choices about exercise and food they consume, and genetics can play a role in a person’s size. However, since the effects of obesity can be so life-changing and damaging, people should not be in the position where they are less able to choose healthy food and therefore are more likely to become obese.
Right to food and food justice
The Universal Declaration of Human Rights includes a right to food. The Food and Agriculture Organization formulated guidelines in 2004 to implement this right to food and the goals include removing food insecurity and malnutrition. Malnutrition includes both over and undernutrition. As discussed earlier, currently not every person in the world is living with the types of food they have a right to. This has led to a movement called food justice, which fights for everyone to have achieved right to food. Food justice has 5 aspects to it: trauma/inequity, land, labour, exchange and democratic process. The one that is involved in this issue is inequity – the fight to end injustices, in this case the inequality between the rich and poor’s access to healthy food.
Currently if a person is living in poverty, they are also more likely to be obese and food insecure. This is due to lack of availability and access, especially due to location and higher cost of healthy food. Sometimes even some solutions to food insecurity do not provide adequate healthy food. As obesity can lead to such severe complications, more focus needs to be put on making healthy food available to people from all socio-economic backgrounds, especially considering that right to food is a fundamental human right.<|endoftext|>
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Democratic Politics (NTSE/Olympiad)
3. Constitution Design
Importance of the Preamble to the Constitution of India
The Constitution of the Republic of India is introduced to us through a wisely formulated Preamble. It is an introductory part of the Constitution, though not a legal section of the Constitution. No one can sue the government in the Court of Law and can say that the government has not enforced the Preamble. Still it has great importance of its own because it shows the way the government ought to run and the kind of system the Constitution wishes to set up in India. It makes the intentions of our Constitution quite clear through the following point :
1. It declares India to be Sovereign Socialist-Secular, Democratic Republic.
2. It envisages justice - Social, Economic and Political - for all the citizens of the republic.
3. It would ensure all types of freedom necessary for the individual i.e., freedom of thought and expression, freedom of faith, freedom of belief and of worship, etc.
4. It would strive for equality of status and opportunity to all individuals and safeguard their dignity irrespective of their religious belief or sect.
5. It would promote a sense of brotherhood among the citizens.
6. Unity of the nation would be the hallmark of the efforts of the Government.
Thus, we find that the Preamble to our Constitution is the real index to the provisions-to-come in the Constitution of the Republic of India.
If you want to give information about online courses to other students, then share it with more and more on Facebook, Twitter, Google Plus. The more the shares will be, the more students will benefit. The share buttons are given below for your convenience.<|endoftext|>
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A hash function takes data (like a string, or a file’s contents) and outputs a hash, a fixed-size string or number.
For example, here’s the MD5 hash (MD5 is a common hash function) for a file simply containing “cake”:
And here’s the hash for the same file after it was edited to be “cakes”:
Notice the hash is completely different, even though the files were similar. Here's the hash for a long film I have on my hard drive:
The hash is the same length as my other hashes, but this time it represents a much bigger file—461Mb.
We can think of a hash as a "fingerprint." We can trust that a given file will always have the same hash, but we can't go from the hash back to the original file. Sometimes we have to worry about multiple files having the same hash value, which is called a hash collision.
Some uses for hashing:
- Dictionaries. Suppose we want a list-like data structure with constant-time lookups, but we want to look up values based on arbitrary "keys," not just sequential "indices." We could allocate a list, and use a hash function to translate keys into list indices. That's the basic idea behind a dictionary!
- Preventing man-in-the-middle attacks. Ever notice those things that say "hash" or "md5" or "sha1" on download sites? The site is telling you, "We hashed this file on our end and got this result. When you finish the download, try hashing the file and confirming you get the same result. If not, your internet service provider or someone else might have injected malware or tracking software into your download!"<|endoftext|>
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# What is 18 5 as a mixed number?
Category: What
Author: Daisy Vargas
Published: 2019-03-27
Views: 347
## What is 18 5 as a mixed number?
18 5 as a mixed number would be read as eighteen and five sixths. It is written as 18 5/6. This is a mixed number because it is a combination of a whole number, 18, and a fraction, 5/6. The numerator, 5, represents the number of parts being considered, and the denominator, 6, represents the total number of parts in the whole. In this case, 5 out of 6 parts of the whole are being considered.
## How do you convert 18 5 to a mixed number?
There are a few steps in converting from an improper fraction to a mixed number. In this case, we are converting 18 5 . Convert 18 5 to a mixed number by following these steps:
1) First, determine the greatest common factor (GCF) of the numerator and denominator. The GCF of 18 and 5 is 1.
2) Next, divide the GCF into the numerator and denominator. 18 ÷ 1 = 18 and 5 ÷ 1 = 5.
3) Now, divide the numerator by the denominator. 18 ÷ 5 = 3 with a remainder of 3.
4) The final step is to write the answer as a mixed number. The whole number is 3 and the remainder is 3, so the answer is 3 3 .
## What is the sum of 18 5 as a mixed number?
The sum of 18 and 5 is a mixed number. The sum of 18 and 5 is 23. The mixed number is 23/5. The sum of 18 and 5 is a mixed number because the sum is not an integer. The sum is a mixed number because the sum is not an integer.
## What is the difference of 18 5 as a mixed number?
18 5 as a mixed number is actually 19 5, which is a different number altogether. The extra one in the mixed number indicates that there is an extra whole number in the final sum. In other words, when you add 18 5 together, you are really adding 19 5, or twenty-four.
## What is the product of 18 5 as a mixed number?
In mathematics, a mixed number is a number that consists of a whole number and a fraction. A mixed number is written as a whole number followed by a fraction. For example, the mixed number twelve and a half can be written as 12 + 1/2.
The product of 18 5 as a mixed number is 90 + 2/5.
## What is the quotient of 18 5 as a mixed number?
When we are dealing with mixed numbers, we are dealing with a whole number and a fraction. In this case, the whole number is 18 and the fraction is 5. To find the quotient, we need to divide the whole number by the fraction. In this case, we would divide 18 by 5. The answer would be 3 5/5, or 3 1/5.
## Is 18 5 a mixed number?
Most people would say that 18 5 is not a mixed number because it is an improper fraction. An improper fraction is a fraction where the numerator (top number) is greater than the denominator (bottom number). In order to make an improper fraction into a mixed number, you would need to divide the numerator by the denominator. For example, if someone had the improper fraction ¾, they could divide 3 by 4 to get the mixed number ¾, which would be equal to 0.75.
While some might say that 18 5 is not a mixed number, others could argue that it is. This is because, when looked at as a whole, the number 18 5 is equal to 18.5. In this case, 5 would be the numerator and 18 would be the denominator. Therefore, when divided, 5 would go into 18 three times, with a remainder of 3. This would make the mixed number 18 5 equal to 18.5.
Ultimately, whether or not 18 5 is considered a mixed number is up to interpretation. However, it is generally agreed upon that 18 5 is not a mixed number, but is instead an improper fraction.
## What is the simplified form of 18 5 as a mixed number?
18 5 can be simplified to 3 5. This can be done by taking the 5 away from 18 and giving it to the 3, making 3 8. Then, 5 can be subtracted from 8, giving 3 3.
## What is an equivalent mixed number for 18 5?
An equivalent mixed number for 18 5 is 20. To get this answer, we need to first understand what a mixed number is. A mixed number is a whole number and a fractional part combined. In this case, the whole number is 18 and the fractional part is 5. To find an equivalent mixed number for 18 5, we need to add the whole number and the fractional part together. This gives us a sum of 23. We then need to find a fraction that is equal to 23. The easiest way to do this is to find a fraction with a denominator of 10. The numerator of this fraction will be 23. Therefore, an equivalent mixed number for 18 5 is 20.
## Related Questions
### What is a mixed number in math?
A mixed number is a whole number and a proper fraction combined, i.e. one and three-quarters. The calculator evaluates the expression or solves the equation with step-by-step calculation progress information.
2 + 1 1/2 = 3
### How do you add mixed numbers?
To add mixed numbers, you first find the LCD of both fractions. The LCD is the lowest number that is evenly divisible by both numbers. In this example, the LCD is 3. Next, you add the whole numbers together and slice off the decimal place that corresponds to the LCD. So in this case, 1+2=3 and 1+3=4.
### How do you find the sum of two whole numbers?
To find the sum of two whole numbers, you would first round off each number to the nearest 10 place value. Next, you would add those rounded values and get the estimated sum of the given numbers.
### What is sum in maths?
In maths, the sum of two or more numbers is the total of those numbers.
### What is 24/18 as a mixed number?
This is 12 divided by 18, or 3.6.
### What is the improper fraction 18 5 in mixed numbers?
When an improper fraction is reduced to its simplest form, the numerator and denominator are both reduced to 1. The fraction in its simplest form (18/5) would be equivalent to 3/1.
### What is 28/5 as a mixed number?
28/5 is equal to 3.2.
### What is 18/5 as a fraction?
18/5 is written as 1.8, so it would be equal to .
### What is 18/8 as a mixed number?
18/8 is a mixed number because the numerator is 18 and the denominator is 8.
### What is 24/18 as a fraction?
24/18 is an improper fraction because the numerator, or top number, is 24 and the denominator, or bottom number, is 18.
### What can the mixed numbers calculator do?
The mixed numbers calculator can add, subtract, multiply and divide mixed numbers and fractions.
### Is a mixed number a whole number?
Yes, a mixed number is a whole number. The numerator and denominator are both whole numbers.
### How do you convert mixed numbers to improper fractions?
1. Multiply the whole number by the denominator. 2. Add the answer from Step 1 to the numerator 3. Write answer from Step 2 over the denominator.
### What is an improper fraction?
An improper fraction is when the numerator (top number) is larger than the denominator (bottom number).<|endoftext|>
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# Question: What Is The GCF Of 16 And 24?
## What is the GCF of 15 and 25?
5The Greatest Common Factor of 15and25 is 5.
(the number itself and 1 do not count).
The factor(s) of 25 is 5 ..
## What are factors of 16 and 24?
Answer and Explanation: The common factors of 16 and 24 are: 1, 2, 4, and 8.
## What is the GCF of 18 and 24?
We found the factors and prime factorization of 18 and 24. The biggest common factor number is the GCF number. So the greatest common factor 18 and 24 is 6.
## What is the GCF of 16 and 20?
We may write the solution symbolically as GCF(20, 16) = 4. This says “the greatest common factor of 20 and 16 is 4.”
## What is the HCF of 24 32?
8Thus, the highest common factor of 24 and 32 is 8. Hence, HCF of 24 and 32= 8.
## What is the HCF of 15 and 20?
5Greatest common factor (GCF) of 15 and 20 is 5. We will now calculate the prime factors of 15 and 20, than find the greatest common factor (greatest common divisor (gcd)) of the numbers by matching the biggest common factor of 15 and 20.
## What is the GCF of 7/15 and 21?
The common factors for 7,15,21 7 , 15 , 21 are 1 . The GCF (HCF) of the numerical factors 1 is 1 .
## What is the GCF of and 16?
The factors for 16 are 1, 2, 4, 8, 16. The two numbers (12 and 16) share common factors (1, 2, 4). The greatest of these is 4 and that is the greatest common factor.
## What is the GCF of 18 and 36?
We found the factors and prime factorization of 18 and 36. The biggest common factor number is the GCF number. So the greatest common factor 18 and 36 is 18.
## What is the GCF of 18 and 27?
Example: Find the GCF of 18 and 27 The factors of 18 are 1, 2, 3, 6, 9, 18. The factors of 27 are 1, 3, 9, 27. The common factors of 18 and 27 are 1, 3 and 9. The greatest common factor of 18 and 27 is 9.
## What is the GCF of 36 63?
9To sum up, the gcf of 36 and 63 is 9. In common notation: gcf (36,63) = 9.
## What is the HCF of 16 and 25?
The factors of 25 are 25, 5, 1. The common factors of 16 and 25 are 1, intersecting the two sets above. In the intersection factors of 16 ∩ factors of 25 the greatest element is 1. Therefore, the greatest common factor of 16 and 25 is 1.
## What is the GCF of 18 and 8?
Greatest common factor (GCF) of 18 and 8 is 2. We will now calculate the prime factors of 18 and 8, than find the greatest common factor (greatest common divisor (gcd)) of the numbers by matching the biggest common factor of 18 and 8.
## What is the GCF for 32 and 24?
The greatest common factor of 24 and 32 is 8. To find this, we first need to identify the factors of both 24 and 32.
## What is the GCF of 16 and 25?
The GCF for 16 and 25 is 1. The greatest common factor (GFC) of two or more numbers is the largest positive integer in common that divides each number…
## What is the GCF of 18 and 30?
We found the factors and prime factorization of 18 and 30. The biggest common factor number is the GCF number. So the greatest common factor 18 and 30 is 6.
## What is the GCF of 18 and 20?
Greatest common factor (GCF) of 20 and 18 is 2.
## What is the GCF of 12 and 18?
In terms of numbers, the greatest common factor (gcf) is the largest natural number that exactly divides two or more given natural numbers. Example 1: 6 is the greatest common factor of 12 and 18.<|endoftext|>
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Humanism is the outcome of a long tradition of free thought that has inspired many of the world’s great thinkers and creative artists and gave rise to science itself.
Humanism affirms the worth, dignity and autonomy of the individual and the right of every human being to the greatest possible freedom compatible with the rights of others. Humanists have a duty of care to all of humanity including future generations. Humanists believe that morality is an intrinsic part of human nature based on understanding and a concern for others, needing no external sanction. These affirmations and beliefs make it axiomatic that Humanism means the taking of an ethical stance on all issues with which humanity has to cope in life.
Humanism is rational. It seeks to use science creatively, not destructively. Humanists believe that the solutions to the world’s problems lie solely in human thought and action. Humanism advocates the application of the methods of science and free inquiry to the problems of human welfare. But Humanists also believe that the application of science and technology must be tempered by human values. Science gives us the means but human values must propose the ends.
Humanism supports democracy and human rights. Humanism aims at the fullest possible development of every human being. It holds that democracy and human development are matters of right. The principles of democracy and human rights can be applied to many human relationships and are not restricted to methods of government.
Humanism insists that personal liberty must be combined with social responsibility. Humanism ventures to build a world on the idea of the free person responsible to society, and recognizes our dependence on and responsibility for the natural world. Humanism is undogmatic, imposing no creed upon its adherents. It is thus committed to education free from indoctrination.
While Humanism is a response to the widespread demand for an alternative to religion and religious belief, Humanists unconditionally take the view, based on their support for democracy and human rights, that belief is a matter for individuals to decide for themselves. On the same basis it opposes those who hold beliefs, religious or otherwise, who seek to impose their world views on all of humanity.
Humanism recognizes that reliable knowledge of the world and ourselves arises through a continuing process of observation, evaluation and revision. Humanism values artistic creativity and imagination and recognizes the transforming power of art. Humanism affirms the importance of literature, music, and the visual and performing arts for personal development and fulfilment.
Humanism is a life stance aiming at the maximum possible fulfilment through the cultivation of ethical and creative living and offers an ethical and rational means of addressing the challenges of our times. Humanism can be a way of life for everyone everywhere.
Humanists consider that their primary task is to make human beings aware in the simplest terms of what Humanism can mean to them and what it commits them to. By utilizing free inquiry, the power of science and creative imagination for the furtherance of peace and in the service of compassion, they have confidence that the means is available to humanity to solve the problems that confront it. Humanists call upon all who share this conviction to associate together in this endeavour.
The above is an edited version of the Amsterdam Declaration of 2000.
Humanism is a positive code of ethics for people who do not believe in a god. Its fundamental principle is that we should base our behaviour on what is best for the people concerned, and for the environment whilst endeavour to avoid harm to others. Humanism springs from the Greek philosophers and reflects Enlightenment thinking. Humanists acknowledge the existence of other ethical codes and prefer co-operation where possible emphasising shared values rather than differences.
Humanists are atheists or agnostics and nearly all live happy and fulfilling lives without reference to the supernatural. But people are not Humanists just because they reject supernatural beliefs; it is a positive step expressing concern for humanity. Humanists derive their morals from human nature and experience, and accept responsibility for their own actions. Human welfare and progress should be guided by this reasoning and motivated only by a concern for humanity.
Humanists welcome a society in which every viewpoint is allowed free expression and development, limited only by the reasonable rights of others. They are opposed to dogmatic teaching and fundamentalist attitues of any kind seeing this as detrimental to human development. Humanists are encouraged to find solutions to life's problems for themselves without recourse to a set of rules in a sacred book or to revered teachers.
Humanists endeavour to approach problems in a rational way, cultivating an attitude of mind founded on enquiry rather than superstition or fear. They draw conclusions from available evidence, but are ready to adjust to new findings even if these point in an unwelcome direction. To a Humanist it is a betrayal of reason to make an article of faith out of something for which there is no observable or logical foundation.
Humanism supports the right of individuals to choose what they want to do with their lives so long as it is not detrimental to others. For this reason most Humanists support the following views:
Freedom of choice for the individual should apply in the teaching of children. Pupils should be taught the many different views and beliefs held by people worldwide, and allowed to make their own decisions. Schools should not adopt specific religious or political positions, but should encourage pupils to examine them all in an unprejudiced way. We oppose state-funded “faith” schools, which we consider culturally divisive.
We support the right of terminally ill people who are suffering unbearably to have the option of a safe, legal, medically-assisted death. Assisted death should be brought about only at the request of the terminally ill person, who must be mentally competent to make the decision. There should be effective safeguards against the decision being influenced by external pressure.
We support the Advanced Decision (formerly Living Will) advocated by Dignity in Dying. This informs medical staff about how you wish to be treated should you no longer be able to communicate your wishes (e.g. if you are in a coma or have dementia).
People's sexual preferences are their own business and there should be no discrimination in law or in public attitudes. Protection of the young and vulnerable can be provided by non-discriminatory legislation. There is a Gay and Lesbian Association. Click here for their website.<|endoftext|>
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June 22, 2006
Source:Blackwell Publishing Ltd.
New research examines how humans process music and its positive effects on our health and humanity.
A recent volume of the Annals of the New York Academy of Sciences takes a closer look at how music evolved and how we respond to it. Contributors to the volume believe that animals such as birds, dolphins and whales make sounds analogous to music out of a desire to imitate each other. This ability to learn and imitate sounds is a trait necessary to acquire language and scientists feel that many of the sounds animals make may be precursors to human music.
Another study in the volume looks at whether music training can make individuals smarter. Scientists found more grey matter in the auditory cortex of the right hemisphere in musicians compared to nonmusicians. They feel these differences are probably not genetic, but instead due to use and practice.
Listening to classical music, particularly Mozart, has recently been thought to enhance performance on cognitive tests. Contributors to this volume take a closer look at this assertion and their findings indicate that listening to any music that is personally enjoyable has positive effects on cognition. In addition, the use of music to enhance memory is explored and research suggests that musical recitation enhances the coding of information by activating neural networks in a more united and thus more optimal fashion.
Other studies in this volume look at music’s positive effects on health and immunity, how music is processed in the brain, the interplay between language and music, and the relationship between our emotions and music.
The Neurosciences and Music II is volume 1060 of the Annals of the New York Academy of Sciences .
The above post is reprinted from materials provided by Blackwell Publishing Ltd.. Note: Content may be edited for style and length.<|endoftext|>
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GCSE Calculator Questions 3 (2013/Higher Tier)
This worksheet contains a sample of practice questions based on the 2013 GCSE Higher Level syllabus. Calculators may be used.
Key stage: KS 4
Curriculum topic: GCSE Practice Papers
Curriculum subtopic: Selection of Topics for Calculator Practice
Difficulty level:
QUESTION 1 of 10
Formula Sheet
Higher Tier
Area of trapezium = ½ (a + b) h
Volume of prism = area of cross-section x length
In any triangle ABC:
Area of triangle = ½ab sinC
Sine rule
a = b = c sin A sin B sin C
Cosine rule
a2 = b2 + c2 - 2bc cos A
The solutions of ax2 + bx + c = 0, where a ≠ 0, are given by
Sam is decorating a room.
3 tins of white emulsion paint at £12.50 per tin.
1 roller at £4.50.
4 paintbrushes at £3.99 each.
2 tins of blue gloss paint at £8.99 per tin.
VAT at 20% must be added to all prices.
Calculate the total cost of Sam's purchases.
What is the smallest integer that satisfies this inequality:
17 - 9x < 6(2 - x)
Calculate the surface area of this cylinder in cm2 to 3 sig. figs.
(Just write the number to 3 sig. figs)
Write the following fraction as a percentage:
11 40
Write the following as an ordinary number:
4.23 x 10-3
Calculate the value of the following to 2 significant figures:
√15.1 + 5.14 0.15
y is directly proportional to x.
When x = 300, y = 24.
Calculate the value of y when x = 250.
Factorise fully:
64x2 - 96x - 64
(64x + 32)(x - 2)
(2x + 1)(32x - 64)
32(2x + 1)(x - 2)
4(16x + 1)(x - 2)
Solve simultaneously:
4x + 3y = 22
3x - 4y = 4
x = -4, y = 2
x = 4, y = -2
x = -4, y = -2
x = 4, y = 2
Solve for a:
a + 4 = a - 3 4 8
(Just write down the numerical value of a)
The diagram shows two fair spinners.
Both spinners are spun and the product of the scores is found.
Calculate the probability that the product of the scores is odd.
Here is a formula:
v = u + at
Calculate t when u = 6.2 x 102, v = 1255 and a = 9.8
Use trial and improvement to find a solution to the equation:
2x3 - 7x = 83
The first step is shown below.
Give the solution to 1 decimal place
x 2x3 - 7x Comment
4 2 × 43 - 7×4 = 128 - 28 = 100 too large
The front face of a V-shaped tent is shown in the diagram below.
Find the height from the ground to the top of the tent in metres to 3 s.f.
(Just write the number)
A hardware shop sells cement mix is sold in three sizes.
Which size of box represents the best value for money?
2 kg
3 kg
7 kg
The mean height of 15 girls in a class is 155 cm and the mean height of 12 boys is 152 cm.
Calculate the mean height in cm of all the children in the class to 1 decimal place.
(Just write the number)
A survey was carried out to record the marks scored by nine students in a test.
The marks were as follows:
26, 41, 18, 28, 51, 31, 31, 24, 31
Which box plot shows this information?
A
B
C
none of them
A string of lights is to be suspended from the tops of two trees as shown in the diagram below.
Calculate the least length the string should be in metres to the nearest cm.
(Just write the number)
Find the positive solution to:
3x2 + 4x = 8
Calculate the area of this triangle to 3 sig. figs.
(Diagram not drawn accurately)
In △ABC above,
c = 12 cm
a = 9.5 cm
∠ABC = 82º
Calculate the value of b to 3 s.f.
(Just write the number)
A semi-circle is shown in the diagram below.
Calculate its perimeter in cm to 1 decimal place.
(Just write the number)
Look at this number pattern:
92 = 81
992 = 9801
9992 = 998001
99992 = 99980001
999992 = 9999800001
Write down the value of:
99999992
Triangle ABC has been cut by the line DE to form a smaller triangle and a trapezium.
Calculate the area of the trapezium EDBC in cm2 to 3 significant figures.
(Just write the number)
A table is 142.6 cm long.
What is the greatest possible error?
5 mm
0.5 mm
0.5 cm
5 cm
50 mm
• Question 1
Sam is decorating a room.
3 tins of white emulsion paint at £12.50 per tin.
1 roller at £4.50.
4 paintbrushes at £3.99 each.
2 tins of blue gloss paint at £8.99 per tin.
VAT at 20% must be added to all prices.
Calculate the total cost of Sam's purchases.
£91.13
£91.13
EDDIE SAYS
Total = 3 × £12.50 + £4.50 + 4 × £3.99 + 2 × £8.99 = £75.94
Add 20% VAT to get 1.2 × £75.94 = £91.128 which rounds up to £91.13
• Question 2
What is the smallest integer that satisfies this inequality:
17 - 9x < 6(2 - x)
2
EDDIE SAYS
17 - 9x < 12 - 6x
-3x + 17 < 12
-3x < -5
3x > 5
x > 5/3 = 1.6666...
Smallest integer greater than 1.6666..... is therefore 2
• Question 3
Calculate the surface area of this cylinder in cm2 to 3 sig. figs.
(Just write the number to 3 sig. figs)
363
EDDIE SAYS
Diameter d = 11cm
Circumference = π × d = π × 11 = 34.5575 cm
Area of side = 34.5575 × 5 = 172.7875..cm²
Area of circular top and bottom = 2 × π × r² = 2 × π × 5.5² = 190.066.. cm²
Total surface area = 190.066 cm² + 172.7875 = 362.85.. cm²
• Question 4
Write the following fraction as a percentage:
11 40
27.5%
EDDIE SAYS
11 ÷ 40 = 0.275 = 27.5%
• Question 5
Write the following as an ordinary number:
4.23 x 10-3
0.00423
EDDIE SAYS
4.23 ÷ 1000 = 0.00423
• Question 6
Calculate the value of the following to 2 significant figures:
√15.1 + 5.14 0.15
4500
EDDIE SAYS
On the calculator, type:
(√15.1 + 5.1^4) ÷ 0.15 =
4536.0398... rounds to 4500 to 2 significant figures.
• Question 7
y is directly proportional to x.
When x = 300, y = 24.
Calculate the value of y when x = 250.
20
EDDIE SAYS
y = kx
24 = 300k so k = 24 ÷ 300 = 0.08
Formula is therefore y = 0.08x
When x = 250, y = 0.08 × 250 = 20
• Question 8
Factorise fully:
64x2 - 96x - 64
32(2x + 1)(x - 2)
EDDIE SAYS
This divides by 32 to get 2x² - 3x - 2
Full factorisation is 32(2x + 1)(x - 2)
• Question 9
Solve simultaneously:
4x + 3y = 22
3x - 4y = 4
x = 4, y = 2
EDDIE SAYS
Multiply first equation by 3 and second equation by 4 to get:
12x + 9y = 66
12x - 16y = 16
Subtract to get:
25y = 50
y = 2
Substitute back into either equation to find x.
• Question 10
Solve for a:
a + 4 = a - 3 4 8
(Just write down the numerical value of a)
-11
EDDIE SAYS
Multiply both fractions by 8 to get:
2a + 8 = a - 3
a = -11
• Question 11
The diagram shows two fair spinners.
Both spinners are spun and the product of the scores is found.
Calculate the probability that the product of the scores is odd.
1/4
EDDIE SAYS
For the product to be odd, we must have two odd numbers to start with. If one or both is even, then the product must be even.
Prob of getting an odd number on first spinner is ½
Prob of getting an odd number on second spinner is also ½
Probability required = ½ × ½ = ¼
• Question 12
Here is a formula:
v = u + at
Calculate t when u = 6.2 x 102, v = 1255 and a = 9.8
64.8
EDDIE SAYS
t = (v - u)/a = (1255 - 620) ÷ 9.8 = 64.7959...
• Question 13
Use trial and improvement to find a solution to the equation:
2x3 - 7x = 83
The first step is shown below.
Give the solution to 1 decimal place
x 2x3 - 7x Comment
4 2 × 43 - 7×4 = 128 - 28 = 100 too large
3.8
EDDIE SAYS
2 × 3.5³ - 7×3.5 = 61.25 too small
2 × 3.8³ - 7×3.8 = 83.144 too large
2 × 3.7³ - 7×3.7 = 75.406 too small
2 × 3.75³ - 7×3.75 = 79.21875 too small
So answer lies between 3.75 and 3.8, which will give a solution of 3.8 to 1 decimal place.
• Question 14
The front face of a V-shaped tent is shown in the diagram below.
Find the height from the ground to the top of the tent in metres to 3 s.f.
(Just write the number)
1.98
EDDIE SAYS
Divide the triangle into two as shown, to form a right-angled triangle with top angle 29° and opposite side of 1.1 m.
Height, h = 1.1/tan29º
• Question 15
A hardware shop sells cement mix is sold in three sizes.
Which size of box represents the best value for money?
7 kg
EDDIE SAYS
The 2 kg box costs £3.50 ÷ 2 = £1.75/kg
The 3 kg box costs £4.50 ÷ 3 = £1.50/kg
The 7 kg box costs £10.00 ÷ 7 = £1.43/kg
• Question 16
The mean height of 15 girls in a class is 155 cm and the mean height of 12 boys is 152 cm.
Calculate the mean height in cm of all the children in the class to 1 decimal place.
(Just write the number)
153.7
EDDIE SAYS
Total of girls' heights = 15 × 155 = 2325 cm.
Total of boys' heights = 12 × 152 = 1824 cm.
Total of all heights = 2325 + 1824 = 4149 cm.
Mean height of 27 children = 4149 ÷ 27 = 153.6666...cm.
• Question 17
A survey was carried out to record the marks scored by nine students in a test.
The marks were as follows:
26, 41, 18, 28, 51, 31, 31, 24, 31
Which box plot shows this information?
none of them
EDDIE SAYS
First place the marks in ascending order
18, 24, 26, 28, 31, 31, 31, 41, 51.
Remember that to find the median of an even number of numbers, we must find the mean of the two middle numbers.
Minimum mark = 18.
Lower quartile = 25
Median = 31
Upper quartile = 36
Maximum mark = 51
No box plot shows an upper quartile of 36
• Question 18
A string of lights is to be suspended from the tops of two trees as shown in the diagram below.
Calculate the least length the string should be in metres to the nearest cm.
(Just write the number)
20.22
EDDIE SAYS
We can form a right-angled triangle with the string of lights as the hypotenuse, h. The triangle has a base of 20 m and a height of (10 - 7) = 3 m.
Using Pythagoras' Theorem h² = 3² + 20² = 9 + 400 = 409
h = √409 = 20.2237... m
• Question 19
Find the positive solution to:
3x2 + 4x = 8
1.10
EDDIE SAYS
Rearrange to read 3x² + 4x - 7 = 0
Use the quadratic formula with a = 3, b = 4, c = -8 to get:
x = (-4 ± √(16 + 96)) ÷ 6
Positive solution will be
x = (-4 + √(16 + 96)) ÷ 6 = 1.097....
• Question 20
Calculate the area of this triangle to 3 sig. figs.
(Diagram not drawn accurately)
26.7
EDDIE SAYS
x = 6 ÷ tan34° = 8.89536...
Area = ½ 8.89536 × 6 = 26.686... units²
• Question 21
In △ABC above,
c = 12 cm
a = 9.5 cm
∠ABC = 82º
Calculate the value of b to 3 s.f.
(Just write the number)
14.2
EDDIE SAYS
Using the cos rule, we get:
b² = 9.5² + 12² - 2 × 9.5 × 12cos82° = 202.5185...
b = √202.5185 = 14.2309.... cm
• Question 22
A semi-circle is shown in the diagram below.
Calculate its perimeter in cm to 1 decimal place.
(Just write the number)
38.6
EDDIE SAYS
Cicumference of whole circle = π ×15 = 47.12 cm
Perimeter of curved part = 47.12 ÷ 2 = 23.56 cm
Perimeter including diameter = 23.56 + 15 = 38.56 cm
• Question 23
Look at this number pattern:
92 = 81
992 = 9801
9992 = 998001
99992 = 99980001
999992 = 9999800001
Write down the value of:
99999992
99999980000001
EDDIE SAYS
Following the pattern there will be six 9s, an 8, six 0s and a 1.
• Question 24
Triangle ABC has been cut by the line DE to form a smaller triangle and a trapezium.
Calculate the area of the trapezium EDBC in cm2 to 3 significant figures.
(Just write the number)
43.7
EDDIE SAYS
So AB = 12 × 10 ÷ 7 = 17.1428...cm
So DB = 17.1428 - 12 = 5.1428 cm
Area of trapezium = ½(7 + 10) × 5.1428 = 43.714 cm²
• Question 25
A table is 142.6 cm long.
What is the greatest possible error?<|endoftext|>
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What is Tonsil Cancer?
There are three types of tonsils in the throat: the nasopharyngeal tonsils known as adenoids that are located behind the nose, the palatine tonsils located on the sides of the throat, and the lingual tonsils that are located on the back of the tongue. Tonsil cancer most often involves the palatine tonsils, which are two oval shaped pads made of white blood cells located at the back and sides of the mouth, or the oropharynx. The tonsils are made of lymphoid tissue and are responsible for helping fight infection and defend your body against germs. Tonsil cancer is caused by an overgrowth of cells of the surface lining of the tonsils. Even if you have had your tonsils removed (tonsillectomy) you can still develop tonsil cancer if some tonsil tissue is left behind after the procedure. There are two main types of tonsil cancer: squamous cell carcinomas which makes up the vast majority of cases, and lymphomas. Because some people do not experience symptoms until much later on in the disease, tonsil cell cancer is sometimes diagnosed when it has spread to other areas in the mouth such as the tongue or lymph nodes in the neck. The following information on tonsil cancer concerns only the palatine tonsils.<|endoftext|>
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Overview of life
Sir Isaac Newton PRS MP (25 December 1642 – 20 March 1727) was an English physicist and mathematician who is widely regarded as one of the most influential scientists of all time and as a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica (“Mathematical Principles of Natural Philosophy”), first published in 1687, laid the foundations for most of classical mechanics. Newton also made seminal contributions to optics and, as mathematician, he shares credit with Gottfried Leibniz for the invention of the infinitesimal calculus. Newton’s Principia formulated the laws of motion and universal gravitation that dominated scientists’ view of the physical universe for the next three centuries. It also demonstrated that the motion of objects on the Earth and that of celestial bodies could be described by the same principles. By deriving Kepler’s laws of planetary motion from his mathematical description of gravity, Newton removed the last doubts about the validity of the heliocentric model of the cosmos. Newton built the first practical reflecting telescope and developed a theory of colour based on the observation that a prism decomposes white light into the many colours of the visible spectrum. He also formulated an empirical law of cooling and studied the speed of sound. In addition to his work on the calculus, as a mathematician Newton contributed to the study of power series, generalised the binomial theorem to non-integer exponents, and developed Newton’s method for approximating the roots of a function. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge. He was a devout but unorthodox Christian and, unusually for a member of the Cambridge faculty, he refused to take holy orders in the Church of England, perhaps because he privately rejected the doctrine of trinitarianism. In addition to his work on the mathematical sciences, Newton also dedicated much of his time to the study of alchemy and biblical chronology, but most of his work in those areas remained unpublished until long after his death. In his later life, Newton became president of the Royal Society. He also served the British government as Warden and Master of the Royal Mint.
Isaac Newton was born (according to the Julian calendar in use in England at the time) on Christmas Day, 25 December 1642, (NS 4 January 1643.) at Woolsthorpe Manor in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. He was born three months after the death of his father, a prosperous farmer also named Isaac Newton. Born prematurely, he was a small child; his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug (≈ 1.1 litres). When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabus Smith, leaving her son in the care of his maternal grandmother, Margery Ayscough. The young Isaac disliked his stepfather and maintained some enmity towards his mother for marrying him, as revealed by this entry in a list of sins committed up to the age of 19: “Threatening my father and mother Smith to burn them and the house over them.” Although it was claimed that he was once engaged, Newton never married. From the age of about twelve until he was seventeen, Newton was educated at The King’s School, Grantham. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, where his mother, widowed by now for a second time, attempted to make a farmer of him. He hated farming. Henry Stokes, master at the King’s School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a schoolyard bully, he became the top-ranked student. The Cambridge psychologist Simon Baron-Cohen considers it “fairly certain” that Newton had Asperger syndrome. In June 1661, he was admitted to Trinity College, Cambridge as a sizar – a sort of work-study role. At that time, the college’s teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers, such as Descartes, and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student, Newton’s private studies at his home in Woolsthorpe over the subsequent two years saw the development of his theories on calculus, optics and the law of gravitation. In 1667, he returned to Cambridge as a fellow of Trinity. Fellows were required to become ordained priests, something Newton desired to avoid due to his unorthodox views. Luckily for Newton, there was no specific deadline for ordination, and it could be postponed indefinitely. The problem became more severe later when Newton was elected for the prestigious Lucasian Chair. For such a significant appointment, ordaining normally could not be dodged. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II (see “Middle years” section below).
Newton never married, and no evidence has been uncovered that he had any romantic relationship. Although it is impossible to verify, it is commonly believed that he died a virgin, as has been commented on by such figures as mathematician Charles Hutton, economist John Maynard Keynes, and physicist Carl Sagan. French writer and philosopher Voltaire, who was in London at the time of Newton’s funeral, claimed to have verified the fact, writing that “I have had that confirmed by the doctor and the surgeon who were with him when he died” (allegedly he stated on his deathbed that he was a virgin). In 1733, Voltaire publicly stated that Newton “had neither passion nor weakness; he never went near any woman”. Newton did have a close friendship with the Swiss mathematician Nicolas Fatio de Duillier, whom he met in London around 1690. Their friendship came to an unexplained end in 1693. Some of their correspondence has survived.
Newton’s work has been said “to distinctly advance every branch of mathematics then studied”. His work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newton’s mathematical papers. The author of the manuscript De analysi per aequationes numero terminorum infinitas, sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins in August of that year as:
Mr Newton, a fellow of our College, and very young … but of an extraordinary genius and proficiency in these things.
Newton later became involved in a dispute with Leibniz over priority in the development of calculus (the Leibniz–Newton calculus controversy). Most modern historians believe that Newton and Leibniz developed calculus independently, although with very different notations. Occasionally it has been suggested that Newton published almost nothing about it until 1693, and did not give a full account until 1704, while Leibniz began publishing a full account of his methods in 1684. (Leibniz’s notation and “differential Method”, nowadays recognised as much more convenient notations, were adopted by continental European mathematicians, and after 1820 or so, also by British mathematicians.) But such a suggestion fails to account for the content of calculus in Book 1 of Newton’s Principia itself (published 1687) and in its forerunner manuscripts, such as De motu corporum in gyrum (“On the motion of bodies in orbit”) of 1684; this content has been pointed out by critics of both Newton’s time and modern times. The Principia is not written in the language of calculus either as we know it or as Newton’s (later) ‘dot’ notation would write it. His work extensively uses calculus in geometric form based on limiting values of the ratios of vanishing small quantities: in the Principia itself, Newton gave demonstration of this under the name of ‘the method of first and last ratios’ and explained why he put his expositions in this form, remarking also that ‘hereby the same thing is performed as by the method of indivisibles’.
Because of this, the Principia has been called “a book dense with the theory and application of the infinitesimal calculus” in modern times and “lequel est presque tout de ce calcul” (‘nearly all of it is of this calculus’) in Newton’s time. His use of methods involving “one or more orders of the infinitesimally small” is present in his De motu corporum in gyrum of 1684 and in his papers on motion “during the two decades preceding 1684”.
Newton had been reluctant to publish his calculus because he feared controversy and criticism. He was close to the Swiss mathematician Nicolas Fatio de Duillier. In 1691, Duillier started to write a new version of Newton’s Principia, and corresponded with Leibniz. In 1693, the relationship between Duillier and Newton deteriorated and the book was never completed.
Starting in 1699, other members of the Royal Society (of which Newton was a member) accused Leibniz of plagiarism. The dispute then broke out in full force in 1711 when the Royal Society proclaimed in a study that it was Newton who was the true discoverer and labelled Leibniz a fraud. This study was cast into doubt when it was later found that Newton himself wrote the study’s concluding remarks on Leibniz. Thus began the bitter controversy which marred the lives of both Newton and Leibniz until the latter’s death in 1716.
Newton is generally credited with the generalised binomial theorem, valid for any exponent. He discovered Newton’s identities, Newton’s method, classified cubic plane curves (polynomials of degree three in two variables), made substantial contributions to the theory of finite differences, and was the first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations. He approximated partial sums of the harmonic series by logarithms (a precursor to Euler’s summation formula) and was the first to use power series with confidence and to revert power series. Newton’s work on infinite series was inspired by Simon Stevin’s decimals.
When Newton received his MA and became a Fellow of the “College of the Holy and Undivided Trinity” in 1667, he made the commitment that “I will either set Theology as the object of my studies and will take holy orders when the time prescribed by these statutes [7 years] arrives, or I will resign from the college.” Up till this point he had not thought much about religion and had twice signed his agreement to the thirty-nine articles, the basis of Church of England doctrine.
He was appointed Lucasian Professor of Mathematics in 1669 on Barrow’s recommendation. During that time, any Fellow of a college at Cambridge or Oxford was required to take holy orders and become an ordained Anglican priest. However, the terms of the Lucasian professorship required that the holder not be active in the church (presumably so as to have more time for science). Newton argued that this should exempt him from the ordination requirement, and Charles II, whose permission was needed, accepted this argument. Thus a conflict between Newton’s religious views and Anglican orthodoxy was averted.
In 1666, Newton observed that the spectrum of colours exiting a prism in the position of minimum deviation is oblong, even when the light ray entering the prism is circular, which is to say, the prism refracts different colours by different angles. This led him to conclude that colour is a property intrinsic to light—a point which had been debated in prior years.
Replica of Newton’s second Reflecting telescope that he presented to the Royal Society in 1672
From 1670 to 1672, Newton lectured on optics. During this period he investigated the refraction of light, demonstrating that the multicoloured spectrum produced by a prism could be recomposed into white light by a lens and a second prism. Modern scholarship has revealed that Newton’s analysis and resynthesis of white light owes a debt to corpuscular alchemy.
He showed that coloured light does not change its properties by separating out a coloured beam and shining it on various objects, and that regardless of whether reflected, scattered, or transmitted, the light remains the same colour. Thus, he observed that colour is the result of objects interacting with already-coloured light rather than objects generating the colour themselves. This is known as Newton’s theory of colour.
Illustration of a dispersive prism decomposing white light into the colours of the spectrum, as discovered by Newton
From this work, he concluded that the lens of any refracting telescope would suffer from the dispersion of light into colours (chromatic aberration). As a proof of the concept, he constructed a telescope using reflective mirrors instead of lenses as the objective to bypass that problem. Building the design, the first known functional reflecting telescope, today known as a Newtonian telescope, involved solving the problem of a suitable mirror material and shaping technique. Newton ground his own mirrors out of a custom composition of highly reflective speculum metal, using Newton’s rings to judge the quality of the optics for his telescopes. In late 1668 he was able to produce this first reflecting telescope. It was about eight inches long and it gave a clearer and larger image. In 1671, the Royal Society asked for a demonstration of his reflecting telescope. Their interest encouraged him to publish his notes, Of Colours, which he later expanded into the work Opticks. When Robert Hooke criticised some of Newton’s ideas, Newton was so offended that he withdrew from public debate. Newton and Hooke had brief exchanges in 1679–80, when Hooke, appointed to manage the Royal Society’s correspondence, opened up a correspondence intended to elicit contributions from Newton to Royal Society transactions, which had the effect of stimulating Newton to work out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector (see Newton’s law of universal gravitation – History and De motu corporum in gyrum). But the two men remained generally on poor terms until Hooke’s death.
Facsimile of a 1682 letter from Isaac Newton to Dr William Briggs, commenting on Briggs’ “A New Theory of Vision”
Newton argued that light is composed of particles or corpuscles, which were refracted by accelerating into a denser medium. He verged on soundlike waves to explain the repeated pattern of reflection and transmission by thin films (Opticks Bk.II, Props. 12), but still retained his theory of ‘fits’ that disposed corpuscles to be reflected or transmitted (Props.13). However, later physicists favoured a purely wavelike explanation of light to account for the interference patterns and the general phenomenon of diffraction. Today’s quantum mechanics, photons, and the idea of wave–particle duality bear only a minor resemblance to Newton’s understanding of light.
In his Hypothesis of Light of 1675, Newton posited the existence of the ether to transmit forces between particles. The contact with the theosophist Henry More, revived his interest in alchemy. He replaced the ether with occult forces based on Hermetic ideas of attraction and repulsion between particles. John Maynard Keynes, who acquired many of Newton’s writings on alchemy, stated that “Newton was not the first of the age of reason: He was the last of the magicians.” Newton’s interest in alchemy cannot be isolated from his contributions to science. This was at a time when there was no clear distinction between alchemy and science. Had he not relied on the occult idea of action at a distance, across a vacuum, he might not have developed his theory of gravity. (See also Isaac Newton’s occult studies.)
In 1704, Newton published Opticks, in which he expounded his corpuscular theory of light. He considered light to be made up of extremely subtle corpuscles, that ordinary matter was made of grosser corpuscles and speculated that through a kind of alchemical transmutation “Are not gross Bodies and Light convertible into one another, … and may not Bodies receive much of their Activity from the Particles of Light which enter their Composition?” Newton also constructed a primitive form of a frictional electrostatic generator, using a glass globe.
In an article entitled “Newton, prisms, and the ‘opticks’ of tunable lasers” it is indicated that Newton in his book Opticks was the first to show a diagram using a prism as a beam expander. In the same book he describes, via diagrams, the use of multiple-prism arrays. Some 278 years after Newton’s discussion, multiple-prism beam expanders became central to the development of narrow-linewidth tunable lasers. Also, the use of these prismatic beam expanders led to the multiple-prism dispersion theory.
Subsequent to Newton, much has been amended. Young and Fresnel combined Newton’s particle theory with Huygens’ wave theory to show that colour is the visible manifestation of light’s wavelength. Science also slowly came to realise the difference between perception of colour and mathematisable optics. The German poet and scientist, Goethe, could not shake the Newtonian foundation but “one hole Goethe did find in Newton’s armour, … Newton had committed himself to the doctrine that refraction without colour was impossible. He therefore thought that the object-glasses of telescopes must for ever remain imperfect, achromatism and refraction being incompatible. This inference was proved by Dollond to be wrong.”
Mechanics and gravitation
Newton’s own copy of his Principia, with hand-written corrections for the second edition
Further information: Writing of Principia Mathematica
In 1679, Newton returned to his work on (celestial) mechanics by considering gravitation and its effect on the orbits of planets with reference to Kepler’s laws of planetary motion. This followed stimulation by a brief exchange of letters in 1679–80 with Hooke, who had been appointed to manage the Royal Society’s correspondence, and who opened a correspondence intended to elicit contributions from Newton to Royal Society transactions. Newton’s reawakening interest in astronomical matters received further stimulus by the appearance of a comet in the winter of 1680–1681, on which he corresponded with John Flamsteed. After the exchanges with Hooke, Newton worked out proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector (see Newton’s law of universal gravitation – History and De motu corporum in gyrum). Newton communicated his results to Edmond Halley and to the Royal Society in De motu corporum in gyrum, a tract written on about nine sheets which was copied into the Royal Society’s Register Book in December 1684. This tract contained the nucleus that Newton developed and expanded to form the Principia.
The Principia was published on 5 July 1687 with encouragement and financial help from Edmond Halley. In this work, Newton stated the three universal laws of motion. Together, these laws describe the relationship between any object, the forces acting upon it and the resulting motion, laying the foundation for classical mechanics. They contributed to many advances during the Industrial Revolution which soon followed and were not improved upon for more than 200 years. Many of these advancements continue to be the underpinnings of non-relativistic technologies in the modern world. He used the Latin word gravitas (weight) for the effect that would become known as gravity, and defined the law of universal gravitation.
In the same work, Newton presented a calculus-like method of geometrical analysis using ‘first and last ratios’, gave the first analytical determination (based on Boyle’s law) of the speed of sound in air, inferred the oblateness of Earth’s spheroidal figure, accounted for the precession of the equinoxes as a result of the Moon’s gravitational attraction on the Earth’s oblateness, initiated the gravitational study of the irregularities in the motion of the moon, provided a theory for the determination of the orbits of comets, and much more.
Newton made clear his heliocentric view of the Solar System—developed in a somewhat modern way, because already in the mid-1680s he recognised the “deviation of the Sun” from the centre of gravity of the Solar System. For Newton, it was not precisely the centre of the Sun or any other body that could be considered at rest, but rather “the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem’d the Centre of the World”, and this centre of gravity “either is at rest or moves uniformly forward in a right line” (Newton adopted the “at rest” alternative in view of common consent that the centre, wherever it was, was at rest).
Newton’s postulate of an invisible force able to act over vast distances led to him being criticised for introducing “occult agencies” into science. Later, in the second edition of the Principia (1713), Newton firmly rejected such criticisms in a concluding General Scholium, writing that it was enough that the phenomena implied a gravitational attraction, as they did; but they did not so far indicate its cause, and it was both unnecessary and improper to frame hypotheses of things that were not implied by the phenomena. (Here Newton used what became his famous expression “hypotheses non-fingo” ).
With the Principia, Newton became internationally recognised. He acquired a circle of admirers, including the Swiss-born mathematician Nicolas Fatio de Duillier.
Classification of cubics
Descartes was the most important early influence on Newton the mathematician. Newton classified the cubic curves in the plane. He found 72 of the 78 species of cubics. He also divided them into four types, satisfying different equations, and in 1717 Stirling, probably with Newton’s help, proved that every cubic was one of these four types. Newton also claimed that the four types could be obtained by plane projection from one of them, and this was proved in 1731.
In the 1690s, Newton wrote a number of religious tracts dealing with the literal and symbolic interpretation of the Bible. A manuscript Newton sent to John Locke in which he disputed the fidelity of 1 John 5:7 and its fidelity to the original manuscripts of the New Testament, remained unpublished until 1785.
Even though a number of authors have claimed that the work might have been an indication that Newton disputed the belief in Trinity, others assure that Newton did question the passage but never denied Trinity as such. His biographer, scientist Sir David Brewster, who compiled his manuscripts for over 20 years, wrote about the controversy in well-known book Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton, where he explains that Newton questioned the veracity of those passages, but he never denied the doctrine of Trinity as such. Brewster states that Newton was never known as an Arian during his lifetime, it was first William Whiston (an Arian) who argued that “Sir Isaac Newton was so hearty for the Baptists, as well as for the Eusebians or Arians, that he sometimes suspected these two were the two witnesses in the Revelations,” while other like Hopton Haynes (a Mint employee and Humanitarian), “mentioned to Richard Baron, that Newton held the same doctrine as himself”.
Later works—The Chronology of Ancient Kingdoms Amended (1728) and Observations Upon the Prophecies of Daniel and the Apocalypse of St. John (1733)—were published after his death. He also devoted a great deal of time to alchemy (see above).
Newton was also a member of the Parliament of England for Cambridge University in 1689–90 and 1701–2, but according to some accounts his only comments were to complain about a cold draught in the chamber and request that the window be closed. He was however noted by Cambridge diarist Abraham de la Pryme as having rebuked students who were frightening local residents by claiming that a house was haunted.
Newton moved to London to take up the post of warden of the Royal Mint in 1696, a position that he had obtained through the patronage of Charles Montagu, 1st Earl of Halifax, then Chancellor of the Exchequer. He took charge of England’s great recoining, somewhat treading on the toes of Lord Lucas, Governor of the Tower (and securing the job of deputy comptroller of the temporary Chester branch for Edmond Halley). Newton became perhaps the best-known Master of the Mint upon the death of Thomas Neale in 1699, a position Newton held for the last 30 years of his life. These appointments were intended as sinecures, but Newton took them seriously, retiring from his Cambridge duties in 1701, and exercising his power to reform the currency and punish clippers and counterfeiters.
As Warden, and afterwards Master, of the Royal Mint, Newton estimated that 20 percent of the coins taken in during the Great Recoinage of 1696 were counterfeit. Counterfeiting was high treason, punishable by the felon being hanged, drawn and quartered. Despite this, convicting even the most flagrant criminals could be extremely difficult. However, Newton proved equal to the task.
Disguised as a habitué of bars and taverns, he gathered much of that evidence himself. For all the barriers placed to prosecution, and separating the branches of government, English law still had ancient and formidable customs of authority. Newton had himself made a justice of the peace in all the home counties—there is a draft of a letter regarding this matter stuck into Newton’s personal first edition of his Philosophiæ Naturalis Principia Mathematica which he must have been amending at the time. Then he conducted more than 100 cross-examinations of witnesses, informers, and suspects between June 1698 and Christmas 1699. Newton successfully prosecuted 28 coiners.
As a result of a report written by Newton on 21 September 1717 to the Lords Commissioners of His Majesty’s Treasury the bimetallic relationship between gold coins and silver coins was changed by Royal proclamation on 22 December 1717, forbidding the exchange of gold guineas for more than 21 silver shillings. This inadvertently resulted in a silver shortage as silver coins were used to pay for imports, while exports were paid for in gold, effectively moving Britain from the silver standard to its first gold standard. It is a matter of debate as whether he intended to do this or not. It has been argued that Newton conceived of his work at the Mint as a continuation of his alchemical work.
Newton was made President of the Royal Society in 1703 and an associate of the French Académie des Sciences. In his position at the Royal Society, Newton made an enemy of John Flamsteed, the Astronomer Royal, by prematurely publishing Flamsteed’s Historia Coelestis Britannica, which Newton had used in his studies.
Coat of arms of the Newton family of Gunnerby, Lincolnshire, afterwards used by Sir Isaac
In April 1705, Queen Anne knighted Newton during a royal visit to Trinity College, Cambridge. The knighthood is likely to have been motivated by political considerations connected with the Parliamentary election in May 1705, rather than any recognition of Newton’s scientific work or services as Master of the Mint. Newton was the second scientist to be knighted, after Sir Francis Bacon.
Newton was one of many people who lost heavily when the South Sea Company collapsed. Their most significant trade was slaves, and according to his niece, he lost around £20,000.
Towards the end of his life, Newton took up residence at Cranbury Park, near Winchester with his niece and her husband, until his death in 1727. His half-niece, Catherine Barton Conduitt, served as his hostess in social affairs at his house on Jermyn Street in London; he was her “very loving Uncle”, according to his letter to her when she was recovering from smallpox.
Newton died in his sleep in London on 20 March 1727 (OS 20 March 1726; NS 31 March 1727) and was buried in Westminster Abbey. Voltaire may have been present at his funeral. A bachelor, he had divested much of his estate to relatives during his last years, and died intestate. His papers went to John Conduitt and Catherine Barton. After his death, Newton’s hair was examined and found to contain mercury, probably resulting from his alchemical pursuits. Mercury poisoning could explain Newton’s eccentricity in late life.
Although it was claimed that he was once engaged, Newton never married. The French writer and philosopher Voltaire, who was in London at the time of Newton’s funeral, said that he “was never sensible to any passion, was not subject to the common frailties of mankind, nor had any commerce with women—a circumstance which was assured me by the physician and surgeon who attended him in his last moments”. The widespread belief that he died a virgin has been commented on by writers such as mathematician Charles Hutton, economist John Maynard Keynes, and physicist Carl Sagan.
Newton did have a close friendship with the Swiss mathematician Nicolas Fatio de Duillier, whom he met in London around 1689. Their intense relationship came to an abrupt and unexplained end in 1693, and at the same time Newton suffered a nervous breakdown. Some of their correspondence has survived.
In September of that year, Newton had a breakdown which included sending wild accusatory letters to his friends Samuel Pepys and John Locke. His note to the latter included the charge that Locke “endeavoured to embroil me with woemen”.
The mathematician Joseph-Louis Lagrange often said that Newton was the greatest genius who ever lived, and once added that Newton was also “the most fortunate, for we cannot find more than once a system of the world to establish.” English poet Alexander Pope was moved by Newton’s accomplishments to write the famous epitaph:
Nature and nature’s laws lay hid in night;
God said “Let Newton be” and all was light.
Newton himself had been rather more modest of his own achievements, famously writing in a letter to Robert Hooke in February 1676:
If I have seen further it is by standing on the shoulders of giants.
Two writers think that the above quotation, written at a time when Newton and Hooke were in dispute over optical discoveries, was an oblique attack on Hooke (said to have been short and hunchbacked), rather than—or in addition to—a statement of modesty. On the other hand, the widely known proverb about standing on the shoulders of giants, published among others by seventeenth-century poet George Herbert (a former orator of the University of Cambridge and fellow of Trinity College) in his Jacula Prudentum (1651), had as its main point that “a dwarf on a giant’s shoulders sees farther of the two”, and so its effect as an analogy would place Newton himself rather than Hooke as the ‘dwarf’.
In a later memoir, Newton wrote:
I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.
In 1816, a tooth said to have belonged to Newton was sold for £730 (us$3,633) in London to an aristocrat who had it set in a ring. The Guinness World Records 2002 classified it as the most valuable tooth, which would value approximately £25,000 (us$35,700) in late 2001. Who bought it and who currently has it has not been disclosed.
Albert Einstein kept a picture of Newton on his study wall alongside ones of Michael Faraday and James Clerk Maxwell. Newton remains influential to today’s scientists, as demonstrated by a 2005 survey of members of Britain’s Royal Society (formerly headed by Newton) asking who had the greater effect on the history of science, Newton or Einstein. Royal Society scientists deemed Newton to have made the greater overall contribution. In 1999, an opinion poll of 100 of today’s leading physicists voted Einstein the “greatest physicist ever;” with Newton the runner-up, while a parallel survey of rank-and-file physicists by the site PhysicsWeb gave the top spot to Newton.
Newton statue on display at the Oxford University Museum of Natural History
Newton’s monument (1731) can be seen in Westminster Abbey, at the north of the entrance to the choir against the choir screen, near his tomb. It was executed by the sculptor Michael Rysbrack (1694–1770) in white and grey marble with design by the architect William Kent. The monument features a figure of Newton reclining on top of a sarcophagus, his right elbow resting on several of his great books and his left hand pointing to a scroll with a mathematical design. Above him is a pyramid and a celestial globe showing the signs of the Zodiac and the path of the comet of 1680. A relief panel depicts putti using instruments such as a telescope and prism. The Latin inscription on the base translates as:
Here is buried Isaac Newton, Knight, who by a strength of mind almost divine, and mathematical principles peculiarly his own, explored the course and figures of the planets, the paths of comets, the tides of the sea, the dissimilarities in rays of light, and, what no other scholar has previously imagined, the properties of the colours thus produced. Diligent, sagacious and faithful, in his expositions of nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and good, and expressed the simplicity of the Gospel in his manners. Mortals rejoice that there has existed such and so great an ornament of the human race! He was born on 25 December 1642, and died on 20 March 1726/7.—Translation from G.L. Smyth, The Monuments and Genii of St. Paul’s Cathedral, and of Westminster Abbey (1826), ii, 703–4.
From 1978 until 1988, an image of Newton designed by Harry Ecclestone appeared on Series D £1 banknotes issued by the Bank of England (the last £1 notes to be issued by the Bank of England). Newton was shown on the reverse of the notes holding a book and accompanied by a telescope, a prism and a map of the Solar System.
Eduardo Paolozzi’s Newton, after William Blake (1995), outside the British Library
A statue of Isaac Newton, looking at an apple at his feet, can be seen at the Oxford University Museum of Natural History. A large bronze statue, Newton, after William Blake, by Eduardo Paolozzi, dated 1995 and inspired by Blake’s etching, dominates the piazza of the British Library in London.
Main article: Religious views of Isaac Newton
Newton’s tomb monument in Westminster Abbey
Although born into an Anglican family, by his thirties Newton held a Christian faith that, had it been made public, would not have been considered orthodox by mainstream Christianity; in recent times he has been described as a heretic.
By 1672 he had started to record his theological researches in notebooks which he showed to no one and which have only recently been examined. They demonstrate an extensive knowledge of early church writings and show that in the conflict between Athanasius and Arius which defined the Creed, he took the side of Arius, the loser, who rejected the conventional view of the Trinity. Newton “recognized Christ as a divine mediator between God and man, who was subordinate to the Father who created him.” He was especially interested in prophecy, but for him, “the great apostasy was trinitarianism.”
Newton tried unsuccessfully to obtain one of the two fellowships that exempted the holder from the ordination requirement. At the last moment in 1675 he received a dispensation from the government that excused him and all future holders of the Lucasian chair.
In Newton’s eyes, worshipping Christ as God was idolatry, to him the fundamental sin. Historian Stephen D. Snobelen says, “Isaac Newton was a heretic. But … he never made a public declaration of his private faith—which the orthodox would have deemed extremely radical. He hid his faith so well that scholars are still unravelling his personal beliefs.” Snobelen concludes that Newton was at least a Socinian sympathiser (he owned and had thoroughly read at least eight Socinian books), possibly an Arian and almost certainly an anti-trinitarian.
In a minority view, T.C. Pfizenmaier argues that Newton held the Eastern Orthodox view on the Trinity. However, this type of view ‘has lost support of late with the availability of Newton’s theological papers’, and now most scholars identify Newton as an Antitrinitarian monotheist.
Although the laws of motion and universal gravitation became Newton’s best-known discoveries, he warned against using them to view the Universe as a mere machine, as if akin to a great clock. He said, “Gravity explains the motions of the planets, but it cannot explain who set the planets in motion. God governs all things and knows all that is or can be done.”
Along with his scientific fame, Newton’s studies of the Bible and of the early Church Fathers were also noteworthy. Newton wrote works on textual criticism, most notably An Historical Account of Two Notable Corruptions of Scripture and Observations upon the Prophecies of Daniel, and the Apocalypse of St. John. He placed the crucifixion of Jesus Christ at 3 April, AD 33, which agrees with one traditionally accepted date.
He believed in a rationally immanent world, but he rejected the hylozoism implicit in Leibniz and Baruch Spinoza. The ordered and dynamically informed Universe could be understood, and must be understood, by an active reason. In his correspondence, Newton claimed that in writing the Principia “I had an eye upon such Principles as might work with considering men for the belief of a Deity”. He saw evidence of design in the system of the world: “Such a wonderful uniformity in the planetary system must be allowed the effect of choice”. But Newton insisted that divine intervention would eventually be required to reform the system, due to the slow growth of instabilities. For this, Leibniz lampooned him: “God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion.”
Newton’s position was vigorously defended by his follower Samuel Clarke in a famous correspondence. A century later, Pierre-Simon Laplace’s work “Celestial Mechanics” had a natural explanation for why the planet orbits do not require periodic divine intervention.
Effect on religious thought
Newton and Robert Boyle’s approach to the mechanical philosophy was promoted by rationalist pamphleteers as a viable alternative to the pantheists and enthusiasts, and was accepted hesitantly by orthodox preachers as well as dissident preachers like the latitudinarians. The clarity and simplicity of science was seen as a way to combat the emotional and metaphysical superlatives of both superstitious enthusiasm and the threat of atheism, and at the same time, the second wave of English deists used Newton’s discoveries to demonstrate the possibility of a “Natural Religion”.
Newton, by William Blake; here, Newton is depicted critically as a “divine geometer”. This copy of the work is currently held by the Tate Collection.
The attacks made against pre-Enlightenment “magical thinking”, and the mystical elements of Christianity, were given their foundation with Boyle’s mechanical conception of the Universe. Newton gave Boyle’s ideas their completion through mathematical proofs and, perhaps more importantly, was very successful in popularising them.
See also: Isaac Newton’s occult studies and eschatology
In a manuscript he wrote in 1704 in which he describes his attempts to extract scientific information from the Bible, he estimated that the world would end no earlier than 2060. In predicting this he said, “This I mention not to assert when the time of the end shall be, but to put a stop to the rash conjectures of fanciful men who are frequently predicting the time of the end, and by doing so bring the sacred prophesies into discredit as often as their predictions fail.”
In the character of Morton Opperly in “Poor Superman” (1951), speculative fiction author Fritz Leiber says of Newton, “Everyone knows Newton as the great scientist. Few remember that he spent half his life muddling with alchemy, looking for the philosopher’s stone. That was the pebble by the seashore he really wanted to find.”
Of an estimated ten million words of writing in Newton’s papers, about one million deal with alchemy. Many of Newton’s writings on alchemy are copies of other manuscripts, with his own annotations. Alchemical texts mix artisanal knowledge with philosophical speculation, often hidden behind layers of wordplay, allegory, and imagery to protect craft secrets. Some of the content contained in Newton’s papers could have been considered heretical by the church.
In 1888, after spending sixteen years cataloging Newton’s papers, Cambridge University kept a small number and returned the rest to the Earl of Portsmouth. In 1936, a descendant offered the papers for sale at Sotheby’s. The collection was broken up and sold for a total of about £9,000. John Maynard Keynes was one of about three dozen bidders who obtained part of the collection at auction. Keynes went on to reassemble an estimated half of Newton’s collection of papers on alchemy before donating his collection to Cambridge University in 1946.
All of Newton’s known writings on alchemy are currently being put online in a project undertaken by Indiana University: “The Chymistry of Isaac Newton”.
Newton’s fundamental contributions to science include the quantification of gravitational attraction, the discovery that white light is actually a mixture of immutable spectral colors, and the formulation of the calculus. Yet there is another, more mysterious side to Newton that is imperfectly known, a realm of activity that spanned some thirty years of his life, although he kept it largely hidden from his contemporaries and colleagues. We refer to Newton’s involvement in the discipline of alchemy, or as it was often called in seventeenth-century England, “chymistry.”
Enlightenment philosophers chose a short history of scientific predecessors – Galileo, Boyle, and Newton principally – as the guides and guarantors of their applications of the singular concept of nature and natural law to every physical and social field of the day. In this respect, the lessons of history and the social structures built upon it could be discarded.
It was Newton’s conception of the universe based upon natural and rationally understandable laws that became one of the seeds for Enlightenment ideology. Locke and Voltaire applied concepts of natural law to political systems advocating intrinsic rights; the physiocrats and Adam Smith applied natural conceptions of psychology and self-interest to economic systems; and sociologists criticised the current social order for trying to fit history into natural models of progress. Monboddo and Samuel Clarke resisted elements of Newton’s work, but eventually rationalised it to conform with their strong religious views of nature.
Reputed descendants of Newton’s apple tree (from top to bottom) at Trinity College, Cambridge, the Cambridge University Botanic Garden, and the Instituto Balseiro library garden
Newton himself often told the story that he was inspired to formulate his theory of gravitation by watching the fall of an apple from a tree. Although it has been said that the apple story is a myth and that he did not arrive at his theory of gravity in any single moment, acquaintances of Newton (such as William Stukeley, whose manuscript account of 1752 has been made available by the Royal Society) do in fact confirm the incident, though not the cartoon version that the apple actually hit Newton’s head. Stukeley recorded in his Memoirs of Sir Isaac Newton’s Life a conversation with Newton in Kensington on 15 April 1726:
we went into the garden, & drank thea under the shade of some appletrees, only he, & myself. amidst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. “why should that apple always descend perpendicularly to the ground,” thought he to him self: occasion’d by the fall of an apple, as he sat in a comtemplative mood: “why should it not go sideways, or upwards? but constantly to the earths centre? assuredly, the reason is, that the earth draws it. there must be a drawing power in matter. & the sum of the drawing power in the matter of the earth must be in the earths center, not in any side of the earth. therefore dos this apple fall perpendicularly, or toward the center. if matter thus draws matter; it must be in proportion of its quantity. therefore the apple draws the earth, as well as the earth draws the apple.”
John Conduitt, Newton’s assistant at the Royal Mint and husband of Newton’s niece, also described the event when he wrote about Newton’s life:
In the year 1666 he retired again from Cambridge to his mother in Lincolnshire. Whilst he was pensively meandering in a garden it came into his thought that the power of gravity (which brought an apple from a tree to the ground) was not limited to a certain distance from earth, but that this power must extend much further than was usually thought. Why not as high as the Moon said he to himself & if so, that must influence her motion & perhaps retain her in her orbit, whereupon he fell a calculating what would be the effect of that supposition.
In similar terms, Voltaire wrote in his Essay on Epic Poetry (1727), “Sir Isaac Newton walking in his gardens, had the first thought of his system of gravitation, upon seeing an apple falling from a tree.”
It is known from his notebooks that Newton was grappling in the late 1660s with the idea that terrestrial gravity extends, in an inverse-square proportion, to the Moon; however it took him two decades to develop the full-fledged theory. The question was not whether gravity existed, but whether it extended so far from Earth that it could also be the force holding the Moon to its orbit. Newton showed that if the force decreased as the inverse square of the distance, one could indeed calculate the Moon’s orbital period, and get good agreement. He guessed the same force was responsible for other orbital motions, and hence named it “universal gravitation”.
Various trees are claimed to be “the” apple tree which Newton describes. The King’s School, Grantham, claims that the tree was purchased by the school, uprooted and transported to the headmaster’s garden some years later. The staff of the (now) National Trust-owned Woolsthorpe Manor dispute this, and claim that a tree present in their gardens is the one described by Newton. A descendant of the original tree can be seen growing outside the main gate of Trinity College, Cambridge, below the room Newton lived in when he studied there. The National Fruit Collection at Brogdale can supply grafts from their tree, which appears identical to Flower of Kent, a coarse-fleshed cooking variety.
Published in his lifetime
- De analysi per aequationes numero terminorum infinitas (1669, published 1711)
- Method of Fluxions (1671)
- Of Natures Obvious Laws & Processes in Vegetation (unpublished, c. 1671–75)
- De motu corporum in gyrum (1684)
- Philosophiæ Naturalis Principia Mathematica (1687)
- Opticks (1704)
- Reports as Master of the Mint (1701–25)
- Arithmetica Universalis (1707)
- The System of the World (1728)
- Optical Lectures (1728)
- The Chronology of Ancient Kingdoms Amended (1728)
- De mundi systemate (1728)
- Observations on Daniel and The Apocalypse of St. John (1733)
- An Historical Account of Two Notable Corruptions of Scripture (1754)<|endoftext|>
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# Logans Logo
Extracts from this document...
Introduction
Mathematics
Portfolio Type 2
LOGAN’S LOGO
January 2009
Maria Stormo
This diagram shows a rectangle which is divided into three regions by two curves. The shaded region between the two curves is a logo which can be represented by two mathematical functions. The rectangle is 6.5cm by 6.1cm with a 0.5mm thick frame and the lines of the curves are 0.5mm thick. The aim for this portfolio is to manage to find mathematical functions to represent the logo, be able to modify the logo so that it fits other types of dimensions, and in the end to be able to calculate the area fraction of the logo on the business card. What we can ask ourselves is if the area fraction of the business card will be of the same value as the original logo.
1. Developing mathematical functions for the logo
1.1. Finding the coordinates
The problem is to find the two functions that model these two curves. One type of function that can represent these curves might be sine functions. To check this thesis out, it is first necessary to know the coordinates of the curves.
A parameter is a quantity that defines characteristics of functions. There are several parameters relevant to this portfolio:
• The thickness of the lines: 0.5 mm
• The length of the logo: 6.5 cm
• The height of the logo: 6.1 cm
• The type of function used
A variable is a value that may vary. There are several variables relevant to this portfolio:
• The x-value
• The measurements, as others may have measured differently
• The size of the card
Middle
2,7
2,3
3
2,7
3,2
3,1
X
Y
3,5
3,6
3,6
3,8
3,8
4,1
4
4,4
4,2
4,7
4,5
5,1
4,9
5,5
5,2
5,7
5,6
5,7
6,1
5,2
6,4
4,6
6,5
4,4
1.2. Sine regression
After listing these coordinates into the list function on my GDC, a Texas TI-84 Plus, I can use the sine regression to find an appropriate function for the points.
The lower curve (g(x)):
y = a ∙ sin ( bx + c ) + d
a = 1.668240274
b = 0.8100728241
c = –2.216031762
d = 1.96525663
g(x) = 1.668240274 sin ( 0.8100728241 x – 2.216031762 ) + 1.96525663
The upper curve (f(x)):
y = a ∙ sin ( bx + c ) + d
a = 2.378686945
b = 0.7859285241
c = –2.565579771
d = 3.202574671
f(x) = 2.378686945 sin ( 0.7859285241 x – 2.565579771 ) + 3.202574671
By comparing the two functions to the points of the logo, it can be noticed that these functions do not represent the two curves in the logo very precisely. This can be explained by that the curves are not sine functions. Evidence for this is that for a function to be a sine function it has to be symmetric, and the curves of the logo do not have that property.
1.3. Cubic regression
Another type of function that may represent the curves can be polynomial function of 3rd order, a cubic function. Using the graphing package, “Graph 4.3” by Ivan Johansen downloaded at http://www.padowan.dk/graph/, I can find the polynomial function that best represent the curves.
The lower curve (g(x)):
y = ax3 + bx2 + cx + d
a = –0.089515466
b = 0.74847837
c = –0.88572438
d = 0.58866081
R2 = 0.9995
g(x) = –0.089515466x3 + 0.74847837x2 – 0.88572438x + 0.58866081
The upper curve (f(x)):
y = ax3 + bx2 + cx + d
a = –0.13020708
b = 1.2745508
c = –2.4169654
Conclusion
=
=
An accurate way to calculate this is to use the integral function on the GDC.
This gives the area(A) beneath the two curves to be:
A(g(x)) =15.5419
A(f(x)) = 23.6114
area = 23.6114 – 15.5419
= 8.0695
Total area of the card: 9∙5 = 45
= 0.179322
It is impossible to simplify this fraction, but if we first simplify the answer to fewer decimals it is possibe to get a simplified fraction.
0.18 =
So the logo occupy of the card surface.
This might be an important aspect of a business card because it is important for a logo to be noticed but at the same time not take up to much space.
4. Conclusion
In the introduction I asked if the area of the logo on the business card was the same as the area of the original logo, and to answer this the area of the initial logo has to be calculated.
Using the GDC the area of the initial logo is given to be:
0.18 =
So the answer is yes, the area of the logo on the business card is the same as the area of the original logo. This means that all the calculations and modifications are correct and that the logo is precicely modified into other dimensions.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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Know the basics
What is leukemia?
Leukemia is cancer of the blood cells. There are red blood cells, white blood cells, and platelets, which are produced by the bone marrow – the soft, spongy tissue found in the central cavities of bones. Unlike other cancers, leukemia does not produce a mass (tumor) but results in the overproduction of abnormal white blood cells. Red blood cells that carry oxygen and other materials to the tissues of the body, white blood cells that fight infection, and platelets that help the blood clot. Hundreds of billions of new blood cells are produced in the bone marrow each day, providing the body with a constant supply of fresh, healthy cells.
Leukemia usually involves the white blood cells – are a part of the immune system, and help our bodies fight infection. As a result, these abnormal cells, called leukemic cells, are unable to fight infection the way healthy white cells can. Come with time, the accumulation of the leukemic cells also interferes with the production of other blood cells. Eventually, the body has too few red cells for supplying oxygen to the body’s tissues, too few platelets for proper clotting and too few healthy white cells for fighting infection. People with leukemia are at risk for bruising, bleeding, and infections.
There are many types of leukemia, which are classified by the specific type of white blood cell involved. The main types of leukemia are myelogenous and lymphocytic, and each type has an acute (rapidly progressing) and a chronic (slowly progressing) form. Acute leukemia mainly affects cells that are immature or not fully developed, prevents them from maturing and functioning normally. Chronic leukemia develops more slowly so that the body still has some healthy cells available to fight infection.
How common is leukemia?
Leukemia is extremely common in children and teens. Also, it affects far more adults. It commonly affects more males than females and more common in Caucasians than in African-Americans. It can affect patients at any age. It can be managed by reducing your risk factors. Please discuss with your doctor for further information.
Know the symptoms
What are the symptoms of leukemia?
The common symptoms of leukemia are:
- Anemia, pale skin. It is caused by having a lower than the normal number of red blood cells.
- Frequent bleeding from the gums, rectum or nose.
- Tiny red spots on your skin (petechiae).
- Susceptibility to infections.
- Frequent fevers or chills.
- A new lump or swollen gland in your neck, under your arm, or in your groin.
- Weight loss.
- Persistent, unexplained fatigue, weakness.
- Bone pain.
- Excessive sweating at night.
- Swelling and pain on the left side of the belly.
There may be some symptoms not listed above. If you have any concerns about a symptom, please consult your doctor.
When should I see my doctor?
If you have any signs or symptoms listed above or have any questions, please consult with your doctor. Everyone’s body acts differently. It is always best to discuss with your doctor what is best for your situation.
Know the causes
What causes leukemia?
Scientists do not know the exact causes of leukemia. It is believed that the trigger is a combination of genetic and environmental factors. In general, leukemia is thought to occur when some blood cells acquire mutations in their DNA — the instructions inside each cell that guide its action. There may be other changes in the cells that have to be fully understood could contribute to leukemia.
Know the risk factors
What increases my risk for leukemia?
There are many risk factors for leukemia, such as:
- Excessive radiation, harmful chemical exposure;
- Treated chemotherapy or radiation therapy;
- Conditions caused by abnormal chromosomes, such as Down syndrome.
Understand the diagnosis & treatment
The information provided is not a substitute for any medical advice. ALWAYS consult with your doctor for more information.
How is leukemia diagnosed?
You may experience the following diagnostic exams:
- Physical exam. Pale skin from anemia, swelling of your lymph nodes, and enlargement of your liver and spleen are considered as a sign of leukemia.
- Blood tests.This test can determine if you have abnormal levels of white blood cells or platelets — which may suggest leukemia.
- Bone marrow test. Specialized tests of your leukemia cells may reveal certain characteristics that are used to determine your treatment options.
How is leukemia treated?
The leukemia treatment options will be based on your age and overall health, the type of leukemia you have, and whether it has spread to other parts of your body.
Common treatments used to fight leukemia include:
- Chemotherapy is the major form of treatment for leukemia. This drug treatment uses chemicals to kill leukemia cells.
- Biological therapy. Biological therapy works by using treatments that help your immune system recognize and attack leukemia cells.
- Targeted therapy. Targeted therapy uses drugs that attack specific vulnerabilities within your cancer cells.
- Radiation therapy. Radiation therapy uses X-rays or other high-energy beams to damage leukemia cells and stop their growth.
- Radiation therapy may be used to prepare for a stem cell transplant.
- Stem cell transplant. A stem cell transplant is a procedure to replace your diseased bone marrow with healthy bone marrow.
Lifestyle changes & Home remedies
What are some lifestyle changes or home remedies that can help me manage leukemia?
The following lifestyles and home remedies might help you cope with leukemia:
- Balanced diets;
- Do exercises;
- Avoid stress;
- Avoid high doses of radiation;
- Protect yourself from toxic chemical such as benzene;
- Do not smoke or tobacco;
- Using a small ginger spice or candy or tea can help for nausea or vomiting;
- Go to bed on time to set up a sleep habit;
- Take off your time to get a plenty rest;
- Get the support you need.Spend time with people who care about you, and let them help you.
If you have any questions, please consult with your doctor to better understand the best solution for you.
Hello Health Group does not provide medical advice, diagnosis or treatment.
Leukemia. http://my.clevelandclinic.org/health/diseases_conditions/hic_Leukemia Accessed September 17, 2016.
Leukemia http://www.mayoclinic.org/diseases-conditions/leukemia/basics/tests-diagnosis/con-20024914 Accessed September 17, 2016.
Leukemia http://www.webmd.com/cancer/tc/leukemia-exams-and-tests Accessed September 17, 2016.
Review Date: January 4, 2017 | Last Modified: January 4, 2017<|endoftext|>
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The Beautiful Story
The Ugly Duckling by Hans Christian Anderson is a popular fairy tale that your students have probably heard before. After sharing this story with them, these lesson plans for The Ugly Ducking will improve students reading and writing skills. The main character in the story, the "duckling," is shunned by others because of his appearance — he doesn't fit in. He tries to find a family, but fails at every attempt. Finally, when he can just not bear it any longer, he throws himself at the mercy of the swans, only to discover he is a swan himself. There are many different versions of this story — most use the storyline of Hans Christian Anderson with various illustrations for the story.
An objective students must often meet in elementary reading curriculum is how the setting and specific character traits affect a story. Can this same story take place in a different setting? Could this same story take place if the main characters were frogs? The answer to that question for The Ugly Duckling is, "Well, that depends." First, students should explain to you what the setting of the story is. If students can't remember what setting means, then you can remind them with a definition and examples from stories you have read this past year. Next, ask students if the story could take place in various settings around the world. So, you could ask, "Could the author set the story in a desert?" Use the think-pair-share strategy for students to answer these setting questions. This way, each student is thinking about the question and sharing their thoughts with at least one other person.
With the desert question, most children should answer NO. Ducks and swans cannot survive in a desert, so this story could not happen. However, the important objective with this lesson is that students understand what setting is and that they can reason whether or not the setting affects the plot. If they can come up with a way it can take place in a desert, then you can accept their logic. You could also ask questions like:
- Could this story take place in Manhattan?
- Could this story take place in our town?
- Could this story take place in Central Park?
Once you have discussed setting with students, then you can also discuss character traits if you want to extend the lesson. Would this story work if the main characters were dogs? Possibly because puppies look so different; perhaps a Basset Hound puppy wouldn't fit in with a family of German Shepherds, and so on.
When you have finished the lesson, give students a chance to do some independent practice. Students can illustrate a twist on The Ugly Duckling with a brief written explanation. For example, maybe one group said the story would work in a busy city, as long as there was a body of water for the ducks and swans. So, the students in that pair/group would draw a picture of the ugly duckling in a city and write about why the story would work in that setting. You can post these illustrations on a bulletin board with the title: "Try these settings for The Ugly Duckling."
With either of these lesson plans for The Ugly Duckling, you can do them separately or together. One is not dependent on completing the other, and you can do them in any order. For the writing lesson plan, students will compare and contrast themselves with the ugly duckling. For younger primary students, you can do this lesson as a shared writing activity. For older students, they can work on it in pairs or individually. You will know the ability of your students and how well they can accomplish the following tasks.
1. Students fill out a Venn diagram comparing and contrasting themselves with the ugly duckling. Even if students are doing this on their own, you can help them think of ideas as a class — especially if this is one of the first times they are working with a Venn diagram or if students seem stuck for ideas.
2. Students will write a five to six sentence paragraph, comparing and contrasting themselves to the main character. The paragraph will have a topic sentence, one or two sentences with comparisons, one or two with contrasts, and a concluding sentence. As the teacher, you can provide students with the topic sentence, especially if writing a paragraph is new for them.
3. Students draw an illustration to go with their paragraphs.
4. In small groups of four or five, students share their paragraphs and illustrations with each other before these are hung up for display.
The Ugly Duckling Lesson
The Ugly Duckling is such a wonderful story, and one that has an obvious lesson for children. But besides this lesson of self-esteem, individuality and longing to fit in, you can use the book to teach reading and writing skills.
My 16 years of teaching experience in preschool and elementary school
Andersen, Hans Christian. The Ugly Duckling. North-South Books, 2008.<|endoftext|>
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The pathogen Yersinia pestis, believed to have originated in Asia, has caused some of the deadliest pandemics in human history, peaking in Europe between 1348 and 1350 in the Great Pestilence with strains continuing until the 1700s. Both the event and the disease are also known by the name given by later writers, The Black Death. Experienced by the whole of Eurasian/Mediterranean civilization to some degree, it so traumatized the human race that the formal name the disease was given in Europe, derived from the Latin words for to strike down, and to lamentnote , is to this day synonymous with both "widespread threat to society" and "lethal contagious disease": The Plague.
Keep in mind that the disease is not called the "bubonic plague"; it's simply "plague". "Bubonic" is merely one way the disease plays out: by infecting the lymph system and colonizing the lymph nodes, which swell up into "bubos".note In coastal areas, the most common form of plague at that time was pneumonic plague, which affects the lungs. Septicemic plague affects the bloodstream. The difference? Pneumonic plague kills all but a handful of sufferers, usually within a day. Septicemic plague is always fatal and kills within hours. Bubonic plague victims, on the other hand, can take days to die, and one-third actually survive with long lasting traumatic damage to their internal organs and immune systems- with the effect of making these victims the most noticeable and horrifying.
The disturbing explanation for the disease's alternate name, the black death, is that in both the septisemic and bubonic presentations, the victims are left in a horrific swollen and decaying state due to a combination of ruptured lymph nodes and frostbite-like patches of black gangrene —before they die. Following the plague pandemic, this image was so burned into Europe's psyche that it spawned our modern visualization of The Undead, a stark contrast to the prior depictions of liches and kin as unusually pale but otherwise unremarkable, animalistic, or totally skeletal.
There have been many other outbreaks of plague other than the 1348-1350 pandemic. The most recent occurred at the beginning of the 20th century, killing tens of millions in India and China, and the earliest outbreak for which we have definitive historical evidence (at least according to some historians) is The Plague of Justinian in the 6th Century. The growing use of antibiotics and the improvement of hygiene conditions have ensured no pandemic of such scale can happen anymore in most modern countries, but there are still limited outbreaks in the areas where there's a lack of these.
When this appears in a story you know things are quickly going to go downhill for the heroes (if there even are heroes). Due to its transcending memories of death, destruction, and desperation, such stories generally have a Downer Ending. It tends to be used because to most cultures, death is feared and a reminder of our own mortality is chilling.
See also The Plague for devastating pandemics in general. For more on the science and history of the plague, see the Other Wiki.
As a Death Trope, all Spoilers will be unmarked ahead. Beware.
- Problem Children Are Coming from Another World, Aren't They? has one of the characters being the Moe Anthropomorphism of The Black Death, Black Percher with her real name being Pestilence/Pest.
- God Child has a small arc that revolved around a woman who was mistaken to be a vampire due to the amount of deaths that have been on the rise and her looking incredibly young for a lady in her 40s at the time, but it's revealed that the people died due to the plague and she had no real part in it.
- In Robin Vol 1 the biological weapon Edmund Dorrance gets his hands on is revealed to be the black death, which an old Nazi scientist had managed to recreate and which Dorrance somehow heard of and sent his hired help to go track down.
- The Day of the Barney Trilogy reveals that Barney is responsible for this.
- Black Death obviously has this as a topic. It shows well how different people responded to the outbreak in 1348.
- The Black Death plays a major thematic role in The Seventh Seal.
- According to Batman Begins, the Black Death was the League of Shadows' doing.
- Lady Snowblood 2: Love Song of Vengeance: The bad guys rather foolishly inject Ransui with the plague before throwing him back into the general population.
- Referenced in Monty Python and the Holy Grail.
- Panic in the Streets is about a random crook who gets murdered over a crooked card game. Things take a turn when the autopsy reveals the dead guy was going to die within 24 hours anyway because he had pneumonic plague. A heroic doctor and the cops then go on a frantic chase to find the killers before they spread plague all over the city.
- Season of the Witch takes place during the Black Death, though it misrepresents its symptoms as being more similar to leprosy (probably to increase its horror value) and ends on the revelation that it was created by a demon to raise an undead army.
- In Miss Mend, anti-Bolshevik terrorists set out to unleash the Black Death on the Soviet Union by concealing ampoules of plague culture inside electrical insulators.
- White Shadows in the South Seas: Dr. Matthew Lloyd, who has become too troublesome to the Evil Colonialist in charge of a south Pacific island, is set adrift on the open ocean in a boat filled with victims of the plague. Luckily for Lloyd the ship runs aground next to an island before he catches it.
- The 2017 Beauty And The Beast reveals that Belle's mother died from this when Belle was too young to remember her.
- Isle of the Dead revolves around the people on an isolated island finding out that septicemic plague is in their midst. Dwindling Party ensues, as does paranoia, fear, and murder as the situation deteriorates.
- In Northern Europe, the effect of the black death was so severe it held the population down for centuries. It didn`t help much that the disease showed up time and again all the way to 1650. In Norway especially, people came to see the plague incarnate as an old hag, clad in dark clothes, wearing a broom and a rake. Her face was either a skull or made of decomposing flesh. Tradition has it that she usually saved some if she used the rake. On the other hand, if she used the broom, no one was spared.
- Based on truth in television in the more remote parts of Norway and possibly Sweden, where the entire population of some valleys were found dead after the plague, and were not repopulated for 200 years. In one particular case, a lone hunter just accidentally stumbled over the local church, still standing in the middle of nowhere. In the meantime, the building was made a hive for bears. The bearskin allegedly still hangs on the wall in this particular church.
- The most known depiction of the Plague Hag (Pesta) was made in the late nineteenth century by Norwegian painter Theodor Kittelsen, who claimed to have met her in a dark wood near his home. And he ran really fast on his way home. He claimed she looked like this◊.
- Boccaccio's Decameron (written a few years after the plague) is about ten wealthy Florentines who decamp to the countryside with their retinue to escape from the plague, and pass their days in storytelling.
- In the Alternate History novel The Years of Rice and Salt, the Black Death causes the extinction of Western civilization.
- At the end of The Name of the Rose (set in 1327) it's mentioned that William eventually died during the Black Death.
- Ken Follett's World Without End includes a section where the plague comes to Kingsbridge and Caris, our heroine, desperately struggles to limit the destruction. Later parts of the book deal with the sociological changes the plague brought.
- When everyone in the Michael Crichton novel Timeline get tired of the Corrupt Corporate Executive, they send him back in time to 14th century Europe at the height of The Plague. It takes him a little while to realize just where he's been sent, but when he puts it together he notes that he's already showing symptoms...
- A Journal of the Plague Year, as its name says, deals with the epidemic of London between 1664 and 1666.
- In The Trolls, the children's usual babysitter is unable to look after them, because she caught a "touch of" the Black Death while vacationing in Europe. Alarmingly, she still offers to show up if the parents really need a babysitter. The mom understandably doesn't take her up on this offer.
- The Dresden Files:
- In Death Masks, it's revealed that the Black Death was originally caused by Fallen Angels using magic. The plot of the book involves them preparing to do it again.
- It is later mentioned again in Cold Days. Harry knocks some jars off a shelf in the home of Mothers Winter and Summer, and when he puts them back, he notices the labels.
The writing on the cracked pot said simply, Wormwood.
The letters began to fade, but I saw some of the others: Typhos. Pox. Atermors. Choleros. Malaros.
Typhus. Smallpox. The Black Death. Cholera. Malaria.
And there were lots of other jars on the shelf.
- Connie Willis' Hugo- and Nebula-award winning novel Doomsday Book is set in a future version of Oxford where time-travel has become possible, but is used mostly by historians. Kivrin Engle, who studies medieval history, convinces history professor Dunworthy to send her back to the 14th century. Unfortunately, something goes (very) wrong, and Kivrin finds herself in the middle of the 1348 Black Death epidemic. Oopsies!
- A good amount of Scandinavian literature cover the period, due to the fact that the demographics and political landscape changed radically in these areas, at least partly because of the plague. And the plague survived in living tradition all over the place.
- Norwegian examples include The Bridegroom, telling the tragic story of a girl who falls in love with a fiddler who dies in the plague, and the story of Guro Heddeli, telling the tragic story of another girl who falls victim to the plague. Later, a children´s book was written on the subject: A Ship Arrived In Bjorgvin In 1349 telling more accurately on the subject.
- In Up the Line, by Robert Silverberg, there is a popular series of time tours tracing the 14th century Black Death epidemic. Protagonist Jud, while a bit depressed, takes one tour in the series of four (it was the one he could get a spot in on short notice).
- A Parcel of Patterns, among other works, tells the true story of a Derbyshire village called Eyam whose inhabitants voluntarily quarantined themselves for over a year when the plague reached them.
- In Horatio Hornblower and Noah's Ark, Midshipman Hornblower is sent to pick up supplies from the city of Oran on the day of a plague outbreak. Facing a three-week quarantine in a plague city with a crew of panicky men and a delay in desperately-needed foodstuffs, Hornblower asks and is allowed to spend the period on the ship—they're more effectively quarantined at sea than anywhere else and it gets them back to the fleet quickly. (Fortunately, they evade contagion.)
- A large portion of The Dwarf is spent with the titular character observing the death and desperation around him as the Black Death strikes his city.
- Both pneumonic and bubonic strains feature in the novel The Plague which details a outbreak of the disease in the French Algerian city of Oran.
- In the Highlander TV series, Amanda died for the first time during the Black Death. She was not sick herself but she was stealing from houses under quarantine and was clubbed to death because people assumed she was infected.
- In an episode of Torchwood, a number of people slip through the time rift into present-day Cardiff — causing, among other things, an outbreak of bubonic plague. Fortunately, Owen recognizes it, and these days it's treatable.
- In the Secret Army episode "Ring of Rosies" La Résistance discover that an Allied airman being sheltered by them caught bubonic plague from his service in Africa, and so they must prevent the other members of his unit from escaping and infecting an occupied population suffering from lack of food and medical care. One man who does so is gunned down and his body burnt by Molotov Cocktail.
- NCIS. Tony opens a letter and gets sprayed by a white powder that they naturally assume is anthrax, but it turns out to be weaponised Y. pestis. There is no cure, but fortunately as a fit, well-nourished male with access to modern medical care Tony's chances of surviving are a lot better.
- In NCIS: New Orleans, an early case involves plague being found on a Navy ship. In a Required Spinoff Crossover, Tony is sent in from DC to assist on the case, because of his previous experience.
- The Collector: The plague features prominently in Morgan's past.
- The patient of the week is infected with this in the House episode "Sleeping Dogs Lie" although she doesn't die from it.
- True Blood: In a flashback, we learn that in the 17th Century, Nora was helping people infected, contracting the illness herself in the process. This led to Godric turning her into a vampire.
- Frontier Circus: In "Incident at Pawnee Gun", Casey finds himself in a Quarantine with Extreme Prejudice situation when peace officers believe his chimpanzee has the bubonic plague.
- Doctor Who: In "The Visitation'', the TARDIS arrives in a village outside of London during the time of the Black Death. The alien invaders plan to use a genetically modified version of the bubonic plague to wipe out humanity.
- The ecoterrorists in the second series of Bron|Broen are developing a supercharged genetically-engineered version of plague, intending to release it at a meeting of the EU member states' environment ministers.
- The second Horatio Hornblower telefilm is based in part on Hornblower and Noah's Ark. While quarantined at sea, one of the sailors starts weaving and swaying, prompting the others to try and toss him overboard using spars until Hornblower steps in. There's a tense moment when Hornblower gets close to the man, but a sniff of his breath shows that he's just drunk. Later, the reckless Dreadnought Foster takes some cattle before the quarantine is up, over Hornblower's voluble protests.
- The Black Death shows up in the first season of Blackadder, in the episode "Witchsmeller Pursuivant".
- In the Father Brown episode "The Alchemist's Secret," Father Brown and his old friend Professor Hilary Ambrose investigate the alleged hiding of an alchemical formula to turn lead into gold, supposedly hidden in a secret chamber at Ambrose's university. There is indeed a formula hidden there, but it's actually for concentrated, weaponized version of the plague that was tested on a nearby village, wiping said village from the map.
- The Outer Limits (1995): In "Last Supper", Jade discovered that she was immortal at 20 years old when everyone else in her village in Spain died of the Black Death and she survived.
- Planet of the Apes: In "The Surgeon", Leander tells Urko that there is an outbreak of the Black Death in the clinic so that he will leave quickly and Galen, Virdon and Burke can escape.
- Seanan McGuire's cheery Filk Song "The Black Death" argues for the theory that the Black Death was not in fact Y. Pestis:
Speaking epidemiologically, bubonic plague doesn't make sense to me.Yersinia pestis gets you dead, it's true, but it isn't as effective as the common flu.If you want to wipe out half of Europe's population, you'll need a better agent for your devastation;You need a viral agent that is tried and tragic — let's take a look at fevers that are hemorrhagic.
- The whole album A Chronicle of the Plague as well as the track "Breath of the Black Plague" from the album Twilight of Europe by the Ukrainian minimalist dark ambient band Dark Ages are all about the subject.
- A scheduled event in Medieval II: Total War. You can have isolated outbreaks of generic plagues at any time, but near the endgame the world is rocked by the historical Black Death. Typically the campaign crashes to a halt as armies lose men faster than replacements can be recruited, royal family members die left and right, and nations' economies tank from all those dead peasants. Of course, an enterprising player can take advantage of this by, say, sneaking a Spy into an afflicted settlement and sending him to infiltrate as many enemy cities as possible before expiring...
- The spread of the Black Death is also one of the few scripted events guaranteed to happen in both Crusader Kings games, where it's almost instantly lethal to any character that catches it and effectively destroys the economy of any provinces it spreads to.
- The most recent expansion for the second game, The Reaper's Due, focuses on plagues and diseases in general. Special attention is given to the Black Death itself, with major announcements as it spreads through various regions and several events related to the nobility and peasantry reacting to the devastation it brings.
- One scenario of Plague Inc. allows you to take control of a modern-day outbreak of the Black Death and evolve it so that it kills off humanity.
- Though Vampyr takes place during the 1918 Spanish flu, the Black Death (more specifically the 1665 Plague of London) is referenced several times in the backstory and are revealed to have be tied: It turns out that both diseases were engineered by the Red Queen to make humanity suffer and sent a Disaster, a Humanoid Abomination in her service, to spread it whenever they went. The Black Death was ended when vampire champion William Marshall fought against the Disaster in 1666 under St. Paul's Cathedral, with him being forced to burn the church down and causing the Great Fire of London to make sure she was dead.
- Maggie in Times Like This is a victim of the Black Death in Ireland at first... but Cassie then takes her to a future time, when vending machines have the cure for any ailment, and gives her life-saving medicine.
- As has been noted elsewhere on this page, the plague is an entirely treatable illness these days (its effects have been likened to "a bad case of the flu with some pneumonia and chicken pox symptoms tossed in for good measure"). Ironically, this has led to the darkly humorous fact that, in modern industrialized first-world countries at least, the most common source of new plague outbreaks is the vaccination itself (about 1 in 1000 recipients of the plague vaccine actually contract the disease from getting vaccinated).<|endoftext|>
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# Trigonometry: Angles
## Trigonometry: Angles
Trigonometry is a branch of mathematics dealing with the measurement of sides and angles of a triangle. We apply trigonometry in Engineering, Surveying, Astronomy, Geology etc.
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10 Math Problems officially announces the release of Quick Math Solver and 10 Math ProblemsApps on Google Play Store for students around the world.
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### Angles:
Let O be the fixed point on OX, the initial line. Let OY be the revolving line. Then the amount of rotation of OY about O with respect to OX is known as the angle between OX and OY. Here the angle formed is XOY.
If the revolving line rotates about the fixed point O in the anticlockwise direction, the angle so formed is said to be positive.
If the revolving line rotates about the fixed point O in the clockwise direction, the angle formed is said to be negative.
#### Measurement of Angles
A line making one complete rotaion makes 360°. When a line makes a quarter tern, it makes 90° or 1 right angle. The size of a right angle is same in every measurement. The following three system are commonly used in the measurement of angles:
(a) Sexagesimal System (Degree System)
(a) Sexagesimal System: This system is also called British System. In this system, the unit of measurement is degree. So, this system also is known as the degree system. In this system, a right angle is divided into 90 equal parts and each part is called a degree. A degree is divided into 60 equal parts and each part is called as one minute. A minute is also divided into 60 equal parts and each part is called as one second. Therefore, we have
60 seconds = 1 minute (60’’ = 1’)
60 minutes = 1 degree (60’ = 1°)
90 degrees = 1 right angle
The degree, minute and second are denoted by (°), (’) and (’’) respectively.
(b) Centesimal System: This system is also called the French System. In this system, the unit of measurement is grade. So, this system also is known as the grade system. In this system, a right angle is divided into 100 equal parts and each part is called a grade. A grade is divided into 100 equal parts and each part is called a minute. A minute is also divided into 100 equal parts and each part is called a second. Therefore, we have
100 seconds = 1 minute (100’’ = 1’)
100 minutes = 1 grade (100’ = 1g)
100 grades = 1 right angles
The grade, minute and second are denoted by (g), (‘) and (‘’) respectively.
(c) Circular System: In this system, the unit of measurement of an angle is a radian. An angle at the centre of a circle subtended by an arc equal to the length of radius of the circle is known as 1 radian. It is denoted by (c).
As the total length of circumference of a circle is 2Ï€r units, the angle subtended by circumference of a circle at the centre is 2Ï€r/r radian i.e. 2Ï€c.
Which is, 4 right angle = 2Ï€c
or, 1 right angle = (Ï€/2)c
Now, from the definition of sexagesimal measure, centicimal measure and circular measure of angles, we have,
1 right angle = 90° = 100g = (Ï€/2)c
#### Theorem: “Radian is a constant angle.”
Proof:-
Let, O be the centre of the circle and OP = r be the radius of the circle. An arc PQ = r is taken. PO and QO are joined. Produce PO to meet circle at R. Then by definition POQ = 1 radian. The diameter PR = 2r, POR = 2 right angles = 180° and arc PQR = ½ × circumference = ½ × 2Ï€r = Ï€r.
Now, since the angles at the centre of a circle are proportional to the corresponding arcs on which the stand.
i.e. POQ/POR = arc PQ/arc PQR
Since 1 radian is independent of the radius of the circle, it is a constant angle.
Proved.
#### Relation between different system of measurement of angles:
Since, 1 right angle = 90° and 1 right angle = 100g
90° = 100g
1° = (10/9)g
Also, 1g = (9/10)°
Again, Ï€c = 180° = 200g
1c = (180/Ï€ and 1c = (200/Ï€)g
Also, 1° = (Ï€/180)c and 1g = (Ï€/200)c
### Workout Examples
Example 1: Reduce 24g 20’ 44’’ into centicimal seconds.
Solution:
The given angle is 24g 20’ 44’’
= 24 × 100 × 100’’ + 20 × 100’’ + 44’’
= 240000’’ + 2000’’ + 44’’
= 242044’’
Example 2: Express 42° 20’ 15’’ into the number of degrees.
Solution:
The given angle is 42° 20’ 15’’
= 42° + (20/60)° + (15/60×60)°
= 42° + 0.333333° + 0.004166°
= 42.337499°
Example 3: Express 48g 54’ 68’’ into degrees, minutes and seconds.
Solution:
The given angle is 48g 54’ 68’’
= (48 + 54/100 + 68/100×100)g
= (48 + 0.54 + 0.0068)g
= 48.5468g
= (48.5468 × 9/10)° [ 1g = (9/10)°]
= 43.69212°
= 43° + 0.69212°
= 43° + (0.69212 × 60)’
= 43° + 41.5272’
= 43° + 41’ + 0.5272’
= 43° + 41’ + (0.5272 × 60)’’
= 43° + 41’ + 31.632’’
= 43° 41’ 31.63’’
Hence, 48g 54’ 68’’ = 43° 41’ 31.63’’
Example 4: Express π/6 radians into sexagesimal and centicimal measures.
Solution:
The given angle is π/6 radians
= (Ï€/6 × 180/Ï€ [ 1c = (180/Ï€)°]
= 30°
Again,
= (Ï€/6 × 200/Ï€)g [ 1c = (200/Ï€)g]
= 33.33g
Example 5: Reduce the following angles into radian measure.
a. 42g75’
b. 42° 15’ 30’’
Solution:
a. 42g 75’
= (42 + 75/100)g
= (42 + 0.75)g
= 42.75g
= (42.75 × Ï€/200)c [ 1g = (Ï€/200)c]
= 0.21375Ï€c
b. 42° 15’ 30’’
= (42 + 15/60 + 30/60×60)°
= (42 + 0.25 + 0.00833)°
= 42.25833°
= (42.25833 × Ï€/180)c [ 1° = (Ï€/180)c]
= 0.2348Ï€c
Example 6: The angles of a triangle are (7x/2)g, (9x/4)° and (Ï€x/50)c. Find the angles of the triangle in degrees.
Solution: Here,
First angle = (7x/2)g = (7x/2 × 9/10)° = (63x/20)°
Second angle = (9x/4)°
Third angle = (Ï€x/50)c = (Ï€x/50 × 180/Ï€)° = (18x/5)°
Now, the sum of all angles of a triangle is 2 right angles.
i.e. (63x/20)° + (9x/4)° + (18x/5)° = 180°
or, {(63x + 45x + 72x)/20}° = 180°
or, (180x/20)° = 180°
or, (9x)° = 180°
or, x° = 20°
Hence, the angles of the triangle are (63×20/20)°, (9×20/4)° and (18×20/5)° i.e 63°, 45° and 72°.
Example 7: Sum of the first and the second angle of a triangle is 150°. The ratio of the number of grades in the first angle to the number of degrees in the second angle is 5:3. Find the angles of the triangle in circular measure.
Solution: Suppose number of grades in the first angle be 5x and the number of degrees in the second angle be 3x.
First angle = 5xg = (5x × 9/10)° = (9x/2)°
Second angle = 3x°
From question,
9x/2 + 3x = 150°
or, 15x/2 = 150°
or, 15x = 300°
or, x = 20°
The first angle = 9x/2 = 9×20/2 = 90°
The second angle = 3x = 3 × 20 = 60°
The third angle = 180° - 150° = 30°
The angles of the triangle in circular measure are,
(90 × Ï€/180)c, (60 × Ï€/180)c and (30 × Ï€/180)c
i.e. (Ï€/2)c, (Ï€/3)c and (Ï€/6)c
You can comment your questions or problems regarding the system of measurement of angles here.<|endoftext|>
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Information sent around the world is encrypted by a variety of encryption protocols, the most popular of which is RSA. But these safeguards on our information and data may not be as secure as is widely believed. Why, and how can they be made more secure?
Perhaps one of the most important aspects of society is communication. It’s what allows diverse populations from across the world to cooperate and socialize, and ideas and opinions to spread around the globe and make us a truly global community.
From writing on papyrus scrolls to sending messages instantly around the world, methods of communication have evolved rapidly over the years. And yet, a concern that has never ceased to exist is that of privacy. The need to keep important messages safe from prying eyes and ears resulted in the field of study now known as cryptography.
The word conjures to mind images of hackers locked in battle with cryptographers, attempting to ferret out secrets of international import. But cryptography has far older origins than one might think. The oldest use of codes can be traced to Egypt in around 1900 BCE, where non-standard hieroglyphs were found carved into stone. Since then, codes and ciphers grew progressively more complicated, from the Caesar Cipher employed by Julius Caesar (simply shift every letter of the alphabet to the left or right by a fixed number of letters) to the supposedly unbreakable Enigma employed by the Germans in WWII. The Allied efforts to crack Enigma, largely aided by the work of Alan Turing, marked the beginning of the era of computers – an era that would see cryptography mutating into a well-defined field of study.
It is in this field that Professor Santanu Sarkar, from the Department of Mathematics, works. His research considers encryption, as he says, “from the attacker’s point of view”. It concerns the use of mathematical constructs called lattices in attempts to break the RSA cryptosystem, one of the most common encryption methods in use today.
Before explaining his research, Professor Sarkar outlines the history of modern cryptography. The foundations of modern cryptography were laid in 1976, when Whitfield Diffie and Martin Hellman published a paper that would lay the foundation for a revolution in cryptography. This paper outlined a new concept – the key-exchange system. Till that point, cryptosystems used symmetric-key encryption. This meant that both the receiver and sender of information used a single, shared key (a term for a large number used in the encryption process) to encrypt plaintext (unencrypted information) and to decipher ciphertext (encrypted information). This necessitates the use of a secure channel for the key to be shared between the sender and the receiver. But the necessity of a secure channel in order to set up a secret key was a insurmountable chicken-egg problem in the real world.
Diffie and Hellman’s paper, on the other hand, posited that a shared key was not necessary. Instead, they proposed that three keys be used – one key known to both parties (traditionally called Alice and Bob in cryptographic lingo), and two others, known only to the respective party. Their method was such that encryption and decryption could be performed without having to share the secret keys.
From the foundations that they laid, a more secure system arose: public-key encryption, which used a public and a private key. The public key would be available to anyone who wanted to communicate securely with a system, while the private key would be known only to the system itself. Encryption would be carried out with the public key and decryption with the private key. It was also mandatory that the private key not be deducible from the public key, as that would compromise the system’s security. Since the private key didn’t need to be shared with anyone via a potentially insecure channel, public-key encryption was clearly a better choice.
Although Diffie and Hellman were able to prove the feasibility of such an encryption system, they weren’t able to come up with a viable system themselves. That was accomplished two years later, by Ron Rivest, Adi Shamir and Leonard Adleman, researchers at MIT. That eponymous cryptosystem – RSA – has since become one of the strongest known encryption standards in the world.
It is now used mainly as part of hybrid encryption methods: data is encrypted using a symmetric-key system, and the shared key is then encrypted using RSA. This is largely because of the RSA algorithm’s computational inefficiency – encrypting the data itself using RSA would take a very long time.
At a basic level, the RSA algorithm is based on the premise that the product of two large prime numbers is very hard to factorize. Put into mathematical terms, consider 2 prime numbers, P and Q, and their product, N. Then, the integer solutions to the equation p(x, y) = N – xy are the factors of N. Trivial (irrelevant) solutions to this equation include (x=1, y=N) and (x=N, y=1). The important solutions, though, are (x=P, y=Q) and (x=Q, y=P). While a computer can solve this equation relatively fast when N is small, larger values of N result in runtimes that render decryption attempts infeasible. For example, it is theorized that a single 1024-bit value of N (i.e, N approx. = 21024) will take approximately 4000 years to factorize!
At this point, Professor Sarkar mentions a caveat. “Till today, the RSA encryption hasn’t been broken in a feasible amount of time by any algorithms that conventional computers can run. However, there is an algorithm, called Shor’s algorithm, that can be used to break RSA encryption using quantum computers.” But since quantum computers are still in nascent stages of development, the RSA algorithm is still considered to be a bastion of cryptography.
The mathematical framework above describes the simplest conceptualization of RSA. The algorithm actually uses the equation ed = 1 + k(N+s), where as before, N=PQ, and s=1-P-Q. e and N are known (and hence, are public keys) and d, k and s are unknown (and hence, private keys). This can be expressed as a polynomial p(x, y, z) = ex – 1 – y(N+z). As before, obtaining the non-trivial solutions of this polynomial is equivalent to breaking the RSA encryption.
Possible avenues of attack on the RSA algorithm were discovered soon after the algorithm was published. Most, though, were inefficient in terms of computing time required. One of the most preferred methods of attack was established by Don Coppersmith, an American mathematician. He postulated and proved a theorem which when used in conjunction with mathematical constructs called lattices, could break the RSA algorithm in polynomial time (This means that the running time is a polynomial function of the input size. It’s largely used to denote programs whose running times don’t blow up too fast). Fortunately for cryptographers around the world, the guarantee of success for such an attack was attached to certain conditions.
The encryption could be broken in a feasible time scale only if d, one of the private keys, was less than N0.292 – which implies that for a secure RSA design, d would have to be greater than N0.292. But Professor Sarkar prefers to think of it as an upper bound for the system to be vulnerable, rather than a lower bound for security. “I always look at the problem from the attacker’s point of view”, he says with a smile. “Hence, I think of values of d for which the system is insecure.” This bound was proven by two cryptographers, Dan Boneh and Glenn Durfee in 1999. For example, if N was a 1000-digit number, the concerned RSA system would be secure as long as d was a 292-or-more digit number. However, the mathematical community conjectured that for the RSA system to be truly secure, d would have to be greater than N0.5. Using the example from before, d would have to have more than 500 digits.
Any increase in the bounds on d would have two consequences. First, it would expose any RSA systems that used values of d below the new bound as insecure. Secondly, an increase in the value of d in any RSA system results in a significant increase in the time taken to decrypt it using Coppersmith’s theorem. Hence, improving the lower bounds on d contributes greatly to improving the security of RSA systems used across the world.
Professor Sarkar goes on to explain that he worked on a further variant of this RSA scheme. “N doesn’t have to be only a product of two prime numbers. It can instead be of the form N=Pr Q, where r is another integer. I worked on proving bounds on d when r=2.” Professor Sarkar’s work improved the bounds on d from d<N0.22 to d<N0.395. Talking about the implications of his work, he says, “The results published will prompt RSA system designers to revise their designs. Since there is a larger range of d over which RSA can be broken, systems will have to be designed with the new bounds in mind.”
I mention to Professor Sarkar that his work seems highly theoretical. He’s quick to point out that it does involve some simulations – he runs attacks on RSA-protected systems using a open-source software called Sage. This serves to validate his results. “My work, and in fact all work since Boneh and Durfee’s paper & Coppersmith’s work involve some implicit mathematical assumptions that no one’s formally proved. I need to run actual attacks in order to validate the bounds I derive.” But, he admits with a rueful grin, “It can get tedious at times. You just have to keep trying the problem from different angles. I also like what I do.”
When I ask him how he decided to venture into cryptography, he points to his alma mater, ISI Kolkata, as his inspiration. “ISI is well known for cryptography. Once I started working in this field, I saw problems of this type, and they interested me. I still work with my colleagues there, as well as with collaborators in China.”
Professor Sarkar is currently attempting to improve the bounds described above even further. He’s also working on problems related to another encryption algorithm, RC4, primarily employed in wireless networks.
Professor Santanu Sarkar received his PhD degree in mathematics from the Indian Statistical Institute. He is currently an Assistant Professor at IIT Madras, India. Before that, he was a guest researcher at the National Institute of Standards and Technology. His main research interests include cryptology and number theory.
Nithin Ramesan is a junior undergraduate in the department of Electrical Engineering. He likes to quiz, write and read Calvin & Hobbes. For bouquets or brickbats, he can be contacted at [email protected].<|endoftext|>
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The Creation Of The State Of Israel:
After the Second World War, the region became one of the scenarios of the Cold War: on the one hand, a Cairo-Damascus axis was developed, closer to the USSR, and, on the other, a more pro-Western camp, led by Saudi Arabia.
In 1947, the UN approved a plan for the partition of Palestine and the creation of two independent states, one Arab and one Jewish, with the city of Jerusalem under international control. A year later, in May 1948, David Ben Gurion proclaimed the establishment of the State of Israel. In May of 1948, the armies of the Arab League (made up of Egypt, Iraq, Syria, Lebanon, Transjordan, Saudi Arabia and Yemen) invaded Israel. The confrontation was decided in favor of the Israelis who, after the conflict, were occupying territories that, according to the division approved by the UN, corresponded to the Arab Palestine.
In 1956, the second Arab-Israeli war broke out as a result of the nationalization of the Suez Canal, undertaken by Egyptian President Gamal Abdul Nasser. In 1967, the third confrontation broke out, the Six Day War, which culminated in the Israeli occupation of the Sinai Peninsula, the Gaza Strip, the Golan and the West Bank.
Palestinian Resistance And Peace Negotiations:
Since the 1950s, different Palestinian armed groups were formed, for example, Fatah, which emerged in 1956. In 1963, Arab governments decided to create the Organization for the Liberation of Palestine (OLP). The PLO rejected the partition of Palestine in 1947, ignored the State of Israel and claimed the entire Palestinian territory. After the defeat of the Arabs in the 1967 war, Al Fatah gradually took control of the PLO.<|endoftext|>
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# Menu. Review of Number Systems EEL3701 EEL3701. Math. Review of number systems >Binary math >Signed number systems
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1 Menu Review of number systems >Binary math >Signed number systems Look into my... 1 Our decimal (base 10 or radix 10) number system is positional. Ex: = 9x x x x10 0 We have a total of (R=10) digits, i.e., {0,1,2,3,4,5,6,7,8,9}; where R = radix Similarly for R=2, 8, 16; R=2 1 is called Binary; R=2 3 is called Octal; R=2 4 is called Hexadecimal (or Hex) For R > 10 we need additional symbols, e.g., for R=16 we need 6 additional symbols 0,1,2,3,4,5,6,7,8,9 and A,B,C,D,E,F for
2 For example, = 1x x x = 1x x x8 0 or = = 1x x x16 0 or = Fractions are also represented positionally as weighted negative powers of the radix or base. Ex: = 0x x x x10-3 Thus, for example, = 0x x x x = 0x x x x16-3 In general, if R>1, any rational number N can be represented in a power series given by: N = (d 4, d 3, d 2, d 1, d 0, d -1, d -2, d -3 ) R N=d 4 xr 4 + d 3 xr 3 + d 2 xr 2 + d 1 xr 1 +d 0 + d -1 xr -1 + d -2 xr -2 + d -3 xr -3 m= # of digits in the integer part - 1 k =# of digits in the fractional part 4 2
3 Decimal Numbers = (d 2, d 1, d 0, d -1, d -2 ) 10 R=10; m=2; k= = 9x x x x x10-2 d i = (9,5,3,7,8) What are: = = = 16 We ll get to this later. An 7-segment display LED can represent all 16 hex symbols. 5 7-Segment LED a b c h GND c g b d f a e h Show 7-segment LED in LogicWorks, 7-Segment Display * See also the LogicWorks 4-output device called Hex Keyboard See also the LogicWorks 4-input device called Hex Display abc defg A = abc defg F =
4 How many bits? Example: Let N=50 and R=2. We can use R=2 symbols, i.e., {0,1} = (d 5, d 4, d 3, d 2, d 1, d 0 ) 2 Why? 2 5 =32 and 2 6 =64 and = d d d d d d 0 Question: What is the range of unsigned numbers you can represent with...? Low High How Many? 4 bits 7 bits 8 bits 7 Q: =? 2 Decimal to Binary To obtain the binary digits (d i s), use long division and save each remainder until the quotient equals 0. d i s = the remainders in reverse order. 50 2: q=25, r=0 25 2: q=12, r=1 12 2: q=6, r=0 6 2: q=3, r=0 3 2: q=1, r=1 1 2: q=0, r=1 d i s = { } 2 STOP! = = =
5 Hex is a grouping of 4 bits Octal is a grouping of 3 bits Ex: =? 16 =? 8 >First regroup the binary as needed >Then convert using table of hex or octal values >For Hex (group in 4 s): = A6 16 >For Octal (group in 3 s): = Decimal to Hex Decimal Binary Hex Octal A B F A 52 9 Decimal to Hex Can convert to hex or octal using the same technique = X 16? R = 16; 50 16: q = 3, r = 2, 3 16: q= 0, r = 3 So = Check: = 48+2 = 50! = X 8? R = 8; 50 8: q = 6, r = 2, 6 8: q= 0, r = 6. So = 62 8 Check: 6 8 = = 50! Now = = 1x x x x x2 1 = = 50 Grouping in 3 bits each = = = 62 8 Therefore converting from Hex to Octal or to Binary is Trivial! Example: = 2? = 8? 10 5
6 Trivial Conversions: It is easy to convert from Binary/Hex/Octal to a Binary or Hex or Octal number Since 16 = 2 4 and 8 = 2 3, then to convert from >Binary to Octal we group bits in groups of three >Binary to Hex we group bits in groups of four 11 Ex: Ex: AC 16 It s easier to convert Dec Hex Bin then Dec Bin directly Ex: = { q=6; r= q r=6} = = = = = = = *8 + 4 = 100 From above you can also see that it may be easier to convert Bin Hex Dec (or Bin Oct Dec) then Bin Dec directly 12 6
7 Can you convert to Hex, Octal and Binary? Binary Codes Q: How many symbols can a four bit binary code represent? Binary Coded Decimal or BCD Code: Choose the first 10 binary values to represent the 10 digits {0,1,2,3,4,5,6,7,8,9}. Each digit will use 4-bits. > Ex: Convert to BCD > Answer: BCD 14 7
8 Ex: Convert BCD to Decimal BCD =? Ex: Convert BCD to Decimal BCD =? Since the weights of the 4 digits are 8, 4, 2 and 1, BCD is often called code. Modern digital computers & micros include instructions to allow us to manipulate BCD numbers as if they were binary numbers. > One common Y2K problem is strongly influenced by BCD 15 Addition, Subtraction, Multiplication Binary : We can manipulate Binary, Octal and Hexadecimal numbers like decimal numbers with respect to: addition, subtraction, multiplication and division. Examples: Dec Dec Dec
9 Division with Binary Numbers Example: =? Answer=110 R10 Dec 6 R =6 R2 17 Decimal Fraction Conversion To convert a decimal fraction to any other base, just multiply the decimal fraction by the radix and save the integer Ex: Convert to binary > 0.37 * 2 = 0.74 => > 0.74 * 2 = 1.48 => > 0.48 * 2 = 0.96 => > 0.96 * 2 = 1.92 => Ex: Convert to hex > 0.37 * 16 = 5.92 => > 0.92 * 16 = => 0.5E 16 > 0.72 * 16 = => 0.5EB 16 > 0.52 * 16 = 8.32 => 0.5EB8 16 Note that conversion to hex also gives binary (or octal): =
10 Signed Number Representations Negative Binary Numbers >Signed-Magnitude: Treats the most significant bit (MSB) as the sign of the number (0 = +, 1 = ) >One s Complement: Changes every bit from 0 to 1 and 1 to 0 (for negative numbers) >Two s Complement: One s Complement + 1 A 4-bit 2 s complement number d 3 d 2 d 1 d 0 has value >(d 3 d 2 d 1 d 0 ) 4-bit 2 s comp = d 3 -(2 3 ) + d d d A 3-bit 2 s complement number d 2 d 1 d 0 has value >(d 2 d 1 d 0 ) 3 bit 2 s comp = d 2 -(2 2 ) + d d Signed Numbers Examples of positive and negative number representations, using only 4 bits Decimal Sign Mag. 1 s Compl. 2 s Compl garbage garbage garbage Note: Given an n-bit binary number N N + N 1 s compl = = 2 n 1 (n ones) N + N 2 s compl = = 2 n (a 1 followed by n zeros) 20 10
11 2 s Comp Numbers To convert N to its 2 s complement form, it may be easier to subtract it from 2 n or from 2 n 1 and add 1 Example: Convert -25 to 8-bit 2 s complement: 25 = = = \$19 Tech 1: Complement then add or or \$E7 Tech 2: Subtract for 2 n = 231 = ( ) + (7) = \$E7 Tech 3: Subtract binary from 11 1, then add = \$FF - \$19 = \$E6 \$E6 + 1 = \$E7 No borrows in this subtraction, so easy 21 Subtraction and 2 s complement Suppose you do the following: 6-7=? You solve this by doing: -(7-6)= -(1)= -1 In 2 s comp. math we don t have to subtract Instead we can just negate and add Example: 6-7=? (using 4-bit 2 s comp.) 6=0110 2, 7=0111 2, -7=1001 2(2 s comp) 6-7 = 6+(-7)= (2 s comp) (2 s comp) 2 s comp. # s (2c) = =-1 Alternative is -(7-6)=-( )=-( ) = 2 s comp(0001)= (2c) 22 11
12 Sign Extension When you want a two s complement signed number to have more bits, you use Sign Extension >Keep the most significant bit for all of the added bits >Examples: Make 5-bit 2 s complement numbers into 8-bit 2 s complement numbers 1 10 = bit 2 s comp bit 2 s comp = bit 2 s comp bit 2 s comp = bit 2 s comp bit 2 s comp = bit 2 s comp bit 2 s comp 23 Signed Operations -vs- Representations 1 s complement and 2 s complement are both operations and representations Operations > 4-bit 1 s comp(1001)=0110 > 4-bit 2 s comp(1001)=0111 Representations > bit 1 s comp MSBit is 1 so negative bit 1 s comp = -(0110) = > bit 2 s comp =? MSBit is 1 so negative Complement: 0110; Add 1: 0111 = bit 2 s comp = Operations > 4-bit 1 s comp(0110)=1001 > 4-bit 2 s comp(0110)=1010 Representations > bit 1 s comp MSBit is 0 so positive bit 1 s comp = +(0110) = 6 10 > bit 2 s comp =? MSBit is 0 so positive bit 2 s comp = 0110 =
13 Overflow (2 s Comp) An overflow occurs when > Two positive numbers are added and the result appears to be negative > Two negative numbers are added and the result appears to be positive > Subtraction with the same strange results as above Example using 4-bit 2 s complement numbers > What is in 4-bit 2 s complement? >3 10 = bit 2 s comp and 7 10 = bit 2 s comp (2c) (2c) (2c) But bit 2 s comp is negative (= = -6) ==> OVERFLOW 25 Alien Number Systems 26 13
14 The End! 27 There are only 10 types of people in the world: those who understand binary and those who don t
15 The End! 29 15
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### Why digital? Overview. Number Systems. Binary to Decimal conversion
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Utah's intriguing history dates back to the Messozoic Era (230 to 65 million years ago), when many types of dinosaurs lived in the eastern and southern parts of what is now known as Utah. Their fossilized remnants are still being discovered and unearthed.
Ancient Puebloan cultures also known as the Anasazi and Fremont Indians had an agricultural lifestyle in southern Utah from about 1 A.D. to 1300. The Utes and the Navajo tribes lived across the area before the arrival of explorers, mountain men and pioneer settlers.
In the late 1700s while residents of the eastern United States were declaring independence from England, Catholic Spanish Explorers and Mexican traders drew journals documenting Utah's terrain, and the native people, as well as plants and animals. In the 1820s "mountain men" like Jedediah Smith, William Ashley and Jim Bridger roamed northern Utah, taking advantage of abundant fur trapping opportunities.
During 1847, members of the Church of Jesus Christ of Latter-day Saints (Mormons) migrated to the Salt Lake Valley seeking religious freedom. Before the first transcontinental railroad was completed at Promontory, Utah in May of 1869, more than 60,000 Mormons had come to the territory by covered wagon or handcart. Utah became America's 45th State on January 4, 1896.<|endoftext|>
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During AMC testing, the AoPS Wiki is in read-only mode. No edits can be made.
Difference between revisions of "2015 AIME I Problems/Problem 1"
Problem
The expressions $A$ = $1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39$ and $B$ = $1 + 2 \times 3 + 4 \times 5 + \cdots + 36 \times 37 + 38 \times 39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$.
Solution 1
We see that
$A=(1\times 2)+(3\times 4)+(5\times 6)+\cdots +(35\times 36)+(37\times 38)+39$
and
$B=1+(2\times 3)+(4\times 5)+(6\times 7)+\cdots +(36\times 37)+(38\times 39)$.
Therefore,
$B-A=-38+(2\times 2)+(2\times 4)+(2\times 6)+\cdots +(2\times 36)+(2\times 38)$
$=-38+4\times (1+2+3+\cdots+19)$
$=-38+4\times\frac{20\cdot 19}{2}=-38+760=\boxed{722}.$
Solution 2 (slower solution)
For those that aren't shrewd enough to recognize the above, we may use Newton's Little Formula to semi-bash the equations.
We write down the pairs of numbers after multiplication and solve each layer:
$2, 12, 30, 56, 90...(39)$
$6, 18, 26, 34...$
$8, 8, 8...$
and
$(1) 6, 20, 42, 72...$
$14, 22, 30...$
$8, 8, 8...$
Then we use Newton's Little Formula for the sum of n terms in a sequence.
Notice that there are 19 terms in each sequence, plus the tails of 39 and 1 on the first and second equations, respectively.
So: (this is as far as my latex knowledge goes)
((19 choose 1) x 2) + ((19 choose 2) x 10) + ((19 choose 3) x 8) + 1
((19 choose 1) x 6) + ((19 choose 2) x 14) + ((19 choose 3) x 8) + 39
Subtracting A from B gives:
((19 choose 1) x 4) + ((19 choose 2) x 4) - 38
Which unsurprisingly gives us $/boxed{722}.$
2015 AIME I (Problems • Answer Key • Resources) Preceded byFirst Problem Followed byProblem 2 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions<|endoftext|>
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Override means having two methods with the same name but doing different tasks. It means that one of the methods overrides the other.
If there is any method in the superclass and a method with the same name in a subclass, then by executing the method, the method of the corresponding class will be executed.
Let's take an example on this.
- Python 2
- Python 3
class Rectangle(): def __init__(self,length,breadth): self.length = length self.breadth = breadth def getArea(self): print(self.length*self.breadth," is area of rectangle") class Square(Rectangle): def __init__(self,side): self.side = side Rectangle.__init__(self,side,side) def getArea(self): print(self.side*self.side," is area of square") s = Square(4) r = Rectangle(2,4) s.getArea() r.getArea()
Since the method from the coressponding class came into action, it means that one overrode the other.
Execution of 'getArea' on the object of Rectangle (r) printed "8 is area of rectangle" from the 'getArea' defined in the Rectangle class whereas, execution of 'getArea' on the object of Square (s) printed "16 is area of square" from the 'getArea' defined in the Square class.
It is very useful because it prevents us from making methods with different names and remembering them all.
There is not much to explain in this chapter. So, it is recommended to solve questions to get things clearer.<|endoftext|>
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The existence of planets outside our solar system was strictly the domain of science fiction until 1992, when scientists confirmed that at least two of them circled a rapidly spinning neutron star called a pulsar in the constellation Virgo. Another extrasolar planet was detected three years later around 51 Pegasi, a Sun-like star, and since then more than 1,000 have been discovered. Most of these, large and boiling hot, are incapable of supporting life as we know it. In 2011, however, NASA announced the discovery of Kepler-22b, the first known planet in the “habitable zone,” where liquid surface water is possible.
Before breaking down earlier this year, the Kepler spacecraft was a particularly important tool in the search for new worlds. It monitored about 170,000 stars in the Cygnus and Lyra constellations, looking for tiny dips in brightness that occur when planets cross in front. At last count, it had identified 3,538 planet candidates, with more certain to come as additional data gets analyzed. Of these, 167 were subsequently confirmed, including Kepler-22b, four other habitable zone planets, a dozen or so planets smaller than Earth and a few that, like Tatooine in the “Star Wars” movies, orbit two stars. Kepler has even established that at least three extrasolar planets, and likely many more, are rocky. “All of a sudden it’s like our blinders have been completely lifted,” said Natalie Batalha, Kepler’s mission scientist. She expressed particular interest in Kepler-62f, a planet smack in the middle of the “Goldilocks” zone and only 40 percent larger than Earth.
The latest and arguably most stunning analysis of Kepler data came Monday, when an independent research team hypothesized that Earth-like planets—defined as those with a diameter one to two times that of Earth and located in the habitable zone—orbit 22 percent of Sun-like stars in the Milky Way galaxy. “If we assume that planets are as prevalent locally as they are in the Kepler field, it means that…the expected distance to the nearest one is about 12 light years,” said Erik Petigura, a Ph.D. candidate at the University of California at Berkeley, who authored the paper along with astronomers Geoffrey Marcy and Andrew Howard. The team furthermore noted that Earth-size planets are much more common than larger, Jupiter-size planets, at least in the warmer regions of solar systems.
In order to reach their conclusions, Petigura and his colleagues examined only Sun-like stars shining with steady brightness. This left them with 42,557 of the 170,000 stars in Kepler’s field of view. With the help of customized software, a supercomputer in Oakland, California, and telescopes on the summit of Hawaii’s Mauna Kea, they identified 10 Earth-like planets among those stars. They then inserted false planets into their program, assuming that the percentage missed by the software would correspond to the percentage of real planets missed. And they made sure to take into account that only a small fraction of planets can be seen crossing their host stars.
If their predictions are correct, then Earth-like planets exist around 11 billion of the roughly 50 billion Sun-like stars in the Milky Way. According to a separate paper published earlier this year, 15 percent of red dwarf stars—or close to 50 percent if the habitable zone is expanded—likewise have an Earth-like planet orbiting them, thereby bringing the approximate number of such planets in the galaxy up to 40 billion. Planets circling red giants and other types of stars, along with certain moons, may also be the size of Earth and in the habitable zone.
Astronomers, of course, have little idea about the composition of these Earth-like planets. Some may be gassy balls like Jupiter, and even the rocky ones may be nothing like our own world. They could, for instance, have a carbon dioxide-based atmosphere like Venus, with temperatures too hot for DNA-like molecules to survive. Or they could have an extremely thin atmosphere like Mars. Nonetheless, Petigura’s findings have stoked the possibility of alien organisms. “I’m sort of wary of speculation,” Petigura said, “but either life is extremely rare, a one in ten billion possibility, or the galaxy is teaming with life. There are no other interpretations.”<|endoftext|>
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It can be difficult to really understand the impact we as individuals and as a species are affecting the planet. When we hear statistics like, “Every year, the United States generates over 230 million tons of trash,” we understand that that’s a lot of garbage, but huge numbers like that can be hard to fathom. What does “230 million tons of trash” look like? What impact does it have? Those questions are almost impossible to answer, leaving us with little understanding of what that number means.
In short, 230 million pounds is a huge amount of garbage and studies have shown that if we keep throwing things away at the rate we do, it will have serious, irreversible negative effects on the environment. So, something has to change. And to give you some motivation to make a change — even a small change will do (check out these two simple tricks to reduce waste) — here are some statistics that are a little easier to understand.
- If even one-tenth of all American newspapers that are thrown away were recycled instead, over 20 million trees would be saved every year.
- Every year, the average American uses two trees’ worth of paper products.
- The average American generates 4.4 pounds of trash every day. If every American made a commitment to reduce the amount of trash we use by one pound each day, we would reduce the amount of trash we use in a year from 230 million tons to around 178 million tons. That’s still a lot, but it would be a significant improvement! Just goes to show how making small changes can have a big impact.
- The amount of paper that offices in America throw away each year could make a 12-foot wall long enough to reach from Los Angeles to New York City.
- If all of the trash that America throws away each year was stacked, it would create a chain long enough to reach the moon and back over 25 times.
- Americans are making progress when it comes to recycling — we recycle over 87 million tons each year, which is a huge step forward when compared with only 60 years ago when we were recycling an average of only 5.6 tons each year.
- Aluminum cans are the most recycled object in America, which is great news because we use about 65 billion of them each year.
- Glass and aluminum are two of many materials that are infinitely recyclable, which means they can be recycled indefinitely without any loss in purity or quality.<|endoftext|>
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## Вход
Забыли?
#### вход по аккаунту
код для вставкиСкачать
```Chapter 7: Variation in repeated samples – Sampling distributions
Recall that a major objective of statistics is to make inferences about a
population from an analysis of information contained in sample data.
Typically, we are interested in learning about some numerical feature of
the population, such as
• the proportion possessing a stated characteristic;
• the mean and the standard deviation.
A numerical feature of a population is called a parameter.
The true value of a parameter is unknown. An appropriate sample-based
quantity is our source about the value of a parameter.
A statistic is a numerical valued function of the sample observations.
Sample mean is an example of a statistic.
The sampling distribution of a statistic
Three important points about a statistic:
• the numerical value of a statistic cannot be expected to give us the
exact value of the parameter;
• the observed value of a statistic depends on the particular sample that
happens to be selected;
• there will be some variability in the values of a statistic over different
occasions of sampling.
Because any statistic varies from sample to sample, it is a random
variable and has its own probability distribution.
The probability distribution of a statistic is called its sampling
distribution.
Often we simply say the distribution of a statistic.
Distribution of the sample mean
Statistical inference about the population mean is of prime practical
mean and its sampling distribution.
An example illustrating the central limit
theorem
Figure 7.4 (p. 275)
Distributions of X for n = 3 and n = 10 in sampling from an asymmetric population.
Example on probability calculations
for the sample mean
Consider a population with mean 82 and standard deviation 12.
If a random sample of size 64 is selected, what is the probability that the sample mean
will lie between 80.8 and 83.2?
Solution: We have μ = 82 and σ = 12. Since n = 64 is large, the central limit theorem
tells us that the distribution of the sample mean is approximately normal with
E ( X ) 82 , sd ( X )
n
12
1 .5
64
Converting to the standard normal variable:
Z
X
n
X 82
1 .5
Thus,
P [ 80 . 8 X 83 . 2 ]
P [( 80 . 8 82 ) / 1 . 5 Z ( 83 . 2 82 ) / 1 . 5 ]
P [ . 8 Z . 8 ] . 7881 . 2119 . 5762
```
1/--страниц
Пожаловаться на содержимое документа<|endoftext|>
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Anyone who has ever lugged a pallet of water down to the basement in preparation for the latest extreme weather event knows that water is heavy.
How heavy? About 8.3 pounds per gallon.
Now, scientists have developed a way to use water's weight to measure just how much of it is sitting up in snow-covered mountains in the western United States.
In states like California, which is currently in the midst of a crippling drought, the more water managers know about how much snow is in the mountains, the better they can plan for the summer to come (Greenwire).
More accurate information about snowpack can help these managers and hydrologists plan for how to fill reservoirs, how much water they might have available during the dry season and how dry the soils might be during fire season. They'll also get a better fix on future levels of hydroelectric reservoirs.
Snow 'pushes' earth down
Donald Argus, a research scientist and geophysicist at NASA's Jet Propulsion Laboratory in Pasadena, Calif., recently published a study outlining the new technique in the journal Geophysical Research Letters.
Since water is so heavy, when there is enough of it on the ground in the form of snow, it actually weighs down the land's surface.
"The weight of the snow and the water pushes the earth down," said Argus. "The earth subsides up to 12 millimeters."
If scientists know how high the land is in summertime, and also how high it is when snow covers it, they can use the difference in height, plus a lot of math, to calculate how much water, in snow form, is sitting on the mountains.
The technique uses a dense network of GPS sites scattered across the West. The sites are part of a project called the Plate Boundary Observatory, run by the science consortium UNAVCO.
"There's over 1,000 sites in California, Oregon, Nevada and Washington," said Argus.
The scientists studied data from these GPS sites going back to 2006. They calculated the average difference in height for each GPS site between the end of the dry season, Oct. 1, and April 1, when the snow is usually at its maximum, and used that difference to calculate how much water was needed to weigh down the ground.
While many Americans use GPS to navigate from their phones or in cars, the GPS receivers used for this work are far more precise, able to measure changes in their location of a few millimeters. Since the GPS network in California is so dense, the scientists are able to get a spatial resolution of about 50 miles, said Argus.
New method complements other measurements
On average, the researchers found the difference in water weight from summer to winter in the Sierra Nevada mountain range equaled about 1.6 feet of water spread over the 26,000-square-mile region.
Water managers already have other tools to track water quantities in the West. These include satellites, hydrologic models, and even another NASA project that uses planes and Lidar to measure snowpack in parts of the Sierra Nevada (ClimateWire, Dec. 10, 2013).
But the airborne surveys are costly and only see one part of the Sierra Nevada, whereas the GPS network is widespread among many states.
And the GPS data is about 50 percent more accurate than one hydrologic model used to estimate snowpack in the Sierra Nevada and roughly on par with another one, Argus said.
Right now, the researchers are able to develop a monthly average for snowpack about 10 days after the end of the month. Their goal, however, is to do that even faster.
"The idea is to make it available in near real time and be able to give it to [water managers] one, two, five days after," said Argus.
Reprinted from Climatewire with permission from Environment & Energy Publishing, LLC. www.eenews.net, 202-628-6500<|endoftext|>
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(Updated September 2008)
If you have heard of a university member diagnosed with meningitis, please read the following information to help answer questions you may have. Be assured, if bacterial meningitis is diagnosed in any patient, the local health department will immediately investigate and be in touch with close personal contacts that need to receive medication as soon as possible. If a diagnosis of viral menigitis is made, there will be no public health information disseminated due to patient confidentiality issues.
Meningitis is an illness in which there is inflammation of the tissues that cover the brain and spinal cord. Viral (aseptic) meningitis, which is the most common type, is caused by an infection with one of several types of viruses. Meningitis can also be caused by infections with several types of bacteria or fungi. In the United States, there are between twenty-five thousand and fifty thousand hospitalizations caused by viral meningitis each year.
The more common symptoms of meningitis are fever, severe headache, stiff neck, bright lights hurting the eyes, drowsiness or confusion, and nausea and vomiting. In babies, the symptoms are more difficult to identify. They may include fever, fretfulness or irritability, difficulty in awakening the baby, or the baby refuses to eat. The symptoms of meningitis may not be the same for every person.
Viral (aseptic) meningitis is serious but rarely fatal in persons with normal immune systems. Usually, the symptoms last from seven to ten days, and the patient recovers completely. Bacterial meningitis, on the other hand, can be very serious and result in disability or death if not treated promptly. Often, the symptoms of viral meningitis and bacterial meningitis are the same. For this reason, if you think you or your child has meningitis, see your doctor as soon as possible.
Many different viruses can cause meningitis. About 90 percent of cases of viral meningitis are caused by members of a group of viruses known as enteroviruses, such as coxsackieviruses and echoviruses. These viruses are more common during summer and fall months. Herpesviruses and the mumps virus can also cause viral meningitis.
Viral meningitis is usually diagnosed by laboratory tests of spinal fluid obtained with a spinal tap. The specific cause of viral meningitis can be determined by tests that identify the virus in specimens collected from the patient, but these tests are rarely done.
No specific treatment for viral meningitis exists at this time. Most patients completely recover on their own. Doctors often will recommend bed rest, plenty of fluids, and medicine to relieve fever and headache.
Enteroviruses, the most common cause of viral meningitis, are most often spread through direct contact with respiratory secretions (e.g., saliva, sputum, or nasal mucus) of an infected person. This usually happens by shaking hands with an infected person or touching something he or she has handled, and then rubbing your own nose or mouth. The virus can also be found in the stool of persons who are infected. The virus is spread through this route mainly among small children who are not yet toilet trained. It can also be spread this way to adults changing the diapers of an infected infant. The incubation period for enteroviruses is usually between three and seven days from the time you are infected until you develop symptoms. You can usually spread the virus to someone else beginning about three days after you are infected until about ten days after you develop symptoms.
The viruses that cause viral meningitis are contagious. Enteroviruses, for example, are very common during the summer and early fall, and many people are exposed to them. However, most infected persons either have no symptoms or develop only a cold or rash with low-grade fever. Only a small portion of infected persons actually develop meningitis. Therefore, if you are around someone who has viral meningitis, you have a moderate chance of becoming infected, but a very small chance of developing meningitis.
Because most persons who are infected with enteroviruses do not become sick, it can be difficult to prevent the spread of the virus. However, adhering to good personal hygiene can help to reduce your chances of becoming infected. If you are in contact with someone who has viral meningitis, the most effective method of prevention is to wash your hands thoroughly and often (see “Hand Washing” in “An Ounce of Prevention: Keeps the Germs Away” on the Center for Disease Control and Prevention website). Also, cleaning contaminated surfaces and soiled articles first with soap and water, and then disinfecting them with a dilute solution of chlorine-containing bleach (made by mixing approximately a quarter cup of bleach with one gallon of water) can be a very effective way to inactivate the virus, especially in institutional settings such as child care centers.
Questions can be directed to the University Health Service.<|endoftext|>
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Lesson 5
Reasoning about Equations and Tape Diagrams (Part 2)
5.1: Algebra Talk: Seeing Structure (10 minutes)
Warm-up
This warm-up parallels the one in the previous lesson. The purpose of this Algebra Talk is to elicit strategies and understandings students have for solving equations. These understandings help students develop fluency and will be helpful later in this unit when students will need to be able to come up with ways to solve equations of this form. While four equations are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.
Students should understand the meaning of solution to an equation from grade 6 work as well as from work earlier in this unit, but this is a good opportunity to re-emphasize the idea.
In this string of equations, each equation has the same solution. Digging into why this is the case requires noticing and using the structure of the equations (MP7). Noticing and using the structure of an equation is an important part of fluency in solving equations.
Launch
Display one equation at a time. Give students 30 seconds of quiet think time for each equation and ask them to give a signal when they have an answer and a strategy. Keep all equations displayed throughout the talk. Follow with a whole-class discussion.
Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization
Student Facing
Solve each equation mentally.
$$x -1 = 5$$
$$2(x-1) = 10$$
$$3(x-1) = 15$$
$$500 = 100(x-1)$$
Activity Synthesis
This discussion may go quickly, because students are likely to recognize similarities between this equation string and the one in the previous day’s warm-up.
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
• “Who can restate ___’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the equation in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)
5.2: More Situations and Diagrams (15 minutes)
Activity
The purpose of this activity is to work toward showing students that some situations can be represented by an equation of the form $$p(x+q)=r$$ (or equivalent). In this activity, students are simply tasked with drawing a tape diagram to represent each situation. In the following activity, they will work with corresponding equations.
For each question, monitor for one student with a correct diagram. Press students to explain what any variables used to label the diagram represent in the situation.
Launch
Ensure students understand that the work of this task is to draw a tape diagram to represent each situation. There is no requirement to write an equation or solve a problem yet.
Arrange students in groups of 2. Give 5–10 minutes to work individually or with their partner, followed by a whole-class discussion.
Action and Expression: Develop Expression and Communication. Maintain a display of important terms, vocabulary, and examples. During the launch, take time to review examples of drawing a tape diagram based on situations from previous lessons that students will need to access for this activity. Consider providing step-by-step directions that generalize the process using student input and ideas.
Supports accessibility for: Memory; Language
Student Facing
Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.
1. Each of 5 gift bags contains $$x$$ pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils.
2. Noah drew an equilateral triangle with sides of length 5 inches. He wants to increase the length of each side by $$x$$ inches so the triangle is still equilateral and has a perimeter of 20 inches.
3. An art class charges each student $3 to attend plus a fee for supplies. Today,$20 was collected for the 5 students attending the class.
4. Elena ran 20 miles this week, which was three times as far as Clare ran this week. Clare ran 5 more miles this week than she did last week.
Launch
Ensure students understand that the work of this task is to draw a tape diagram to represent each situation. There is no requirement to write an equation or solve a problem yet.
Arrange students in groups of 2. Give 5–10 minutes to work individually or with their partner, followed by a whole-class discussion.
Action and Expression: Develop Expression and Communication. Maintain a display of important terms, vocabulary, and examples. During the launch, take time to review examples of drawing a tape diagram based on situations from previous lessons that students will need to access for this activity. Consider providing step-by-step directions that generalize the process using student input and ideas.
Supports accessibility for: Memory; Language
Student Facing
Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.
1. Each of 5 gift bags contains $$x$$ pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils.
2. Noah drew an equilateral triangle with sides of length 5 inches. He wants to increase the length of each side by $$x$$ inches so the triangle is still equilateral and has a perimeter of 20 inches.
3. An art class charges each student $3 to attend plus a fee for supplies. Today,$20 was collected for the 5 students attending the class.
4. Elena ran 20 miles this week, which was three times as far as Clare ran this week. Clare ran 5 more miles this week than she did last week.
Activity Synthesis
Select one student for each situation to present their correct diagram. Ensure that students explain the meaning of any variables used to label their diagram. Possible questions for discussion:
• “For the situations with no $$x$$, how did you decide what quantity to represent with variable?” (Think about which amount is unknown but has a relationship to one or more other amounts in the story.)
• “What does the variable you used to label the diagram represent in the story?”
• “Did any situations have the same diagrams? How can you tell from the story that the diagrams would be the same?” (Same number of equal parts, same amount for the total.)
Speaking, Representing, Reading: MLR1 Stronger and Clearer Each Time. Ask students to explain to a partner how they created the tape diagram to represent the situation “An art class charges each student $3 to attend plus a fee for supplies. Today,$20 was collected for the 5 students attending the class.” Ask listeners to press for details in the arrangement of the grouped quantities (e.g., “Explain how you chose what values go in each box.”). When roles are switched, listeners can press for details in what “$$x$$” represents in the diagram. Allow students to revise their diagrams, if necessary, based on the feedback they received from their partner. Once their revision is complete, invite students to turn to a new partner to explain their revised diagram. This will help students productively engage in discussion as they make connections between written situations and visual diagrams.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
5.3: More Situations, Diagrams, and Equations (10 minutes)
Activity
This activity is a continuation of the previous one. Students match each situation from the previous activity with an equation, solve the equation by any method that makes sense to them, and interpret the meaning of the solution. Students are still using any method that makes sense to them to reason about a solution. In later lessons, a hanger diagram representation will be used to justify more efficient methods for solving.
For each equation, monitor for a student using their diagram to reason about the solution and a student using the structure of the equation to reason about the solution.
Launch
Keep students in the same groups. 5 minutes to work individually or with a partner, followed by a whole-class discussion.
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as “To find the solution, first, I _____ because...”, “I made this match because I noticed...”, “Why did you...?”, or “I agree/disagree because…”
Supports accessibility for: Language; Social-emotional skills
Speaking, Representing: MLR2 Collect and Display. As students share their ideas about how the equations match the situations, listen for and collect students’ description of the situation (e.g., “5 gift bags, $$x$$ pencils, adds 3 more, 20 pencils”) with the corresponding equation. Remind students to borrow language from the displayed examples while describing what each solution tells about the situation, after the matching is complete. This will help students make connection between language, diagrams, and equations.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Each situation in the previous activity is represented by one of the equations.
• $$(x+3) \boldcdot 5 = 20$$
• $$3(x+5)=20$$
1. Match each situation to an equation.
3. What does each solution tell you about its situation?
Student Facing
Han, his sister, his dad, and his grandmother step onto a crowded bus with only 3 open seats for a 42-minute ride. They decide Han’s grandmother should sit for the entire ride. Han, his sister, and his dad take turns sitting in the remaining two seats, and Han’s dad sits 1.5 times as long as both Han and his sister. How many minutes did each one spend sitting?
Activity Synthesis
For each equation, ask one student who reasoned with the diagram and one who reasoned only about the equation to explain their solutions. Display the diagram and the equation side by side, drawing connections between the two representations. If no students bring up one or both of these approaches, demonstrate the maneuvers on a diagram side by side with the maneuvers on the corresponding equation. For example, “I divided the number of gift bags by 5, leaving me with 4 pencils per gift bag. Since Tyler added 3 pencils to each gift bag, there must have been 1 pencil in each gift bag to start,” can be shown on a tape diagram and on a corresponding equation. It is not necessary to invoke the more abstract language of “doing the same thing to each side” of an equation yet.
Lesson Synthesis
Lesson Synthesis
Display one of the situations from the lesson and its corresponding equation. Ask students to explain:
• “What does each number and letter in the equation represent in the situation?”
• “What is the reason for each operation (multiplication or addition) used in the equation?”
• “What is the solution to the equation? What does it mean to be a solution to an equation? What does the solution represent in the situation?”
5.4: Cool-down - More Finding Solutions (5 minutes)
Cool-Down
Each bag in the first story has an unknown number of toys, $$x$$, that is increased by 2. Then ten groups of $$x+2$$ give a total of 140 toys. An equation representing this situation is $$10(x+2)=140$$. Since 10 times a number is 140, that number is 14, which is the total number of items in each bag. Before Lin added the 2 items there were $$14 - 2$$ or 12 toys in each bag.
The executive in the second story knows that the size of each team of $$y$$ employees has been increased by 10. There are now 2 teams of $$y+10$$ each. An equation representing this situation is $$2(y+10)=140$$. Since 2 times an amount is 140, that amount is 70, which is the new size of each team. The value of $$y$$ is $$70-10$$ or 60. There were 60 employees on each team before the increase.<|endoftext|>
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- Leaves are a plant’s food factory. Leaves make glucose using the energy of the sun, water and carbon dioxide through the process of photosynthesis. Oxygen is also a product of photosynthesis.
- Chlorophyll is the green pigment that gives leaves their green color and helps make photosynthesis possible.
- Deciduous trees are loss their leaves in the fall. These leaves are generally large and flat.
- Evergreens stay green year round, as their name implies. Their leaves are actually needles.
- As the days get shorter and cooler in the fall, the process of photosynthesis slows down and the tree begins to use some of its stored energy. As a result, the true color of the leaves can show through.
- That’s right! The beautiful colors of the fall are the actual colors of the leaves, but the chlorophyll is masking the colors during the spring and summer while photosynthesis is occurring.
Fall Leaf Hunt and Sorting Activity
Leaf Memory Game
Preschool Leaf Pack
Parts of a Leaf
How Water Travels Through Leaves Experiment
Photosynthesis Leaf Model
Map a Leaf
Photosynthesis in Action
Leaf Color Change Lesson and Chromatography Experiment<|endoftext|>
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This marmot has a yellowy-brown back with light tips on all the hairs. Its feet and legs are dark brown and there are white-grey patches across its head and nose. It has short little round ears and a dark brown tails. The belly is a reddish-yellow colour. It has sharp front teeth and claws, and can grow to about 60 cm long.
Range & Habitat
They are found from south central BC to California and New Mexico. They always live around large rocks, and sometimes near grasslands and valley bottoms.
Diet & Behaviour
Marmots eat lots of green plants, and will sometimes eat dead animals. They sun themselves in the morning to warm up, and disappear into their rock burrows to keep cool during the day. The have a loud whistling chirp that will sound if they are alarmed, and sit on their hind legs, showing their yellow belly.
Lifecycle & Threats
Three to eight young are born in June in the mother’s burrow, because each adult marmot has a separate den. All the marmot dens are usually in a similar area, and the marmots living there are called a colony. Marmots hibernate in the winter, and marmots who can’t find good winter dens won’t make it through the cold.
COSEWIC: Not at Risk
Photo: J N Stuart<|endoftext|>
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Tracing Worksheet: Curved Lines Tracing
Tracing curved lines preschool worksheets are an important way to improve your child’s fine motor skills and promote healthy handwriting habits. Excite your child for drawing and writing with this adorable worksheet featuring characters from our popular app!
Working on this worksheet will help your child:
- Improve fine motor skills to prepare for writing
- Promote proper handwriting habits through practicing pencil grip
- To sharpen hand-to-eye coordination
- Encourage focus and stamina to complete a tricky task.
Your little learner will draw and write throughout his or her life. Help your child get a jump start on one of life’s most important skills by utilizing this worksheet to offer meaningful tracing practice.<|endoftext|>
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History of Veterans Day
Observed annually in the United States on November 11, Veterans Day is a federal holiday, honoring the service of all U.S. military veterans. Along with Remembrance Day, Veterans Day was also originally Armistice Day - which officially started in 1918 to mark the end of the First World War. For more information, visit our articles on Remembrance Day and Armistice Day.
Veterans Day became an official holiday in 1954, after an 9 year campaign to expand Armistice Day to celebrate all United States Veterans, by the "Father of Veterans Day" World War II veteran Raymond Weeks. President Dwight D. Eisenhower signed a bill into law on May 26, 1954, which accomplished Weeks' goals. On June 1, 1954, Congress amended the bill, replacing "Armistice" with "Veterans"; giving the holiday its current name of Veterans Day.
Veterans Day Traditions
Many cities around the United States have Veterans Day parades, a time for the community to come together to celebrate peace, and honor the veterans who have served their country in the Armed Forces, and many schools hold Veterans Day activities to honor American veterans throughout the week and on Veterans Day.
Citizens are encouraged to fly the flag of a Military branch, or a POW or MIA flag on Veterans Day to show support to United States Veterans.
A Veterans Day parade in Beaufort, South Carolina.<|endoftext|>
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# How do you solve log x^2 = (log x)^2?
Dec 1, 2015
$x = 100$
#### Explanation:
Rewrite as:
$2 \log x = {\left(\log x\right)}^{2}$
$0 = {\left(\log x\right)}^{2} - 2 \log x$
$0 = \log x \left(\log x - 2\right)$
Set each portion equal to $0$.
$\implies \log x = 0$
$x$ does not exist.
$\implies \log x - 2 = 0$
$\log x = 2$
${\log}_{10} x = 2$
${10}^{{\log}_{10} x} = {10}^{2}$
$x = 100$<|endoftext|>
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# SOLUTION: When using the quadratic formula to solve a quadratic equation ax2 + bx + c = 0, the discriminant is b2 - 4ac. This discriminant can be positive, zero, or negative. (When the discr
Algebra -> -> SOLUTION: When using the quadratic formula to solve a quadratic equation ax2 + bx + c = 0, the discriminant is b2 - 4ac. This discriminant can be positive, zero, or negative. (When the discr Log On
Ad: Mathway solves algebra homework problems with step-by-step help! Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!
Click here to see ALL problems on Quadratic Equations Question 76249: When using the quadratic formula to solve a quadratic equation ax2 + bx + c = 0, the discriminant is b2 - 4ac. This discriminant can be positive, zero, or negative. (When the discriminant is negative, then we have the square root of a negative number. This is called an imaginary number, sqrt(-1) = i. ) Explain what the value of the discriminant means to the graph of y = ax2 + bx + c. Hint: Chose values of a, b and c to create a particular discriminant. Then, graph the corresponding equation?Answer by Edwin McCravy(9717) (Show Source): You can put this solution on YOUR website!``` When using the quadratic formula to solve a quadratic equation , the discriminant is . This discriminant can be positive, zero, or negative. (When the discriminant is negative, then we have the square root of a negative number. This is called an imaginary number, sqrt(-1) = i. ) Explain what the value of the discriminant means to the graph of y = ax2 + bx + c. Hint: Chose values of a, b and c to create a particular discriminant. Then, graph the corresponding equation? Choose a = 1, b = 4, c = -21 Then the equation is y = x² + 4x - 21 and the discriminant is (4)² - 4(1)(-21) = 16 + 84 = 100, which is positive. The graph intersects the x-axis twice, once at x = -7 and again at x = 3. There are two real zeros, -7, and 3 ----------------- Now choose a = 1, b = 4, c = 4 Then the equation is y = x² + 4x + 4 and the discriminant is (4)² - 4(1)(4) = 16 - 16 = 0. The graph just touches the x-axis at -2. There is just one real zeros, -2. [This zero is said to have multiplicity 2 because people like to think of the graph as "crossing the x-axis twice at the same point", and "both its two zeros are the same, i.e., 'merging' into one".] ------------------- Finally choose a = 1, b = 4, c = 6 Then the equation is y = x² + 4x + 6 and the discriminant is (4)² - 4(1)(6) = 16 - 24 = -8, which is negative. The graph does not cross or touch the x-axis. Therefore it has no real zeros, which means that both its solutions are imaginary. Edwin```<|endoftext|>
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# where is wrong in the sum of series $\frac{1}{3}+\frac{1}{4}\cdot\frac{1}{2!}+\frac{1}{5}\cdot\frac{1}{3!}+\cdots$
I came through two types of solutions of the series $\frac{1}{3}+\frac{1}{4}\cdot\frac{1}{2!}+\frac{1}{5}\cdot\frac{1}{3!}+\cdots$ \begin{align*} \frac{1}{3}+\frac{1}{4}\cdot\frac{1}{2!}+\frac{1}{5}\cdot\frac{1}{3!}+\cdots &=\sum_{n=1}^{\infty}\frac{1}{n+2}\cdot\frac{1}{n!}\\ &=\sum_{n=1}^{\infty}\frac{n+1}{(n+2)!}\\ &=\sum_{n=1}^{\infty}[\frac{1}{(n+1)!}-\frac{1}{(n+2)!}]\\ &=\frac{1}{2} \end{align*} \begin{align*} \frac{1}{3}+\frac{1}{4}\cdot\frac{1}{2!}+\frac{1}{5}\cdot\frac{1}{3!}+\cdots &=\sum_{n=1}^{\infty}\frac{1}{n+2}\cdot\frac{1}{n!}\\ &=\sum_{n=1}^{\infty}\int_{0}^{1}\frac{x^{n+1}}{n!}dx\\ &=\int_{0}^{1}\sum_{n=1}^{\infty}\frac{x^{n+1}}{n!}dx\\ &=\int_{0}^{1}x\sum_{n=1}^{\infty}\frac{x^{n}}{n!}dx\\ &=\int_{0}^{1}x(e^x-1)dx\\ &=\frac{-1}{2} \end{align*} where am I getting wrong please help!
• Probably that only a constant is supposed to be taken out...?
– P.K.
Dec 28, 2012 at 17:02
• $x(e^x-1)$ is positive for $x\in(0,1)$, so the value of your last integral can't be negative. Dec 28, 2012 at 17:03
• Actually, $x(e^x-1)$ integrates to $1/2$, not $-1/2$.
– fbg
Dec 28, 2012 at 17:04
You have $$\int^{1}_{0}x(e^{x}-1)dx=[e^{x}(x-1)-\frac{x^{2}}{2}]|^{1}_{0}=\frac{1}{2}$$
• This is a comment, not an answer. The evaluation of the integral is done by the fundamental theorem of calculus:$$\int_0^1 x \left( \mathrm{e}^x-1\right) \mathrm{d}x = \left. \left(\left(x-1\right) \mathrm{e}^{x} - \frac{x^2}{2} \right)\right|_{x=0}^{x=1} = \frac{1}{2}$$ Dec 28, 2012 at 17:08<|endoftext|>
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# Question: What is the measure of each exterior angle of a polygon?
Exterior angles of polygons The measure of each exterior angle =360°/n, where n = number of sides of a polygon. An important property of the outer angles of a regular polygon is that the sum of the dimensions of the outer angles of a polygon is always 360°.
## What is the measure of the outer corner of a polygon?
The sum of the outer angles of a polygon is 360°. The formula to calculate the size of an outer angle in a regular polygon is: 360. If you know the outer angle, you can use the formula to find the inner angle: inner angle + outer angle = 180°
## What is the size of each outside corner?
It doesn’t matter how many sides a regular polygon has, the measure of any outer angle is the same, and the sum of the measures of all the outer angles of the regular polygon is equal to 360°.
## What is the size of each outer corner of a pentagon?
Answer: The measure of any outer angle of a regular pentagon is 72°. In a regular pentagon, all angles are the same size and all sides are the same length.
## What is the size of each outer angle of a 9-sided regular polygon*?
Answer: An outer angle of a regular polygon with 9 sides is 40°.
## What is the outer angle of a 40-sided polygon?
Exterior angles (in a regular polygon) add up to 360°. So you would do 360/40 to get an answer of 9.
## What is the outer angle of an 18-sided polygon?
It further explains the formula by taking an 18-sided regular polygon as an example and calculating the outside angle as 360/18 or 20 degrees.
## How many sides does a polygon with an exterior angle of 36 have?
The sum of the exterior angles of a regular polygon is 360 degrees. The outer angles of a regular polygon are also equal. Allow s to be the number of sides of the polygon. Answer: The polygon has 10 sides.
## How do you find the exterior angle of a 12-sided polygon?
In a regular n-gon, the dimension of an outer angle is 360˚n. Since a dodecagon has 12 sides, the exterior angle is 360˚12=30˚.
## Why are the exterior angles of a polygon 360?
Due to the congruence of vertical angles, it doesn’t matter which side is extended; The outside corner is the same. The sum of the exterior angles of any polygon (remember, only convex polygons are covered here) is 360 degrees.<|endoftext|>
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Labyrinthitis is an inflammation of the inner ear. The labyrinth is a structure of fluid-filled sacs and tubes just inside the skull. It contains two important organs: the cochlea, which is necessary for hearing, and the balance organs (vestibular system), which tell people which way is up and down, even when their eyes are shut. Either one or both of these organs can be affected. The entire labyrinth is less than a half inch (1.25 cm) across, so infection can easily spread throughout.
The inner ear also contains a bundle of nerves leading from these organs to the brain. Inflammation of these nerves is called vestibular neuronitis (or vestibular neuritis). It's sometimes difficult to find out whether it is the labyrinth or the nerves that are inflamed.<|endoftext|>
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The Individualized Education Plan (IEP) is a document developed for special needs children by the parents and the educators who work with them. The plan is provided at no cost to parents, and is required in all public and private schools receiving public funds for the education of children with disabilities.
Development of the plan is mandated by the Individuals with Disabilities Education Act (IDEA). This Federal Law defines and governs special needs services, and requires that the plan describe how the student learns; how the student best demonstrates that learning; and what teachers and service providers will do to help the student toward better academic achievement.
In order to determine eligibility, the school must conduct a full evaluation of a child in all areas of suspected disability noted by the parent or classroom teacher. However, having a disability is not sufficient for eligibility.
When a disability is identified, it must be determined how or if the disability is preventing the child from learning. This step helps to identify needed services and answers the question of whether special education is appropriate even if the child is eligible. If simple modifications to the traditional general education classroom there is no need for special education or an individualized education plan. However, special education with full inclusion in the traditional classroom as the least restrictive environment gives you the best of both worlds.
In this placement the individualized education plan will state that the child will remain in the traditional classroom and the special education teacher will go to the child's classroom to provide special instruction in needed areas, or the child may go to the special teacher for a few minutes each day for special instruction in areas of weakness.
The law requires that the Individualized Education Plan be completed before placement decisions are made so that the child's educational needs determine the services outlined by the plan. Schools may not develop a student's IEP to fit into a pre-existing program for a particular classification of disability. The individualized plan is written to fit the student. The placement is chosen to fit the needed services in the plan.
The goal of the the law is that, to the maximal extent possible, children are to be educated in the same classroom as the child's non-disabled peers in the school nearest the child's home.
Parents are crucial participants during the development of the Individualized Education Plan (IEP) in both public and private schools because they bring a unique perspective based upon their knowledge of, and insight into, the child's developmental history and environmental experiences.
For this reason, the law states that parents must be present and involved in any and all meetings that discuss the identification, evaluation, IEP development and educational placement of their children.
The law grants parents permission to ask questions, dispute points, and request modifications to the IEP along with all other members of the team.
Although teams are required to work toward consensus, school personnel are, ultimately, responsible for ensuring that the student's IEP includes the services that the student needs.
It is during the development of the individualized education plan that parents can best wield the power granted by Title IV IDEA to insure that their child receives needed services in an appropriate program.
During this activity, the parent is granted more power than the school. The process cannot continue without the parent's signature on the individulized plan. So the IEP should not be signed until the parent has reviewed it, gotten a second opinion, and made certain that they are satisfied that it meets the needs of their child.
The law requires that the school reevaluate the child's eligibility for special education every three years, but the parent may ask for a review of the special education IEP at any time if the child does not seem to be achieving as planned. At a minimum the plan should be reviewed annually.
Parental preparation and participation will guarantee the child an appropriate education with all the latest modifications and accommodations to insure that he is not only successful in achieving his special education goals for academic achievement, but that he is prepared to go on to college, or to become employed after graduation from high school.
The IEP should state these additional goals and describe how special education will help the child to achieve them.
Readers and book reviewers state that every parent should have a copy of the LD Reference Book Learning Disabilities, Understanding the Problem, Managing the Challenges before beginning this process.
The reference book takes you step-by-step through the entire process of designing the IEP. And, the tips and techniques will guide you through what to say, when to say it, what to do, when to do it. You will learn to interact with the school in a way that will be very helpful when claiming your rights under the law.
If your child needs special services not provided by your school district, then the school district, has to transport the child to a location where the services are provided at no cost to the parent.
Some students have been placed out of state in private schools at costs up to $100,000 per year per child. A parent will need an attorney, lots of money and some pull to make that happen, but some parents have accomplished it.
Parents just need to know their rights and learn how to advocate for them. The only challenge will be to justify the need. The required help is available in our LD Reference Book.
And, if you want someone to do it for you, someone to discuss the process, or just answer your questions, personal support is available by fax, phone, or email free of charge to parents who purchase the reference book.<|endoftext|>
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# Pipe and Cistern (5 Questions) -01
Question 01
Two pipes A and B can fill a tank in 20 and 30 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?
Given:
Pipe A can fill the whole tank in 20 mins
So, Part of the tank filled by Pipe A in 1 minute = 1/20
Pipe B can fill the whole tank in 30 mins
So, Part of the tank filled by Pipe B in 1 minute = 1/30
If both pipes are used together, part of the tank filled in 1 minute = 1/20 + 1/30 = = 5/60 = 1/12
So, 12 minutes are required to fill the tank
Question 02
A cistern can be filled by a tap in 4 hours while it can be emptied by another tap in 9 hours. If both the taps are opened simultaneously, then after how much time will the cistern get filled?
Let there are two taps A and B
Tap A can fill the tank in 4 hours
Part of the tank filled by first tap in 1 hour = ¼
Similarly,
Part of the tank emptied by second tap in 1 hour = 1/9
When both taps are opened simultaneously, part of the cistern filled in 1 hour = ¼ – 1/9 = = 5/36
So, the total time taken = 36/5 = 7.2 hours
Question 03
Pipes A and B can fill a tank in 5 and 6 hours respectively. Pipe C can empty it in 12 hours. If all the three pipes are opened together, then the tank will be filled in?
Sol:
Pipe A can fill the tank in 5 hours
Part of the tank filled by Pipe A in 1 hour = 1/5
Pipe B fill the tank in 6 hours
Part of the tank filled by Pipe B in 1 hour = 1/6
Pipe C empty the tank in 12 hours
Part of the tank emptied by Pipe C in 1 hour = 1/12
If all three pipes are opened together, part of the filled in 1 hour = 1/5 + 1/6 – 1/12 = 17/60 hours
So, the tank will be filled in = 60/17 hours
Question 04
Three pipes A, B and C can fill a tank from empty to full in 30 minutes, 20 minutes and 10 minutes respectively. When the tank is empty, all the three pipes are opened. A, B and C discharge chemical solutions P, Q and R respectively. What is the proportion of solution R in the liquid in the tank after 3 minutes?
Pipe A can fill the tank in 30 minutes
Part of the tank filled by Pipe A in 1 minute =1/30
Pipe B can fill the tank in 20 minutes
Part of the tank filled by Pipe B in 1 minute =1/20
Pipe C can fill the tank in 10 minutes
Part of the tank filled by Pipe C in 1 minute =1/10
Part filled by (A + B + C) in 3 minutes = 3 ( 1/30 + 1/20 + 1/10) = 3 × 11/60 = 11/20
Thus, Part filled by C in 3 minutes = 3/10
Therefore, Required ratio = 3/10 × 20/11 = 6/11
Question 05
Two pipes A and B can separately fill a cistern in 60 minutes and 75 minutes respectively. There is a third pipe in the bottom of the cistern to empty it. If all the three pipes are simultaneously opened, then the cistern is full in 50 minutes. In how much time, the third pipe alone can empty the cistern?
Sol:
Pipe A can fill the tank in 60 minutes
Part of the tank filled by Pipe A in 1 minute =1/60
Pipe B can fill the tank in 75 minutes
Part of the tank filled by Pipe A in 1 minute =1/75
Let the third pipe alone can empty the cistern in x minutes.
As per question- 1/60 + 1/75 – 1/x = 1/50
1/x = 1/60 + 1/75 – 1/50
1/x = 3/300
1/x = 1/100
x = 100 minutes
Thus, the third pipe alone can empty the cistern in 100 minutes<|endoftext|>
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Abraham Lincoln in Civil War era lithography
In recent weeks, historical sites and societies throughout the country have been marking the 151st anniversary of the end of the Civil War and the death of Abraham Lincoln. In the Harry T. Peters America on Stone Collection there are numerous prints pertaining to the Civil War, including several on the assassination, death, and commemoration of Abraham Lincoln. The lithographs selected below are just a few examples of how the country used visual print culture to absorb, personalize, and make sense of a national tragedy.
Just weeks before his death, Lincoln's features had been mocked by his political adversaries. After the assassination his visage was cherished. According to Page Smith, author of Trial by Fire, some contemporary sources mentioned that his face was "calm and striking." After Lincoln's death, demand for images of the late president dramatically increased, though prints and lithographs of Lincoln were readily available. In the North, many families had hung Lincoln's portrait in prominent spots in their homes during the war. After his death, these portraits were draped in black cloth, as if the families were mourning the death of a family member.
Lincoln's assassination drew almost instant analogies, as in the print above, in which John Wilkes Booth is compared to the murderous Macbeth in Shakespeare's tragedy. Others saw religious connections. It did not go unnoticed that Lincoln's assassination took place on Good Friday. That Easter Sunday, dark mourning crepe replaced vibrant flowers, and clergymen across the country began to compare Lincoln's assassination to the crucifixion of Jesus Christ. In fact, Booth became known as a "second Judas" for having betrayed the nation and killing a man some believed to be favored by God.
Because access to the dying president was limited, there are only a few accounts of his final hours. While as many as 55 people visited the Lincoln's bedside, not all were present at any given time. The bedroom was only 9.5 feet wide by 17 feet long, so it would have been impossible to fit a crowd of 16 adult males and a child around the bed, an artistic license taken in several contemporary depictions. This exaggerated use of space was common in lithography. Though photography existed in the period, photographs were not reproduced in newspapers to report stories. The use of lithographs allowed for audiences to sometimes gain more insight into an event, even if the depictions were overdramatized.
Images like the one above demonstrate how commemorative memorial prints were produced to capture and capitalize on the nation's grief. Symbols of Lincoln's great accomplishments surround his headstone. A weeping figure of Liberty drapes herself over the stone memorial, as the slain dragon representing the rebellion of the Civil War lies at its base. In the years after his death, linking symbols to his name and death was necessary to ensure that Lincoln's legacy and sacrifice were not forgotten.
Lincoln's signing of the Emancipation Proclamation in January 1863 was a contentious act in the politically divided North. The president's supporters likened him to a new Moses, and between 1863 and 1864 there was a dramatic increase in the production of prints related to the Emancipation Proclamation, which would have found a ready place in many Republican households. After his assassination in 1865, calligraphy prints, such as the one above, were widely circulated, as they literally infused the words of emancipation with the image of Abraham Lincoln. The legacy becomes the man.
Sarah Crosswy is a graduate of the College of Wooster and a volunteer in the Division of Home and Community Life. She previously interned with curator Debbie Schaefer-Jacobs and is currently assisting in cataloging the Dr. Richard Lodish American School Collection, a recent acquisition.<|endoftext|>
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# The Altitude of a Circular Cylinder is Increased Six Times and the Base Area is Decreased One-ninth of Its Value. the Factor by Which the Lateral Surface of the Cylinder Increases, is - Mathematics
MCQ
The altitude of a circular cylinder is increased six times and the base area is decreased one-ninth of its value. The factor by which the lateral surface of the cylinder increases, is
#### Options
• $\frac{2}{3}$
• $\frac{1}{2}$
• $\frac{3}{2}$
• 2
#### Solution
$\text{ Let the original radius of the base of cylinder = r }$
$\text{ Let the original height of cylinder = h }$
$\text{ Now, original base area of the cylinder,} S = \pi r^2$
$\text{ Now, original LSA of cylinder, } A = 2\pi rh . . . . . . . . . . . . \left( 1 \right)$
$\text{ When the altitude is increased to 6 times of its initial value and base area is decreased one - ninth of its initial value: }$
$\text{ Let the new height of the cylinder } = h'$
$\text{ Let the new radius of base of cylinder } = r'$
$\text{ Now, new base area of cylinder, } S' = \pi \left( r' \right)^2$
$\text{ Now, it is given that, }$
$\text{ new height of cylinder } = 6 \times \text{ original height of cylinder }$
$\Rightarrow h' = 6h$
$\text{ Also, new base area of cylinder } = \frac{1}{9}\left( \text{ original base area of the cylinder } \right)$
$\Rightarrow S' = \frac{1}{9}S$
$\Rightarrow \pi \left( r' \right)^2 = \frac{1}{9}\left( \pi r^2 \right)$
$\Rightarrow \left( r' \right)^2 = \left( \frac{r}{3} \right)^2$
$\Rightarrow r' = \frac{r}{3}$
$\text{ Now, new LSA of cylinder } , A' = 2\pi r'h'$
$\Rightarrow A' = 2\pi \times \left( \frac{r}{3} \right) \times \left( 6h \right) = 2 \times \left( 2\pi rh \right)$
$\Rightarrow A' = 2 A \left[ \text{ Using } \left( 1 \right) \right]$
$\text{ Hence, lateral surface area of the cylinder becomes twice of the original } .$
Concept: Surface Area of Cylinder
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Mathematics for Class 9
Chapter 19 Surface Areas and Volume of a Circular Cylinder
Exercise 19.4 | Q 20 | Page 30
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Creating rock pools and studying seashore life
- Learning outcome
To collect and sort seashore creatures and plants.
- Key Stubbington focus
Environment - seashore
- For Key Stage 2
- Duration 1 hours (whole morning or afternoon)
- Success criteria
- I understand that living things have different habitats
- I can create an artificial rock pool that will provide each creature with all it needs to survive
- I can plan, make and review my work with my group
- Upper KS2: I can discuss a few adaptations of creatures that live in the intertidal zone
- Session plan
Introduction (20 minutes)
In Great Chamber, use PowerPoint to introduce session on whiteboard.
- Discuss the habitat of an intertidal zone, what the animals need and how they have adapted to living in this habitat (hard shells, movement, closing tightly or withdrawing into shells, etc.)
- On some beaches, the tide goes out but some of the water is trapped in puddles (rock pools)
- Explain that we will be creating our own ‘rock pools’ as we don’t have any natural rock pools to study. The children will be collecting live sea creatures that have been stranded in the intertidal zone and putting them into a tray filled with seawater. Don’t collect crabs, they can get crushed in the bucket and might wander out of rock pool and get squashed by feet!
- Give examples of a univalve and bivalve and explain how the children will know if they are alive or dead
- Discuss resources and their responsibilities.
All children to go to the toilet, put on wellies, an extra layer and waterproofs if needed.
Main (approx 90 minutes)
Once on the beach, show children the boundaries.
Set Up Main Activity 1 (20-30 mins)
- Trays filled with water by adults (using buckets maybe so the trays are stationary and less likely to be spilled)
- Working in groups of 3, children collect animals and identify them using their key
- Animals to be put down one end and seaweed at the other end to see if there is any movement/travelling of animals
- Discuss with each group why the animals might move and therefore which animals are most likely to move
- Trays to be left to settle
Teachers to choose two activities from the following ‘Seashore Activities’ list. These activities are a mix of developing teamwork and learning about the seashore.
Seashore Activities (all about 15 minutes)
- Seashore Rainbow – Collecting natural objects in different colours and textures to make a group seashore rainbow on the beach
- Seashore Scavenger Hunt – Using the keys, children try to find as many of the items as they can
- Fossil Search – Focussed search for fossils along the shore line
- Sandcastle challenge
- Best decorated
- The Limpet Game – A game that helps the children understand about limpets and life in the intertidal zone
- Migration Tag – A game that introduces and helps the children to understand bird migration
Plenary (20 mins)
When return to the activity, children check to see if there is any movement.
Discuss the results. Was this a fair test? (No, it may not have been the right type of seaweed, the tide was artificial i.e. was it created accurately, we may not have left the creatures long enough, we put them with predators that may have effected their behaviour, etc.)
Release the animals sensibly and safely.
Less able children:
- Give them a physical example of what they are to collect and help them find the first few
- Limit the specimens they need to find
More able children:
- Children can collect a wider variety of specimens
- They can discuss the intertidal zone as a habitat including a few problems that are faced by the animals that live there and how they have adapted to deal with these problems
Prior activities in school:
Find out what creatures on the beach would need to survive.
Follow up activities:
- Create a sea creature/monster that lives in the intertidal zone
- Explain how it survives (oxygen, food, shelter)
- A visit to Titchfield Haven to see and identify birds including migrating species
- Health and safety checks
- Inhalers and epipens to be taken
- Appropriate clothing should be worn including lots of layers in cold weather; hats and sun cream in hot weather
- At all times, children should wear wellington boots
- Stubbington member of staff will carry a mobile phone to enable incoming & outgoing contact
- Stubbington member of staff will carry a whistle, throw bag and first aid kit
- Children must not eat or put hands near faces. Hands to be washed immediately on return
- Children must not throw stones
- Children should be accompanied by a member of school staff when using public toilets
- Regular head counts to be taken
- There must be an adult to child ratio of 1:12 for all offsite visits (aged 8 and over)
- Awareness of public
- No paddling in water (supervised cleaning out of buckets at the end of the session)
- Ensure boundaries for collecting are made clear to adults and children
- Check games area for potential hazards as children are running on the sand<|endoftext|>
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# Difference between revisions of "Rope around the Earth"
Rope around the Earth
Field: Geometry
Image Created By: Harrison Tasoff
Rope around the Earth
The puzzle of lengthening a rope tied taut around the equator so that, if made to levitate, there is a one foot gap at all points between the rope and the Earth.
# Basic Description
A question similar to this appeared in William Whiston's The Elements of Euclid circa 1702. Suppose a rope was tied taut around the Earth's equator. It would have the same circumference as the Earth (24,901.55 miles). The question is: by how much would the rope have to be lengthened such that, if made to hover, it would be one foot off the ground at all points around the Earth?
Despite the enormous size of the Earth, and the 1 foot gap around the entire circumference, the rope would have to be lengthened by a mere 2π feet, or roughly 6.28 feet.
In fact, this result is independent of the size of the ball around which the rope is wrapped.
# A More Mathematical Explanation
Note: understanding of this explanation requires: *High-school algebra and High-school geometry
The Circumference of a circle is given by the equation: $C=2\pi\,\!r$ Where r is t [...]
The Circumference of a circle is given by the equation: $C=2\pi\,\!r$ Where r is the radius.
In the image to the right:
Lrope is the length of the rope.
Cearth is the Circumference of the Earth.
Rearth is the radius of the Earth.
When the rope is taut around the globe, its length equals the circumference of the Earth.
$L_{rope}=C_{earth}=2\pi\,\!R_{earth}$
Lengthening the rope so that it is 1 foot off the ground at all points simply means changing the radius of the circle it forms from:
Rrope 1= Rearth
to
Rrope 2= Rearth+1 ft.
So: $L_{rope 2}=2\pi\,\!(R_{earth}+1)$
Distributing the 2 π yields:
$L_{rope 2}=2\pi\,\!R_{earth}+2\pi\,\!$
Now it is clear that new length of the rope is merely 2 π feet longer than he original length. Indeed, one can see that the additional 2 π is a result of extending the radius of the rope circle by one foot, an extension that will by definition be the same no matter the initial radius of the object being enclosed.
### Maximum Height of Rope
Were the new rope again to be held taught, by raising it at an arbitrary point (as shown in the picture to the right), what would the distance form this point to the surface of the earth be?
In the diagram to the right:
x1 is the distance from the horizon to the highest point on the taut rope.
xo is the ground distance from the point where the rope leaves the globe, to the point below the apex of the rope.
R is the radius of the globe.
h is the height of the apex of the rope above the ground.
The Pythagorean Theorem: $C^2=A^2+B^2$ where A and B are the legs of a right triangle, and C is the .
Using this theorem, we know that: $(R+h)^2=R^2+x_1^2$, which is equivalent to $x_1=\sqrt{(R+h)^2-R^2}$.
To find the length of xo, we must remember what the length of an arc is:
$L_{arc}=r\theta\,\!$
Where θ is the angle formed between two radii from the center of the circle to the endpoints of the arc.
Thus, $x_o=R cos^{-1}(\frac{R}{R+h})$.
Where cos-1 represents the angle whose cosine is$\tfrac{R}{R+h}$.
Since we know that we lengthened the rope by 2 π feet, we know that 2x1= 2xo + 2 π, because 2x1 is the extra slack put in to the rope.
Thus: $x_1=x_o+\pi\,\!$
Running the three equations above through a numerical calculator resulted in h = 614.771 ft as the height that a 2π extension would yield.
# Why It's Interesting
Though it may seem that this is minuscule amount of extra rope needed to to produce such a considerable result, a look at the ratios will show otherwise.
The radius of the Earth is roughly 20,920,000 feet. There is 1 foot of difference between the radius of the circle made by the lengthened rope and the radius of the Earth. This foot of difference is a mere fraction of the radius of the Earth: about five one-hundred millionths, or .000000047, of the Earth's radius. A foot doesn't seem so large anymore.
Similarly, 2 π feet is 4.7 x 10-8 of the circumference of the Earth (which is about 131,000,000 feet). And, unsurprisingly, the ratio of 1 foot to the Earth's radius is the same as that of 2 π feet to the Earth's circumference.
So, in this perspective, a small change in the length of the rope yields a proportionally equivalent small change in the radius of the rope circle.<|endoftext|>
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University Physics Volume 1
# 12.2Examples of Static Equilibrium
University Physics Volume 112.2 Examples of Static Equilibrium
## Learning Objectives
By the end of this section, you will be able to:
• Identify and analyze static equilibrium situations
• Set up a free-body diagram for an extended object in static equilibrium
• Set up and solve static equilibrium conditions for objects in equilibrium in various physical situations
All examples in this chapter are planar problems. Accordingly, we use equilibrium conditions in the component form of Equation 12.7 to Equation 12.9. We introduced a problem-solving strategy in Example 12.1 to illustrate the physical meaning of the equilibrium conditions. Now we generalize this strategy in a list of steps to follow when solving static equilibrium problems for extended rigid bodies. We proceed in five practical steps.
## Problem-Solving Strategy
### Static Equilibrium
1. Identify the object to be analyzed. For some systems in equilibrium, it may be necessary to consider more than one object. Identify all forces acting on the object. Identify the questions you need to answer. Identify the information given in the problem. In realistic problems, some key information may be implicit in the situation rather than provided explicitly.
2. Set up a free-body diagram for the object. (a) Choose the xy-reference frame for the problem. Draw a free-body diagram for the object, including only the forces that act on it. When suitable, represent the forces in terms of their components in the chosen reference frame. As you do this for each force, cross out the original force so that you do not erroneously include the same force twice in equations. Label all forces—you will need this for correct computations of net forces in the x- and y-directions. For an unknown force, the direction must be assigned arbitrarily; think of it as a ‘working direction’ or ‘suspected direction.’ The correct direction is determined by the sign that you obtain in the final solution. A plus sign $(+)(+)$ means that the working direction is the actual direction. A minus sign $(−)(−)$ means that the actual direction is opposite to the assumed working direction. (b) Choose the location of the rotation axis; in other words, choose the pivot point with respect to which you will compute torques of acting forces. On the free-body diagram, indicate the location of the pivot and the lever arms of acting forces—you will need this for correct computations of torques. In the selection of the pivot, keep in mind that the pivot can be placed anywhere you wish, but the guiding principle is that the best choice will simplify as much as possible the calculation of the net torque along the rotation axis.
3. Set up the equations of equilibrium for the object. (a) Use the free-body diagram to write a correct equilibrium condition Equation 12.7 for force components in the x-direction. (b) Use the free-body diagram to write a correct equilibrium condition Equation 12.11 for force components in the y-direction. (c) Use the free-body diagram to write a correct equilibrium condition Equation 12.9 for torques along the axis of rotation. Use Equation 12.10 to evaluate torque magnitudes and senses.
4. Simplify and solve the system of equations for equilibrium to obtain unknown quantities. At this point, your work involves algebra only. Keep in mind that the number of equations must be the same as the number of unknowns. If the number of unknowns is larger than the number of equations, the problem cannot be solved.
5. Evaluate the expressions for the unknown quantities that you obtained in your solution. Your final answers should have correct numerical values and correct physical units. If they do not, then use the previous steps to track back a mistake to its origin and correct it. Also, you may independently check for your numerical answers by shifting the pivot to a different location and solving the problem again, which is what we did in Example 12.1.
Note that setting up a free-body diagram for a rigid-body equilibrium problem is the most important component in the solution process. Without the correct setup and a correct diagram, you will not be able to write down correct conditions for equilibrium. Also note that a free-body diagram for an extended rigid body that may undergo rotational motion is different from a free-body diagram for a body that experiences only translational motion (as you saw in the chapters on Newton’s laws of motion). In translational dynamics, a body is represented as its CM, where all forces on the body are attached and no torques appear. This does not hold true in rotational dynamics, where an extended rigid body cannot be represented by one point alone. The reason for this is that in analyzing rotation, we must identify torques acting on the body, and torque depends both on the acting force and on its lever arm. Here, the free-body diagram for an extended rigid body helps us identify external torques.
## Example 12.3
### The Torque Balance
Three masses are attached to a uniform meter stick, as shown in Figure 12.9. The mass of the meter stick is 150.0 g and the masses to the left of the fulcrum are $m1=50.0gm1=50.0g$ and $m2=75.0g.m2=75.0g.$ Find the mass $m3m3$ that balances the system when it is attached at the right end of the stick, and the normal reaction force at the fulcrum when the system is balanced.
Figure 12.9 In a torque balance, a horizontal beam is supported at a fulcrum (indicated by S) and masses are attached to both sides of the fulcrum. The system is in static equilibrium when the beam does not rotate. It is balanced when the beam remains level.
### Strategy
For the arrangement shown in the figure, we identify the following five forces acting on the meter stick:
$w1=m1gw1=m1g$ is the weight of mass $m1;m1;$ $w2=m2gw2=m2g$ is the weight of mass $m2;m2;$
$w=mgw=mg$ is the weight of the entire meter stick; $w3=m3gw3=m3g$ is the weight of unknown mass $m3;m3;$
$FSFS$ is the normal reaction force at the support point S.
We choose a frame of reference where the direction of the y-axis is the direction of gravity, the direction of the x-axis is along the meter stick, and the axis of rotation (the z-axis) is perpendicular to the x-axis and passes through the support point S. In other words, we choose the pivot at the point where the meter stick touches the support. This is a natural choice for the pivot because this point does not move as the stick rotates. Now we are ready to set up the free-body diagram for the meter stick. We indicate the pivot and attach five vectors representing the five forces along the line representing the meter stick, locating the forces with respect to the pivot Figure 12.10. At this stage, we can identify the lever arms of the five forces given the information provided in the problem. For the three hanging masses, the problem is explicit about their locations along the stick, but the information about the location of the weight w is given implicitly. The key word here is “uniform.” We know from our previous studies that the CM of a uniform stick is located at its midpoint, so this is where we attach the weight w, at the 50-cm mark.
Figure 12.10 Free-body diagram for the meter stick. The pivot is chosen at the support point S.
### Solution
With Figure 12.9 and Figure 12.10 for reference, we begin by finding the lever arms of the five forces acting on the stick:
$r1=30.0cm+40.0cm=70.0cmr2=40.0cmr=50.0cm−30.0cm=20.0cmrS=0.0cm(becauseFSis attached at the pivot)r3=30.0cm.r1=30.0cm+40.0cm=70.0cmr2=40.0cmr=50.0cm−30.0cm=20.0cmrS=0.0cm(becauseFSis attached at the pivot)r3=30.0cm.$
Now we can find the five torques with respect to the chosen pivot:
$τ1=+r1w1sin90°=+r1m1g(counterclockwise rotation, positive sense)τ2=+r2w2sin90°=+r2m2g(counterclockwise rotation, positive sense)τ=+rwsin90°=+rmg(gravitational torque)τS=rSFSsinθS=0(becauserS=0cm)τ3=−r3w3sin90°=−r3m3g(clockwise rotation, negative sense)τ1=+r1w1sin90°=+r1m1g(counterclockwise rotation, positive sense)τ2=+r2w2sin90°=+r2m2g(counterclockwise rotation, positive sense)τ=+rwsin90°=+rmg(gravitational torque)τS=rSFSsinθS=0(becauserS=0cm)τ3=−r3w3sin90°=−r3m3g(clockwise rotation, negative sense)$
The second equilibrium condition (equation for the torques) for the meter stick is
$τ1+τ2+τ+τS+τ3=0.τ1+τ2+τ+τS+τ3=0.$
When substituting torque values into this equation, we can omit the torques giving zero contributions. In this way the second equilibrium condition is
$+r1m1g+r2m2g+rmg−r3m3g=0.+r1m1g+r2m2g+rmg−r3m3g=0.$
12.17
Selecting the $+y+y$-direction to be parallel to $F→S,F→S,$ the first equilibrium condition for the stick is
$−w1−w2−w+FS−w3=0.−w1−w2−w+FS−w3=0.$
Substituting the forces, the first equilibrium condition becomes
$−m1g−m2g−mg+FS−m3g=0.−m1g−m2g−mg+FS−m3g=0.$
12.18
We solve these equations simultaneously for the unknown values $m3m3$ and $FS.FS.$ In Equation 12.17, we cancel the g factor and rearrange the terms to obtain
$r3m3=r1m1+r2m2+rm.r3m3=r1m1+r2m2+rm.$
To obtain $m3m3$ we divide both sides by $r3,r3,$ so we have
$m3=r1r3m1+r2r3m2+rr3m=7030(50.0g)+4030(75.0g)+2030(150.0g)=316.023g≃317g.m3=r1r3m1+r2r3m2+rr3m=7030(50.0g)+4030(75.0g)+2030(150.0g)=316.023g≃317g.$
12.19
To find the normal reaction force, we rearrange the terms in Equation 12.18, converting grams to kilograms:
$FS=(m1+m2+m+m3)g=(50.0+75.0+150.0+316.7)×10−3kg×9.8ms2=5.8N.FS=(m1+m2+m+m3)g=(50.0+75.0+150.0+316.7)×10−3kg×9.8ms2=5.8N.$
12.20
### Significance
Notice that Equation 12.17 is independent of the value of g. The torque balance may therefore be used to measure mass, since variations in g-values on Earth’s surface do not affect these measurements. This is not the case for a spring balance because it measures the force.
Repeat Example 12.3 using the left end of the meter stick to calculate the torques; that is, by placing the pivot at the left end of the meter stick.
In the next example, we show how to use the first equilibrium condition (equation for forces) in the vector form given by Equation 12.7 and Equation 12.8. We present this solution to illustrate the importance of a suitable choice of reference frame. Although all inertial reference frames are equivalent and numerical solutions obtained in one frame are the same as in any other, an unsuitable choice of reference frame can make the solution quite lengthy and convoluted, whereas a wise choice of reference frame makes the solution straightforward. We show this in the equivalent solution to the same problem. This particular example illustrates an application of static equilibrium to biomechanics.
## Example 12.4
### Forces in the Forearm
A weightlifter is holding a 50.0-lb weight (equivalent to 222.4 N) with his forearm, as shown in Figure 12.11. His forearm is positioned at $β=60°β=60°$ with respect to his upper arm. The forearm is supported by a contraction of the biceps muscle, which causes a torque around the elbow. Assuming that the tension in the biceps acts along the vertical direction given by gravity, what tension must the muscle exert to hold the forearm at the position shown? What is the force on the elbow joint? Assume that the forearm’s weight is negligible. Give your final answers in SI units.
Figure 12.11 The forearm is rotated around the elbow (E) by a contraction of the biceps muscle, which causes tension $T→M.T→M.$
### Strategy
We identify three forces acting on the forearm: the unknown force $F→F→$ at the elbow; the unknown tension $T→MT→M$ in the muscle; and the weight $w→w→$ with magnitude $w=50lb.w=50lb.$ We adopt the frame of reference with the x-axis along the forearm and the pivot at the elbow. The vertical direction is the direction of the weight, which is the same as the direction of the upper arm. The x-axis makes an angle $β=60°β=60°$ with the vertical. The y-axis is perpendicular to the x-axis. Now we set up the free-body diagram for the forearm. First, we draw the axes, the pivot, and the three vectors representing the three identified forces. Then we locate the angle $ββ$ and represent each force by its x- and y-components, remembering to cross out the original force vector to avoid double counting. Finally, we label the forces and their lever arms. The free-body diagram for the forearm is shown in Figure 12.12. At this point, we are ready to set up equilibrium conditions for the forearm. Each force has x- and y-components; therefore, we have two equations for the first equilibrium condition, one equation for each component of the net force acting on the forearm.
Figure 12.12 Free-body diagram for the forearm: The pivot is located at point E (elbow).
Notice that in our frame of reference, contributions to the second equilibrium condition (for torques) come only from the y-components of the forces because the x-components of the forces are all parallel to their lever arms, so that for any of them we have $sinθ=0sinθ=0$ in Equation 12.10. For the y-components we have $θ=±90°θ=±90°$ in Equation 12.10. Also notice that the torque of the force at the elbow is zero because this force is attached at the pivot. So the contribution to the net torque comes only from the torques of $TyTy$ and of $wy.wy.$
### Solution
We see from the free-body diagram that the x-component of the net force satisfies the equation
$+Fx+Tx−wx=0+Fx+Tx−wx=0$
12.21
and the y-component of the net force satisfies
$+Fy+Ty−wy=0.+Fy+Ty−wy=0.$
12.22
Equation 12.21 and Equation 12.22 are two equations of the first equilibrium condition (for forces). Next, we read from the free-body diagram that the net torque along the axis of rotation is
$+rTTy−rwwy=0.+rTTy−rwwy=0.$
12.23
Equation 12.23 is the second equilibrium condition (for torques) for the forearm. The free-body diagram shows that the lever arms are $rT=1.5in.rT=1.5in.$ and $rw=13.0in.rw=13.0in.$ At this point, we do not need to convert inches into SI units, because as long as these units are consistent in Equation 12.23, they cancel out. Using the free-body diagram again, we find the magnitudes of the component forces:
$Fx=Fcosβ=Fcos60°=F/2Tx=Tcosβ=Tcos60°=T/2wx=wcosβ=wcos60°=w/2Fy=Fsinβ=Fsin60°=F3/2Ty=Tsinβ=Tsin60°=T3/2wy=wsinβ=wsin60°=w3/2.Fx=Fcosβ=Fcos60°=F/2Tx=Tcosβ=Tcos60°=T/2wx=wcosβ=wcos60°=w/2Fy=Fsinβ=Fsin60°=F3/2Ty=Tsinβ=Tsin60°=T3/2wy=wsinβ=wsin60°=w3/2.$
We substitute these magnitudes into Equation 12.21, Equation 12.22, and Equation 12.23 to obtain, respectively,
$F/2+T/2−w/2=0F3/2+T3/2−w3/2=0rTT3/2−rww3/2=0.F/2+T/2−w/2=0F3/2+T3/2−w3/2=0rTT3/2−rww3/2=0.$
When we simplify these equations, we see that we are left with only two independent equations for the two unknown force magnitudes, F and T, because Equation 12.21 for the x-component is equivalent to Equation 12.22 for the y-component. In this way, we obtain the first equilibrium condition for forces
$F+T−w=0F+T−w=0$
12.24
and the second equilibrium condition for torques
$rTT−rww=0.rTT−rww=0.$
12.25
The magnitude of tension in the muscle is obtained by solving Equation 12.25:
$T=rwrTw=13.01.5(50 lb)=43313lb≃433.3lb.T=rwrTw=13.01.5(50 lb)=43313lb≃433.3lb.$
The force at the elbow is obtained by solving Equation 12.24:
$F=w−T=50.0lb−433.3lb=−383.3lb.F=w−T=50.0lb−433.3lb=−383.3lb.$
The negative sign in the equation tells us that the actual force at the elbow is antiparallel to the working direction adopted for drawing the free-body diagram. In the final answer, we convert the forces into SI units of force. The answer is
$F=383.3lb=383.3(4.448N)=1705N downwardT=433.3lb=433.3(4.448N)=1927N upward.F=383.3lb=383.3(4.448N)=1705N downwardT=433.3lb=433.3(4.448N)=1927N upward.$
### Significance
Two important issues here are worth noting. The first concerns conversion into SI units, which can be done at the very end of the solution as long as we keep consistency in units. The second important issue concerns the hinge joints such as the elbow. In the initial analysis of a problem, hinge joints should always be assumed to exert a force in an arbitrary direction, and then you must solve for all components of a hinge force independently. In this example, the elbow force happens to be vertical because the problem assumes the tension by the biceps to be vertical as well. Such a simplification, however, is not a general rule.
### Solution
Suppose we adopt a reference frame with the direction of the y-axis along the 50-lb weight and the pivot placed at the elbow. In this frame, all three forces have only y-components, so we have only one equation for the first equilibrium condition (for forces). We draw the free-body diagram for the forearm as shown in Figure 12.13, indicating the pivot, the acting forces and their lever arms with respect to the pivot, and the angles $θTθT$ and $θwθw$ that the forces $T→MT→M$ and $w→w→$ (respectively) make with their lever arms. In the definition of torque given by Equation 12.10, the angle $θTθT$ is the direction angle of the vector $T→M,T→M,$ counted counterclockwise from the radial direction of the lever arm that always points away from the pivot. By the same convention, the angle $θwθw$ is measured counterclockwise from the radial direction of the lever arm to the vector $w→.w→.$ Done this way, the non-zero torques are most easily computed by directly substituting into Equation 12.10 as follows:
$τT=rTTsinθT=rTTsinβ=rTTsin60°=+rTT3/2τw=rwwsinθw=rwwsin(β+180°)=−rwwsinβ=−rww3/2.τT=rTTsinθT=rTTsinβ=rTTsin60°=+rTT3/2τw=rwwsinθw=rwwsin(β+180°)=−rwwsinβ=−rww3/2.$
Figure 12.13 Free-body diagram for the forearm for the equivalent solution. The pivot is located at point E (elbow).
The second equilibrium condition, $τT+τw=0,τT+τw=0,$ can be now written as
$rTT3/2−rww3/2=0.rTT3/2−rww3/2=0.$
12.26
From the free-body diagram, the first equilibrium condition (for forces) is
$−F+T−w=0.−F+T−w=0.$
12.27
Equation 12.26 is identical to Equation 12.25 and gives the result $T=433.3lb.T=433.3lb.$ Equation 12.27 gives
$F=T−w=433.3lb−50.0lb=383.3lb.F=T−w=433.3lb−50.0lb=383.3lb.$
We see that these answers are identical to our previous answers, but the second choice for the frame of reference leads to an equivalent solution that is simpler and quicker because it does not require that the forces be resolved into their rectangular components.
Repeat Example 12.4 assuming that the forearm is an object of uniform density that weighs 8.896 N.
## Example 12.5
### A Ladder Resting Against a Wall
A uniform ladder is $L=5.0mL=5.0m$ long and weighs 400.0 N. The ladder rests against a slippery vertical wall, as shown in Figure 12.14. The inclination angle between the ladder and the rough floor is $β=53°.β=53°.$ Find the reaction forces from the floor and from the wall on the ladder and the coefficient of static friction $μsμs$ at the interface of the ladder with the floor that prevents the ladder from slipping.
Figure 12.14 A 5.0-m-long ladder rests against a frictionless wall.
### Strategy
We can identify four forces acting on the ladder. The first force is the normal reaction force N from the floor in the upward vertical direction. The second force is the static friction force $f=μsNf=μsN$ directed horizontally along the floor toward the wall—this force prevents the ladder from slipping. These two forces act on the ladder at its contact point with the floor. The third force is the weight w of the ladder, attached at its CM located midway between its ends. The fourth force is the normal reaction force F from the wall in the horizontal direction away from the wall, attached at the contact point with the wall. There are no other forces because the wall is slippery, which means there is no friction between the wall and the ladder. Based on this analysis, we adopt the frame of reference with the y-axis in the vertical direction (parallel to the wall) and the x-axis in the horizontal direction (parallel to the floor). In this frame, each force has either a horizontal component or a vertical component but not both, which simplifies the solution. We select the pivot at the contact point with the floor. In the free-body diagram for the ladder, we indicate the pivot, all four forces and their lever arms, and the angles between lever arms and the forces, as shown in Figure 12.15. With our choice of the pivot location, there is no torque either from the normal reaction force N or from the static friction f because they both act at the pivot.
Figure 12.15 Free-body diagram for a ladder resting against a frictionless wall.
### Solution
From the free-body diagram, the net force in the x-direction is
$+f−F=0+f−F=0$
12.28
the net force in the y-direction is
$+N−w=0+N−w=0$
12.29
and the net torque along the rotation axis at the pivot point is
$τw+τF=0.τw+τF=0.$
12.30
where $τwτw$ is the torque of the weight w and $τFτF$ is the torque of the reaction F. From the free-body diagram, we identify that the lever arm of the reaction at the wall is $rF=L=5.0mrF=L=5.0m$ and the lever arm of the weight is $rw=L/2=2.5m.rw=L/2=2.5m.$ With the help of the free-body diagram, we identify the angles to be used in Equation 12.10 for torques: $θF=180°−βθF=180°−β$ for the torque from the reaction force with the wall, and $θw=180°+(90°−β)θw=180°+(90°−β)$ for the torque due to the weight. Now we are ready to use Equation 12.10 to compute torques:
$τw=rwwsinθw=rwwsin(180°+90°−β)=−L2wsin(90°−β)=−L2wcosβτF=rFFsinθF=rFFsin(180°−β)=LFsinβ.τw=rwwsinθw=rwwsin(180°+90°−β)=−L2wsin(90°−β)=−L2wcosβτF=rFFsinθF=rFFsin(180°−β)=LFsinβ.$
We substitute the torques into Equation 12.30 and solve for $F:F:$
$−L2wcosβ+LFsinβ=0F=w2cotβ=400.0N2cot53°=150.7N−L2wcosβ+LFsinβ=0F=w2cotβ=400.0N2cot53°=150.7N$
12.31
We obtain the normal reaction force with the floor by solving Equation 12.29: $N=w=400.0N.N=w=400.0N.$ The magnitude of friction is obtained by solving Equation 12.28: $f=F=150.7N.f=F=150.7N.$ The coefficient of static friction is $μs=f/N=150.7/400.0=0.377.μs=f/N=150.7/400.0=0.377.$
The net force on the ladder at the contact point with the floor is the vector sum of the normal reaction from the floor and the static friction forces:
$F→floor=f→+N→=(150.7 N)(−i^)+(400.0N)(+j^)=(−150.7i^+400.0j^)N.F→floor=f→+N→=(150.7 N)(−i^)+(400.0N)(+j^)=(−150.7i^+400.0j^)N.$
Its magnitude is
$Ffloor=f2+N2=150.72+400.02N=427.4NFfloor=f2+N2=150.72+400.02N=427.4N$
and its direction is
$φ=tan−1(N/f)=tan−1(400.0/150.7)=69.3°above the floor.φ=tan−1(N/f)=tan−1(400.0/150.7)=69.3°above the floor.$
We should emphasize here two general observations of practical use. First, notice that when we choose a pivot point, there is no expectation that the system will actually pivot around the chosen point. The ladder in this example is not rotating at all but firmly stands on the floor; nonetheless, its contact point with the floor is a good choice for the pivot. Second, notice when we use Equation 12.10 for the computation of individual torques, we do not need to resolve the forces into their normal and parallel components with respect to the direction of the lever arm, and we do not need to consider a sense of the torque. As long as the angle in Equation 12.10 is correctly identified—with the help of a free-body diagram—as the angle measured counterclockwise from the direction of the lever arm to the direction of the force vector, Equation 12.10 gives both the magnitude and the sense of the torque. This is because torque is the vector product of the lever-arm vector crossed with the force vector, and Equation 12.10 expresses the rectangular component of this vector product along the axis of rotation.
### Significance
This result is independent of the length of the ladder because L is cancelled in the second equilibrium condition, Equation 12.31. No matter how long or short the ladder is, as long as its weight is 400 N and the angle with the floor is $53°,53°,$ our results hold. But the ladder will slip if the net torque becomes negative in Equation 12.31. This happens for some angles when the coefficient of static friction is not great enough to prevent the ladder from slipping.
For the situation described in Example 12.5, determine the values of the coefficient $μsμs$ of static friction for which the ladder starts slipping, given that $ββ$ is the angle that the ladder makes with the floor.
## Example 12.6
### Forces on Door Hinges
A swinging door that weighs $w=400.0Nw=400.0N$ is supported by hinges A and B so that the door can swing about a vertical axis passing through the hinges Figure 12.16. The door has a width of $b=1.00m,b=1.00m,$ and the door slab has a uniform mass density. The hinges are placed symmetrically at the door’s edge in such a way that the door’s weight is evenly distributed between them. The hinges are separated by distance $a=2.00m.a=2.00m.$ Find the forces on the hinges when the door rests half-open.
Figure 12.16 A 400-N swinging vertical door is supported by two hinges attached at points A and B.
### Strategy
The forces that the door exerts on its hinges can be found by simply reversing the directions of the forces that the hinges exert on the door. Hence, our task is to find the forces from the hinges on the door. Three forces act on the door slab: an unknown force $A→A→$ from hinge $A,A,$ an unknown force $B→B→$ from hinge $B,B,$ and the known weight $w→w→$ attached at the center of mass of the door slab. The CM is located at the geometrical center of the door because the slab has a uniform mass density. We adopt a rectangular frame of reference with the y-axis along the direction of gravity and the x-axis in the plane of the slab, as shown in panel (a) of Figure 12.17, and resolve all forces into their rectangular components. In this way, we have four unknown component forces: two components of force $A→A→$ $(Ax(Ax$ and $Ay),Ay),$ and two components of force $B→B→$ $(Bx(Bx$ and $By).By).$ In the free-body diagram, we represent the two forces at the hinges by their vector components, whose assumed orientations are arbitrary. Because there are four unknowns $(Ax,(Ax,$ $Bx,Bx,$ $Ay,Ay,$ and $By),By),$ we must set up four independent equations. One equation is the equilibrium condition for forces in the x-direction. The second equation is the equilibrium condition for forces in the y-direction. The third equation is the equilibrium condition for torques in rotation about a hinge. Because the weight is evenly distributed between the hinges, we have the fourth equation, $Ay=By.Ay=By.$ To set up the equilibrium conditions, we draw a free-body diagram and choose the pivot point at the upper hinge, as shown in panel (b) of Figure 12.17. Finally, we solve the equations for the unknown force components and find the forces.
Figure 12.17 (a) Geometry and (b) free-body diagram for the door.
### Solution
From the free-body diagram for the door we have the first equilibrium condition for forces:
$inx-direction:−Ax+Bx=0⇒Ax=Bx iny-direction:+Ay+By−w=0⇒Ay=By=w2=400.0N2=200.0N.inx-direction:−Ax+Bx=0⇒Ax=Bx iny-direction:+Ay+By−w=0⇒Ay=By=w2=400.0N2=200.0N.$
We select the pivot at point P (upper hinge, per the free-body diagram) and write the second equilibrium condition for torques in rotation about point P:
$pivot atP:τw+τBx+τBy=0.pivot atP:τw+τBx+τBy=0.$
12.32
We use the free-body diagram to find all the terms in this equation:
$τw=dwsin(−β)=−dwsinβ=−dwb/2d=−wb2τBx=aBxsin90°=+aBxτBy=aBysin180°=0.τw=dwsin(−β)=−dwsinβ=−dwb/2d=−wb2τBx=aBxsin90°=+aBxτBy=aBysin180°=0.$
In evaluating $sinβ,sinβ,$ we use the geometry of the triangle shown in part (a) of the figure. Now we substitute these torques into Equation 12.32 and compute $Bx:Bx:$
$pivot atP:−wb2+aBx=0⇒Bx=wb2a=(400.0N)12·2=100.0N.pivot atP:−wb2+aBx=0⇒Bx=wb2a=(400.0N)12·2=100.0N.$
Therefore the magnitudes of the horizontal component forces are $Ax=Bx=100.0N.Ax=Bx=100.0N.$ The forces on the door are
$at the upper hinge:F→Aon door=−100.0Ni^+200.0Nj^at the lower hinge:F→Bon door=+100.0Ni^+200.0Nj^.at the upper hinge:F→Aon door=−100.0Ni^+200.0Nj^at the lower hinge:F→Bon door=+100.0Ni^+200.0Nj^.$
The forces on the hinges are found from Newton’s third law as
$on the upper hinge:F→door onA=100.0Ni^−200.0Nj^on the lower hinge:F→door onB=−100.0Ni^−200.0Nj^.on the upper hinge:F→door onA=100.0Ni^−200.0Nj^on the lower hinge:F→door onB=−100.0Ni^−200.0Nj^.$
### Significance
Note that if the problem were formulated without the assumption of the weight being equally distributed between the two hinges, we wouldn’t be able to solve it because the number of the unknowns would be greater than the number of equations expressing equilibrium conditions.
Solve the problem in Example 12.6 by taking the pivot position at the center of mass.
A 50-kg person stands 1.5 m away from one end of a uniform 6.0-m-long scaffold of mass 70.0 kg. Find the tensions in the two vertical ropes supporting the scaffold.
A 400.0-N sign hangs from the end of a uniform strut. The strut is 4.0 m long and weighs 600.0 N. The strut is supported by a hinge at the wall and by a cable whose other end is tied to the wall at a point 3.0 m above the left end of the strut. Find the tension in the supporting cable and the force of the hinge on the strut.<|endoftext|>
| 4.4375 |
351 |
This is also called, The “Eye of Africa”, a marvelous geological phenomenon in the Sahara Desert of Mauritania, a country in northern Africa, peers up in this satellite image. The Richat structure, as it is also recognized, and very much resembles to bull's-eye peering out of the sand. The magnificent structure is 30 miles in diameter, very large in the featureless Sahara that the earliest space missions used it as a landmark.
This was well believed, that eye was formed by a meteor impact, but with the passage of time, it is believed to be result of geological uplift that has been exposed over time by wind and water erosion. Therefore, different rates of erosion on the varying rock types have formed concentric ridges, and then further erosion resistant rocks form high ridges while the non-resistant rocks form valleys. A plateau of sedimentary rocks forms the darker regions surrounding the Richat structure, which is roughly stands 656 above the surrounding sand.
Mauritania’s (The Kediet ej jill Mountain) highest peak is a magnetic mountain standing almost 3281 feet can be seen here. It appears very composed in blue colors a true natural magnetic substance. Whenever you’re flying over Mauritania, or passing above Africa, you must have a visit out the window and see if you can spot the Richat Structure. It shouldn’t be too difficult the thing is 30 miles wide. A lot of scientists truly believed that the Eye of the Sahara was first spotted from space in the mid-'60s. The Richat Structure, or blue eye of Africa, is a prominent geological circular feature in the Sahara desert in Mauritania near Ouadane.<|endoftext|>
| 3.859375 |
761 |
## ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 1 Knowing Our Numbers Ex 1.1
Question 1.
Write the smallest natural number. Can you write the largest natural number?
Solution:
Smallest natural number = 1
No, we can not write the largest natural number.
Question 2.
Fill in the blanks:
(i) 1 lakh = … ten thousand
(ii) 1 million = … hundred thousand
(iii) 1 crore = … ten lakh
(iv) 1 billion =… hundred million.
Solution:
(i) 1 lakh = ten ten thousand
(ii) 1 million = ten hundred thousand
(iii) 1 crore = ten ten lakh
(iv) 1 billion = ten hundred million.
Question 3.
Insert commas suitably and write each of the following numbers in words in the Indian system and the International system of numeration.
(i) 506723
(ii) 180018018
Solution:
(i) 506723 = 5,06,723
Five lakh six thousand seven hundred and twenty three.
(ii) 180018018 = 18,00,18,018
Eighteen crore eighteen thousand and eighteen.
Question 4.
Write the following numbers in expanded form:
(i) 750687
(ii) 5032109
Solution:
(i) 750687=700000 + 50000 + 600 + 80 + 7
(ii) 5032109 = 5000000 + 30000 + 2000 + 100 + 9
Question 5.
Write the following number in figures:
(i) Seven lakh three thousand four hundred twenty.
(ii) Eighty crore twenty three thousand ninety three.
Also write the above numbers in the place value chart.
Solution:
(i) 7,03,420
(ii) 80,00,23,093
Question 6.
Write each of the following numbers in numeral form and place commas correctly:
(i) Seventy three lakh seventy thousand four hundred seven.
(ii) Nine crore five lakh forty one.
(iii) Fifty eight million four hundred twenty three thousand two hundred two.
Solution:
(i) 73,73,407
(ii) 9,05,00,041
(iii) 58,423,202
Question 7.
Write the face value and place value of the digit 6 in the number 756032.
Solution:
756032
Face value of 6 = 6
and place value = 6000
Question 8.
Find the difference between the place value and the face value of the digit 9 in the number 229301.
Solution:
Place value of 9
Question 9.
Determine the difference of the place value of two 7’s in 37014472 and write it in words in International system.
Solution:
The given number in International system can be written as 37,014,472
The place value of 7 at ten’s place = 7 × 10 = 70
The place value of 7 at 7 million’s place = 7 × 1,000,000 = 7,000,000
The required difference = 7,000,000 – 70 = 6,999,930
Six million nine hundred ninty thousand nine hundred thirty.
Question 10.
Determine the product of place value and the face value of the digit 4 in the number 5437.
Solution:
Place value of 4
Question 11.
Find the difference between the number 895 and that obtained on reversing its digits.
Solution:
First number = 895
Reversed number = 598
difference 297<|endoftext|>
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Earth’s formation is one of the most profound events in our history. Meteorites provide information about some of the primordial materials from which Earth formed. However, evidence of volatile elements such as carbon, nitrogen, and hydrogen has long since vanished from these celestial time capsules. Addressing the quantities and origins of volatile elements on Earth is therefore extremely challenging.
Surface reservoirs of carbon are generally well documented, but Earth’s deep interior may contain large hidden inventories. Indeed, two recent studies suggest that some of Earth’s carbon may be hidden in the core, making it the planet’s largest carbon reservoir [1,2].
It is now widely accepted that Earth's inner core consists of crystalline iron alloyed with a small amount of nickel and some lighter elements. However, seismic waves called S waves travel through the inner core at about half the speed expected for most iron-rich alloys under relevant pressures.
Some researchers have attributed the S-wave velocities to the presence of liquid, calling into question the solidity of the inner core. In recent years, the presence of various light elements—including sulfur, carbon, silicon, oxygen and hydrogen—has been proposed to account for the density deficit of Earth's core.
DCO’s Jackie Li (University of Michigan, USA), Bin Chen (University of Hawaii, USA) and colleagues suggest that iron carbide, Fe7C3, provides a good match for the density and sound velocities of Earth's inner core under relevant conditions .
"The model of a carbide inner core is compatible with existing cosmochemical, geochemical, and petrological constraints, but this provocative and speculative hypothesis still requires further testing," Li said. "Should it hold up to various tests, the model would imply that as much as two-thirds of the planet's carbon is hidden in its center sphere, making it the largest reservoir of carbon on Earth."
In a complementary paper that formed part of his doctoral thesis, Clemens Prescher (Universität Bayreuth, Germany) and colleagues simulated the high pressures and temperatures of Earth’s core in a diamond anvil cell . They showed that at these conditions, carbon dissolves in iron to form a stable phase. Although the iron carbide phase is the same Fe7C3 predicted by Li and colleagues, the structure was found to be different.
They also show that under high temperatures and pressures, Fe7C3 exhibits the elastic, almost “rubbery,” properties predicted from direct seismological observations of Earth’s core.
“If carbon were the only light element in the core, there would be more than one hundred times more carbon in the core than in all of the Earth’s surface regions and rocks,” noted DCO’s Catherine McCammon, one of the senior co-authors.
Li and McCammon both presented their work at the recent DCO International Science Meeting in Munich, 26-28 March 2015.<|endoftext|>
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Dr Julie Banfield
The Australian National University
Super-massive black holes are a fundamental building block of galaxies like our own Milky Way. However, astronomers and physicists know little about how they form, grow to be a billion times the mass of our own Sun, and affect the formation of galaxies, stars, planets, and life. Dr Banfield’s research contributes to understanding the role of super-massive black holes in the life cycles of galaxies and stars by studying the outflows of electrons from their centre. Her recent study of super-massive black holes shows outflows directly interacting with the galactic gas cloud that will create stars. This study illustrates how super-massive black holes can create stars and not simply destroy them. It will lead to a more complete understanding of these galactic interactions and is likely to reveal the role and physical processes that super-massive black holes play in the creation and destruction of galaxies and stars.
She is a co-leader of Radio Galaxy Zoo, an international citizen science project to enlist the general public to identify the host galaxy of super-massive black holes. This project enables the general public to become involved in the scientific process. There are over 11,000 volunteers and over 1.8 million classifications. She has also been the Astronomer in Residence at the Ayers Rock Resort.<|endoftext|>
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Moses and Aaron, the Old Testament tells us, had to make bricks without straw before their people could leave Egypt and begin the journey to the Promised Land.
Bob and Dick have to figure out how to make bricks from regolith before their people can leave Earth and begin colonizing Mars.
Regolith is the Martian equivalent of what would be called dirt or soil on our planet. Bob and Dick are Robert Orwoll and Richard Kiefer, professors emeriti in William & Mary’s Department of Chemistry. They are collaborating with a private firm, International Scientific Technologies, Inc. (IST) of Radford, Virginia. IST has received a Small Business Innovation Research (SBIR) award from NASA to help find ways to protect astronauts from space radiation.
Kiefer and Orwoll have a dual task. They are developing a procedure to bind Martian regolith with a polymer securely enough to serve as a building material, while acting as a radiation shield as well. There are many challenges to establishing a human beachhead on Mars, including finding a water source and dealing with the travel time necessary for making the trip. But radiation is one of the biggest challenges.
“Radiation is a very serious problem for NASA,” Orwoll said. “You don’t read about it much in the papers, but they’ve got a ways to go yet on solving the radiation problem before they can send anybody to Mars.”
They make a good team. Orwoll is a polymer chemist, specializing in the structure and synthesis of the useful macromolecules. Kiefer is a nuclear chemist, studying the interactions of deep-space radiation with polymers and other materials.
“We have a background in doing this,” Kiefer said. He and Orwoll worked on similar NASA projects involving lunar regolith and developing polymer radiation shielding.
“The most serious kind of radiation is called galactic cosmic radiation — GCR. It is particulate radiation, made up of just the nuclei of lots of elements,” Orwoll said.
GCR particles are essentially cosmic shrapnel from supernova events. Orwoll said that GCR particles are mostly hydrogen and helium nuclei, but can include nuclei of atoms all the way up to iron on the Periodic Table.
“It’s a highly penetrating radiation, and very damaging, both to DNA and to electrical components on a space mission,” Orwoll added.
Previous ventures into space have given scientists a good picture of the nature of how much GCR Martian astronauts can expect.
“When they sent up (Mars rover) Curiosity, they had radiation detectors on the thing,” Kiefer explained. “They tracked the radiation on the full trip to Mars, and they confirmed that the dose was very high for humans.”
Kiefer went on to say that the Red Planet doesn’t offer the protections against cosmic radiation that have made life as we know it possible on Earth.
“Mars has a tenth of the atmosphere that we have. And it doesn’t have magnetic fields,” he explained. “So once you get to Mars, you’re not shielded very much.”
Consequently, the first humans to set foot on Mars will have to take at least some of their shielding with them. But most of the raw material will come from underfoot.
“I think people have contemplated for years using the dirt from Mars — the regolith,” Orwoll said. “Think about hauling a bunch of construction stuff to Mars in a spacecraft! That just wouldn’t be possible.”
“Even Home Depot isn’t there yet,” Kiefer added.
“So it would be neat if a lot of the construction work could be done with material available on Mars,” Orwoll said.
Kiefer and Orwoll have no Martian regolith to experiment with, but they do have a pretty good substitute. Curiosity, Pathfinder and other Mars missions have analyzed the chemical composition of regolith and scientists have found that there’s regolith-like rock here on earth.
“It comes from a volcano in Hawaii. On the Big Island,” Kiefer said. “A company in Wisconsin sells it. ”
The rock, marketed as Martian Regolith Simulant, is quarried from the cinder cone of Pu’u Nene, a volcano between Mauna Loa and Mauna Kea. Scientists have been using the stuff to prepare for Mars since 1998. Kiefer and Orwoll have bags of it, in two different sizes, in their lab. Its chemical makeup is pretty similar to the regolith of Mars.
“There’s a lot of silicate, like sand, in regolith. There’s a lot of iron; that’s why it’s red,” Kiefer said. “It’s a combination of a lot of things, you know, just like dirt here on Earth. But regolith doesn’t have any organic material. It would be more like you find in a desert.”
Kiefer and Orwoll are trying to find the best polymer binder for humans to bring Mars. They experiment with different formulas, making regolith-simulant bricks in the lab. They’ve found kitchen-supply stores to be a good source of molds and so some of their rego-bricks resemble slightly overbaked loaves of banana bread.
The “best” polymer will be easily mixed, an effective binder and the most efficient by weight. Kiefer said they figure a good regolith brick should be no more than 10 percent binder, but of course they’d like to do better.
The final polymer binder would probably — but not necessarily — be in powder form. The collaborators have made progress in figuring out the most efficient type of polymers for shielding.
“We want polymers that have a lot of hydrogen in them,” Kiefer said. “The reason is that hydrogen is the best shield for these galactic cosmic rays.”
“On a per-gram basis,” Orwoll added.
The collaborators pointed out that lead, a venerable and effective earthly radiation shield, is an obvious non-starter because of the weight. And surprisingly, hydrogen is more effective as a radiation shield than lead.
“You’re better off with a pound of hydrogen than you are with a pound of lead,” Orwoll said.
“The first time I heard that, I said ‘Wait a minute,’” Kiefer said. “But a scientist from NASA showed me his calculations. He convinced me.”
Hydrogen’s effectiveness against cosmic radiation is a matter of defense by electron. Electrons carry a negative charge. Each hydrogen atom has only a single electron, but Kiefer pointed out that hydrogen, which occupies the leadoff spot in the Periodic Table, has one electron per one unit of mass. (Carbon, by comparison, is half as electron-rich, with six electrons per 12 units of mass.)
Kiefer explained that the particles of GCR are mostly naked protons, which carry a positive charge.
“So they interact with a negative electron by moving it aside,” he said, “losing energy in the process.”
Hydrogen has another virtue as a radiation shielding component. Kiefer continued to explain that some of the incoming GCR protons do their damage by shattering atoms at high energy. So, when a particle zooming in hits the nucleus of an atom of traditional shielding material, it flies apart, releasing considerable energy.
“And you end up with more radiation behind your shield than you had in front of it,” he added. It’s not a problem with hydrogen, whose one-proton nucleus won’t split apart.
In fact, liquid hydrogen would make the very best cosmic radiation shield, Kiefer said, “but not very practical.” Pure regolith adobe-style walls, with no polymer binder, would offer shielding as well, but would have to be a foot or so thick.
“But to do that, to make walls that thick, you are going to need some heavy equipment,” Kiefer said. “So here’s the trade-off: Do you take stuff to make bricks or do you take along a bulldozer?”
Topics like the bricks-or-bulldozer question come up often in conversations with Orwoll and Kiefer. Recondite details of hydrogen bonds in polymers and the interactions of cosmic rays are peppered with Martian speculations and what-ifs. Their discussion is reminiscent of the development section of a science fiction novel.
Among the ideas that get tossed around: The rego-bricks could be mass-produced at a landing site by robotic devices placed by an unmanned mission years before the first humans land. And it doesn’t have to be bricks —regolith could be mixed with polymer bonder and extruded into logs. A bladder, protected by rego-brick outer walls, could be pressurized to Earth conditions, allowing Martian colonists to get inside and take off their space suits. A spacecraft designed to store their polymer binder in the fuselage walls of the cabin could provide radiation shielding for the trip.
Anything is possible, or at least a large number of things are, but first the chemists need to perfect their polymer binder. Both chemists are aware that their work has a high-tech, even a sci-fi, aspect. At the same time, they realize it’s all about making bricks, something humans have been doing since before the time of Moses.“If you get to an alien planet, low tech is all you’re going to be able to do — at first,” Kiefer said.<|endoftext|>
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A trebuchet is a type of catapult that usually uses a weight, throwing arm, and sling to hurl rocks or even disease ridden animal carcasses at or over protective walls during sieges throughout the Middle Ages? Recently, the South Bend physics and construction students got a brief history lesson, connecting the sling and trebuchet, as they prepared to build their own trebuchets. Although there was no castle siege during the project, there were some exciting results!
After combining their classes for the first time last spring to build bridges, Ryan McMurry and Joel Bale were excited to bring their construction and physics classes together again, respectively. The two teachers spent time researching trebuchets and building a couple models before moving forward with the project. The first model had a one-foot throwing arm and fell short of being impressive. The second model was twice as big as the first and was enough to make you smile as the projectile soared through the air! Of course, the students would build the bigger version!
Building the trebuchets was fun, but the students also had some goals to keep in mind during construction. Scoring would be based on several factors: teamwork, graphing, accuracy, and maximum distance. Physics students and construction students were grouped together in twos and threes, would have to figure out how to work efficiently together and exhibit quality craftsmanship, and give constructive criticism to other groups. In order for a team to be able to hit particular targets, each team made a graph.
To make their graphs, students used three-pound bags to launch their projectiles various distances. Once it was time to hit a target at a particular distance, students could look at their best-fit line to predict the weight that would be necessary to deliver their BB-filled duct tape projectile to the target. The final test was the distance test, during which all students used the same weight and projectile.
McMurry and Bale were amazed by the results. Almost all the students worked very well together. During the accuracy test, students had to determine the weight needed to hit 38 feet, 52 feet, and 73 feet. Amazingly, most students were within a couple feet and several students were hitting the exact distance! We thought they’d do well but didn’t really expect the students to be hitting the exact marks because there were so many variables! The distance test was equally impressive as Mitch Edwards, Myranda Curtis and Jordan Stigall’s trebuchet launched the standardized projectile 134 feet. Teams with top scores for teamwork and the competition got a pizza feed as a reward!
In the end, Bale said that he would call the project a success. “The results of the tests were unbelievable,” Bale said. “The students worked very well together even though they were all a little out of their element. Just like life after high school the teams encountered difficulties and had to figure out how to solve their problems. There were a couple of groups that had major problems that I thought might be enough to make them give up, but instead of giving up, they tackled the problems with renewed energy. It was great to see so many students working well together, problem solving when issues came up, and keeping a positive attitude even when things were falling apart. Many of the students would have made an employer proud.”McMurry and Bale both had a great time and look forward to working together in the future.<|endoftext|>
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The Laws of the Indies, or Las Leyes de los Indios, dictated the Spanish colonies of the Hispanic Americas. These laws directives provided guideance as to how to construct new societies. In a paradoxical way the conquestedors, were to build cities in a way that would reflect human rights. You can see these in places from Savannah, GA, to Santa Fe, NM and Jackson Hole, WY.
They all had common themes, which *should* and *do* apply today:
- Find a suitable place of land
- Construct the plaza first
- Display a governmental building on the plaza
- Build a church
- Two sides for shops with a covered arcade and continued four blocks on two main streets for the "convience of the shop owners", i.e. just in case in rains or snows
- Repeat and connect schools, districts and systems
Asucion Mita, Guatemala
On a recent trip to El Salvador, I randomly stopped in a small town along the highway to buy bricks for my mothers garden. This lead me to the center of the City in search of an ATM, where I found the best living example I have ever seen of one of these cities in effect.
I was so excited taking the video that I forgot to mention the significance of the government near the park: democracy. It gives the people a direct place to demonstrate their grievances with the government. But in some cases, the government had other ideas ... :/
There lies the great debate amongst the urbanists with the public and among ourselves. Does our built environmental reflect how we live our lives? Would it add significance to the idea of democracy if government was physically surrounded by the public [out of their cars]?
Do you think where you lives determines how you live your life?<|endoftext|>
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# Recursion
Welcome back to the Digilent Blog! Today we’re going to go over recursion! Recursion is when a function calls itself directly, or through another function. Sometimes we can’t solve a problem using loops (iteration), so we have to use recursion. Recursion is slower than iteration, difficult to debug, and it uses up more of the stack. But recursion can also have simpler code, so in some cases, the benefits outweigh the problems.
There are two parts of a recursive function: the base case and the recursive call. The base case is when we reduce our function to the most basic case of our function, and then returns. The recursive call is when we call the recursive function again. This is confusing to read, so the best way to show it is with code. While we call the recursive statement, our function is pushing another call of itself with a reduced parameter input (otherwise the function will never end) onto the stack, and these calls are popped when we reach the base case and start to return.
Let’s check out the first function, theĀ factorial. Factorial is simulating the “!” operator in mathematics. We have the parameter and return values be an integer, with parameter named as “n”. The first thing we see is the base case. If n <= 1 (n should never be 0 or less, but just in case, we have the “<” symbol) then we return 1.
Secondly, we have the recursive step, which is were we call factorial again in the return statement. Let’s use the example of “n” being 4, or recursive call will be return(4 * factorial(3)), with (n-1) being our reduction. Because the “n” being passed as the parameter is not our base case, we will do another recursive call, but this time it will be return(3 * factorial(2)).
Another good example case of recursion is the Fibonacci sequence. The Fibonacci sequence is the summation of the two previous entrees up to “n”. What’s new here is that we have two base cases that can happen, because our fibonacci uses the two previous numbers, meaning that it will have to call fibonacci twice for each recursive call. With each call of fibonacci puts two other calls of fibonacci onto the stack, so you can see how recursion can potentially cause stack problems!
All the recursive code
Here’s a small main function that lets users input a value. For this example, we’re limiting the user’s input values to be between 0 and 19. This is because for our factorial function, 20! and beyond is larger than the size of int (check out why here!).
Be the 1st to vote.<|endoftext|>
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- Explain how the triangle of meaning describes the symbolic nature of language.
- Distinguish between denotation and connotation.
- Discuss the function of the rules of language.
- Describe the process of language acquisition.
The relationship between language and meaning is not a straightforward one. One reason for this complicated relationship is the limitlessness of modern language systems like English (Crystal, 2005). Language is productive in the sense that there are an infinite number of utterances we can make by connecting existing words in new ways. In addition, there is no limit to a language’s vocabulary, as new words are coined daily. Of course, words aren’t the only things we need to communicate, and although verbal and nonverbal communication are closely related in terms of how we make meaning, nonverbal communication is not productive and limitless. Although we can only make a few hundred physical signs, we have about a million words in the English language. So with all this possibility, how does communication generate meaning?
You’ll recall that “generating meaning” was a central part of the definition of communication we learned earlier. We arrive at meaning through the interaction between our nervous and sensory systems and some stimulus outside of them. It is here, between what the communication models we discussed earlier labeled as encoding and decoding, that meaning is generated as sensory information is interpreted. The indirect and sometimes complicated relationship between language and meaning can lead to confusion, frustration, or even humor. We may even experience a little of all three, when we stop to think about how there are some twenty-five definitions available to tell us the meaning of word meaning! (Crystal, 2005) Since language and symbols are the primary vehicle for our communication, it is important that we not take the components of our verbal communication for granted.
Language Is Symbolic
Our language system is primarily made up of symbols. A symbol is something that stands in for or represents something else. Symbols can be communicated verbally (speaking the word hello), in writing (putting the letters H-E-L-L-O together), or nonverbally (waving your hand back and forth). In any case, the symbols we use stand in for something else, like a physical object or an idea; they do not actually correspond to the thing being referenced in any direct way. Unlike hieroglyphics in ancient Egypt, which often did have a literal relationship between the written symbol and the object being referenced, the symbols used in modern languages look nothing like the object or idea to which they refer.
The symbols we use combine to form language systems or codes. Codes are culturally agreed on and ever-changing systems of symbols that help us organize, understand, and generate meaning (Leeds-Hurwitz, 1993). There are about 6,000 language codes used in the world, and around 40 percent of those (2,400) are only spoken and do not have a written version (Crystal, 2005). Remember that for most of human history the spoken word and nonverbal communication were the primary means of communication. Even languages with a written component didn’t see widespread literacy, or the ability to read and write, until a little over one hundred years ago.
The symbolic nature of our communication is a quality unique to humans. Since the words we use do not have to correspond directly to a “thing” in our “reality,” we can communicate in abstractions. This property of language is called displacement and specifically refers to our ability to talk about events that are removed in space or time from a speaker and situation (Crystal, 2005). Animals do communicate, but in a much simpler way that is only a reaction to stimulus. Further, animal communication is very limited and lacks the productive quality of language that we discussed earlier.
As I noted in Chapter 1 “Introduction to Communication Studies”, the earliest human verbal communication was not very symbolic or abstract, as it likely mimicked sounds of animals and nature. Such a simple form of communication persisted for thousands of years, but as later humans turned to settled agriculture and populations grew, things needed to be more distinguishable. More terms (symbols) were needed to accommodate the increasing number of things like tools and ideas like crop rotation that emerged as a result of new knowledge about and experience with farming and animal domestication. There weren’t written symbols during this time, but objects were often used to represent other objects; for example, a farmer might have kept a pebble in a box to represent each chicken he owned. As further advancements made keeping track of objects-representing-objects more difficult, more abstract symbols and later written words were able to stand in for an idea or object. Despite the fact that these transitions occurred many thousands of years ago, we can trace some words that we still use today back to their much more direct and much less abstract origins.
For example, the word calculate comes from the Latin word calculus, which means “pebble.” But what does a pebble have to do with calculations? Pebbles were used, very long ago, to calculate things before we developed verbal or written numbering systems (Hayakawa & Hayakawa, 1990). As I noted earlier, a farmer may have kept, in a box, one pebble for each of his chickens. Each pebble represented one chicken, meaning that each symbol (the pebble) had a direct correlation to another thing out in the world (its chicken). This system allowed the farmer to keep track of his livestock. He could periodically verify that each pebble had a corresponding chicken. If there was a discrepancy, he would know that a chicken was lost, stolen, or killed. Later, symbols were developed that made accounting a little easier. Instead of keeping track of boxes of pebbles, the farmer could record a symbol like the word five or the numeral 15 that could stand in for five or fifteen pebbles. This demonstrates how our symbols have evolved and how some still carry that ancient history with them, even though we are unaware of it. While this evolution made communication easier in some ways, it also opened up room for misunderstanding, since the relationship between symbols and the objects or ideas they represented became less straightforward. Although the root of calculate means “pebble,” the word calculate today has at least six common definitions.
The Triangle of Meaning
The triangle of meaning is a model of communication that indicates the relationship among a thought, symbol, and referent and highlights the indirect relationship between the symbol and referent (Richards & Ogden, 1923). As you can see in Figure 3.1 “Triangle of Meaning”, the thought is the concept or idea a person references. The symbol is the word that represents the thought, and the referent is the object or idea to which the symbol refers. This model is useful for us as communicators because when we are aware of the indirect relationship between symbols and referents, we are aware of how common misunderstandings occur, as the following example illustrates: Jasper and Abby have been thinking about getting a new dog. So each of them is having a similar thought. They are each using the same symbol, the word dog, to communicate about their thought. Their referents, however, are different. Jasper is thinking about a small dog like a dachshund, and Abby is thinking about an Australian shepherd. Since the word dog doesn’t refer to one specific object in our reality, it is possible for them to have the same thought, and use the same symbol, but end up in an awkward moment when they get to the shelter and fall in love with their respective referents only to find out the other person didn’t have the same thing in mind.
Being aware of this indirect relationship between symbol and referent, we can try to compensate for it by getting clarification. Some of what we learned in Chapter 2 “Communication and Perception”, about perception checking, can be useful here. Abby might ask Jasper, “What kind of dog do you have in mind?” This question would allow Jasper to describe his referent, which would allow for more shared understanding. If Jasper responds, “Well, I like short-haired dogs. And we need a dog that will work well in an apartment,” then there’s still quite a range of referents. Abby could ask questions for clarification, like “Sounds like you’re saying that a smaller dog might be better. Is that right?” Getting to a place of shared understanding can be difficult, even when we define our symbols and describe our referents.
Definitions help us narrow the meaning of particular symbols, which also narrows a symbol’s possible referents. They also provide more words (symbols) for which we must determine a referent. If a concept is abstract and the words used to define it are also abstract, then a definition may be useless. Have you ever been caught in a verbal maze as you look up an unfamiliar word, only to find that the definition contains more unfamiliar words? Although this can be frustrating, definitions do serve a purpose.
Words have denotative and connotative meanings. Denotation refers to definitions that are accepted by the language group as a whole, or the dictionary definition of a word. For example, the denotation of the word cowboy is a man who takes care of cattle. Another denotation is a reckless and/or independent person. A more abstract word, like change, would be more difficult to understand due to the multiple denotations. Since both cowboy and change have multiple meanings, they are considered polysemic words. Monosemic words have only one use in a language, which makes their denotation more straightforward. Specialized academic or scientific words, like monosemic, are often monosemic, but there are fewer commonly used monosemic words, for example, handkerchief. As you might guess based on our discussion of the complexity of language so far, monosemic words are far outnumbered by polysemic words.
Connotation refers to definitions that are based on emotion- or experience-based associations people have with a word. To go back to our previous words, change can have positive or negative connotations depending on a person’s experiences. A person who just ended a long-term relationship may think of change as good or bad depending on what he or she thought about his or her former partner. Even monosemic words like handkerchief that only have one denotation can have multiple connotations. A handkerchief can conjure up thoughts of dainty Southern belles or disgusting snot-rags. A polysemic word like cowboy has many connotations, and philosophers of language have explored how connotations extend beyond one or two experiential or emotional meanings of a word to constitute cultural myths (Barthes, 1972).Cowboy, for example, connects to the frontier and the western history of the United States, which has mythologies associated with it that help shape the narrative of the nation. The Marlboro Man is an enduring advertising icon that draws on connotations of the cowboy to attract customers. While people who grew up with cattle or have family that ranch may have a very specific connotation of the word cowboy based on personal experience, other people’s connotations may be more influenced by popular cultural symbolism like that seen in westerns.<|endoftext|>
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Researchers have discovered a unique method of integrating solar cells into fabric that could be used to develop lighter, more power-efficient wearable electronics.
A team of scientists from Fudan University and Tongii University in Shanghai created the flexible solar cells using a stacking method that eliminated many of the barriers faced in developing such technology.
The research, published in Angewandte Chemie, demonstrates that stacking electrodes into layers allows the creation of longer, more efficient fabrics, compared to previous methods.
Thread-like solar cells that have been produced in the past have had little practical application as they rely on twisting two electrically conducting fibres together as electrodes.
This method means any solar cell fabric is both time-consuming to create and difficult to make into any length greater than a few millimetres.
By stacking the textile electrodes into layers, the researchers were able to twist the material into a strong enough thread to be woven into a textile.
"A metal–textile electrode that was made from micrometer-sized metal wires was used as a working electrode, while the textile counter electrode was woven from highly aligned carbon nanotube fibers with high mechanical strengths and electrical conductivities," the abstract to the research paper reads.
"This stacked textile unexpectedly exhibited a unique deformation from a rectangle to a parallelogram, which is highly desired in portable electronics."
The researchers have already used small patches of the solar cell fabric to power an LED light.<|endoftext|>
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The circulatory system is the system of vessels and organs through which blood flows and nutrients are transported throughout the body. It is also known as the cardiovascular system.
See the fact file below for more information on the circulatory system or alternatively, you can download our 25-page Circulatory System worksheet pack to utilise within the classroom or home environment.
Key Facts & Information
What is the Circulatory System?
- The primary function of the circulatory system is to keep blood flowing throughout the body in order to fight diseases and maintain homeostasis (healthy internal balance).
- According to the US National Library of Medicine, the circulatory system is made up of three independent systems: the cardiovascular, pulmonary, and systemic systems.
- The cardiovascular system consists of the heart.
- The pulmonary system consists of the lungs.
- The systemic system consists of the arteries, veins, coronary vessels, and portal vessels.
- Pulmonary circulation occurs when blood is oxygenated through the lungs.
- Systemic circulation occurs when oxygenated blood is sent throughout the body.
- The main parts of the circulatory system are the heart, the blood, and the blood vessels.
- The heart sends blood through the blood vessels to cells.
- Blood transports oxygen, carbon dioxide, amino acids, blood cells, hormones, and other gases throughout the body.
- Blood consists of red blood cells, white blood cells, platelets, and plasma.
- The heart beats 60 to 100 times per minute.
- Blood is sent throughout the body with every heartbeat.
- Blood begins to circulate when the heart relaxes between two heartbeats.
- The blood then releases oxygen and nutrients to the cells and takes on carbon dioxide and other waste.
- The deoxygenated blood returns to the heart after transporting nutrients and oxygen.
- The heart pumps blood back to the lungs to pick up more oxygen, then the cycle repeats.
- The heart has two top and two bottom chambers.
- The two top chambers are called right atrium and left atrium, which are divided by a wall called the interatrial septum.
- The two bottom chambers are called right ventricle and left ventricle, which are divided by a wall called the interventricular septum.
- The right and left atriums receive the blood entering the heart.
- The right and left ventricles pump blood out of the heart.
- The top and bottom chambers are divided by atrioventricular valves.
- There are four types of atrioventricular valves: the aortic valve, the pulmonic valve, the tricuspid valve, and the mitral valve.
- The aortic valve separates the aorta from the left ventricle.
- The pulmonic valve separates the pulmonary artery from the right ventricle.
- The tricuspid valve separates the right ventricle from the right atrium.
- The mitral valve separates the left ventricle from the left atrium.
- Arteries carry blood away from the heart.
- The main artery is called the aorta, which carries the blood to the body.
- The pulmonary artery carries blood to the lungs.
- Arteries branch into small pathways called arterioles which then branch into capillaries.
- Capillaries are tiny thin-walled blood vessels that merge into smaller veins called venules which then merge into larger veins.
- Veins carry blood back to the heart.
- There are two major veins that empty into the right atrium of the heart: the superior vena cava and the inferior vena cava.
- The superior vena cava drains the areas above the heart.
- The inferior vena cava drains the areas below the heart.
- Electrical signals in the heart prompts the heart to beat.
- Two nodes are important in sending and relaying these signals: the sinus node and the atrioventricular node.
- Found in the wall of the right atrium, the sinus node is the pacemaker of the heart.
- The sinus node sets the rate of the heartbeat by sending electrical impulses that make the atria contract.
- The atrioventricular (AV) node receives these impulses which make the ventricles contract.
- One heartbeat is made up of two phases: the sistole and the diastole.
- During systole, the ventricles contract and blood is pumped into the aorta.
- During diastole, the ventricles relax and get filled with blood to flow from the atria in preparation for the next heartbeat.
- Circulatory diseases are classified into cardiovascular diseases (affecting the cardiovascular system) and lymphatic diseases (affecting the lymphatic system).
- Cardiovascular diseases are the leading cause of death in the United States according to the American Heart Association.
- Arteriosclerosis is one of the top common diseases, which occurs when the walls of the arteries are thickened and stiffened by fatty deposits such as fat and cholesterol.
- Strokes occur when blood vessels going into the brain are blocked.
- Hypertension, also known as high blood pressure, is a circulatory disease that can cause kidney complications, a heart attack, or a stroke.
- Peripheral arterial disease (PAD) occurs when there is a blockage or narrowing within an artery.
- Diseases that concern the heart are treated by cardiologists.
- Operations on the heart are done by cardiothoracic surgeons.
- Operations on the vascular system are done by vascular surgeons.
A Healthy Heart
- To keep your heart and the rest of your circulatory system healthy, you have to get enough exercise, eat a well-balanced and nutritious diet, not smoke, and get regular medical check-ups.
Circulatory System Worksheets
This is a fantastic bundle which includes everything you need to know about circulatory system across 25 in-depth pages. These are ready-to-use Circulatory System worksheets that are perfect for teaching students about the circulatory system which is the system of vessels and organs through which blood flows and nutrients are transported throughout the body. It is also known as the cardiovascular system.
Complete List Of Included Worksheets
- Circulatory System Facts
- System of Systems
- Quick Questions
- Define the Differences
- Skip A Beat
- Walls, Vessels, Veins
- Heart Parts
- Odd One Out
- Hows of a Heartbeat
- Decode the Disease
- A Healthy Heart
Link/cite this page
If you reference any of the content on this page on your own website, please use the code below to cite this page as the original source.
Link will appear as Circulatory System Facts & Worksheets: https://kidskonnect.com - KidsKonnect, January 7, 2019
Use With Any Curriculum
These worksheets have been specifically designed for use with any international curriculum. You can use these worksheets as-is, or edit them using Google Slides to make them more specific to your own student ability levels and curriculum standards.<|endoftext|>
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Question Video: Solving Trigonometric Equations by Squaring | Nagwa Question Video: Solving Trigonometric Equations by Squaring | Nagwa
# Question Video: Solving Trigonometric Equations by Squaring Mathematics • First Year of Secondary School
## Join Nagwa Classes
By first squaring both sides, or otherwise, solve the equation 4 sin 𝜃 − 4 cos 𝜃 = √3, where 0° < 𝜃 ⩽ 360°. Be careful to remove any extraneous solutions. Give your answers to two decimal places.
06:47
### Video Transcript
By first squaring both sides, or otherwise, solve the equation four sin 𝜃 minus four cos 𝜃 is equal to the square root of three, where 𝜃 is greater than zero degrees and less than or equal to 360 degrees. Be careful to remove any extraneous solutions. Give your answers to two decimal places.
The question advises that we approach the problem by first squaring both sides of the equation. Doing so, we obtain four sin 𝜃 minus four cos 𝜃 all squared is equal to root three squared. Distributing the parentheses and then collecting like terms on the left-hand side gives us 16 sin squared 𝜃 minus 32 sin 𝜃 cos 𝜃 plus 16 cos squared 𝜃. On the right-hand side, root three squared is equal to three.
Next, we recall the Pythagorean identity, which states that sin squared 𝜃 plus cos squared 𝜃 is equal to one. Simplifying the left-hand side further gives us 16 multiplied by sin squared 𝜃 plus cos squared 𝜃 minus 32 sin 𝜃 cos 𝜃. Replacing sin squared 𝜃 plus cos squared 𝜃 with one gives us the equation 16 minus 32 sin 𝜃 cos 𝜃 equals three. Subtracting 16 from both sides of this equation gives us negative 32 sin 𝜃 cos 𝜃 is equal to negative 13. We can then divide through by negative 32 such that sin 𝜃 cos 𝜃 equals 13 over 32.
We now have two equations in the two variables sin 𝜃 and cos 𝜃. This means that the system of equations can be solved simultaneously. Adding four cos 𝜃 to both sides of our original equation, we have four sin 𝜃 is equal to root three plus four cos 𝜃. Dividing both sides of this equation by four, we have sin 𝜃 is equal to root three plus four cos 𝜃 all divided by four.
After clearing some space, we will now consider how we can solve these two simultaneous equations. We will begin by substituting the expression for sin 𝜃 in equation two into equation one. This gives us root three plus four cos 𝜃 over four multiplied by cos 𝜃 is equal to 13 over 32. We can simplify this equation by firstly distributing the parentheses. We can then multiply through by 32, giving us eight root three cos 𝜃 plus 32 cos squared 𝜃 equals 13. Finally, subtracting 13 from both sides of this equation, we have the quadratic equation in terms of cos 𝜃 as shown.
This can be solved using the quadratic formula, where 𝑎 is 32, 𝑏 is eight root three, and 𝑐 is negative 13. Substituting in these values and then simplifying gives us cos of 𝜃 is equal to negative three plus or minus the square root of 29 all divided by eight. Taking the inverse cosine of both sides with positive root 29 gives us 𝜃 is equal to 62.829 and so on. To two decimal places, this is equal to 62.83 degrees. Taking the inverse cosine of our equation with negative root 29 gives us 𝜃 is equal to 152.829 and so on. This rounds to 152.83 degrees to two decimal places.
We were asked to give all solutions that are greater than or equal to zero degrees and less than or equal to 360 degrees. We therefore need to consider the symmetry of the cosine function such that the cos of 𝜃 is equal to the cos of 360 degrees minus 𝜃. Subtracting each of our values from 360 degrees gives us further solutions 297.17 degrees and 207.17 degrees to two decimal places.
We have therefore found four possible solutions to the given equation. However, we were reminded in the question to remove any extraneous solutions, these extra solutions that were created when we squared our original equation. We need to substitute each of our four solutions into the initial equation to check they are valid.
The initial equation was four sin 𝜃 minus four cos 𝜃 equals root three. Substituting 𝜃 equals 62.83 into the left side of our equation gives us an answer of root three. This means that this is a valid solution. However, when we substitute 𝜃 is equal to 152.83 degrees into the left-hand side of our equation, we do not get root three. This means that this is not a valid solution. Repeating this process for 207.17 degrees and 297.17 degrees, we see that 207.17 is a valid solution, whereas the fourth answer of 297.17 is not. We can therefore conclude that there are two solutions that satisfy the equation in the given interval of 𝜃, which are 62.83 and 207.17 degrees.
## Join Nagwa Classes
Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!
• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions<|endoftext|>
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# How do you multiply root3a (root3(a^2) + root3(81a^2)) ?
Oct 2, 2015
I found: $a \left(1 + 3 \sqrt[3]{3}\right)$
#### Explanation:
We can multiply first:
$\sqrt[3]{a} \cdot \sqrt[3]{{a}^{2}} + \sqrt[3]{a} \cdot \sqrt[3]{81 {a}^{2}} =$ and rearrange:
$= \sqrt[3]{a \cdot {a}^{2}} + \sqrt[3]{a \cdot 81 {a}^{2}} =$
$= \sqrt[3]{{a}^{3}} + \sqrt[3]{81 {a}^{3}} =$ taking cube roots:
$= a + a \sqrt[3]{81} = a + a \sqrt[3]{27 \cdot 3} = = a + 3 a \sqrt[3]{3} =$
$= a \left(1 + 3 \sqrt[3]{3}\right)$<|endoftext|>
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was the King of Lagash
, a fertile town nestled between the Tigris
and the Euphrates
. While his domain was prosperous, Eannatum wanted more.
This ambitious king, upon receiving his power, understood that Lagash’s security relied on its water supply from the Shatt al-Gharraf
. Unfortunately his neighbor, the city-state of Umma, also bordered this very important channel on the western bank. The chief cause of hostility between these important cities is unknown according to some historians, and while we can never be certain, it seems obvious to us that the conflict was over water.
held this one strategic advantage over Lagash. Cutting the water supply to the city would hinder its domestic produce and trade via waterway, effectively crippling commerce in Lagash and sending prices upward on all commodities.
The Enmetena Cone, in the Louvre Museum. Photograph courtesy of Trevor Eccles
Knowing all this then, it might not be surprising that conflict between Lagash and Umma was common. We even have primary sources citing the fact. Enmetena, son of Eannatum II and nephew of the famed conqueror Eannatum I, recorded the history of the battles on a large rock known as the “Enmetena Cone
.” The engraving describes the first war between the two powers, fighting for possession for the fertile fields of Guedena, located between the two great city-states.
But who decided the border between Lagash and Umma? King or God… or both?
It was, according to our historical cone, Enlil
, who was considered the king of all the lands and father of all the gods. While Mesalim
, king of Kish
, confirmed the decision by placing a mark and a stele on the borderline.
The actual script reads:
“Enlil, king of all the lands, father of all the gods,
by his righteous command, for Ningirsu and Shara,
demarcated the (border) ground.
Mesalim, king of Kish, by the command of Ishtaran,
laid the measuring line upon it, and on that place he erected a stele.”
This inscription is an entanglement of religion and the state. Enlil was the main Sumerian god. Therefore, he was the judge, jury, and executioner. Enlil was the god who fixed the boundaries and terrestrial estates of the lesser gods. His will could not be changed and his decisions final, regardless of divine assembly.
However, each city-state had a patron god. The god Ningirsu represented the city of Lagash, while Umma worshipped the god Shara.
Lagash made the argument that the borders were already set in place and Enlil was in favor of them retaining control over Guedena
, our attractive fertile field. Umma saw things differently. A mediator, therefore, was needed to settle the dispute. That mediator would be none other than Mesalim, king of Kish, our second name on the historical cone.
The title “King of Kish” actually means “King of the world or King of Kings.” Mesalim was the supreme overseer of the Sumerian lands, which was the civilized world to these people. Mesalim’s decision was final… regardless of the moral argument.
What did this super King conclude? His ultimate directive was to build a trench, along with a levee, on either side to separate the two territories. Finally, the stele was erected at the border indicating his decision. Mesalim’s ruling, however, favored Lagash more so than Umma when it came to the water rights and the fertile fields of the Guedena.
The reason for this decision is unfortunately unknown. Of course we can never be sure, but could it be possible that Lagash was more powerful than Umma?
According to Mesalim, Enlil, the father of the gods, favored the stronger of the two city-states. However, all deities aside, Mesalim likely chose Lagash because it had a much stronger economy and military and could provide more to the loosely knit confederation of the Sumerian city-states in a time of crisis than Umma.
Therefore, in essence, the King of Kish picked the winners and losers of Sumer.
a Sumerian battle scene by HongNian Zhang
This was not the end of the border dispute between the two city-states. Later, Ush, ruler of Umma, marched to the border, smashed Mesalim’s stele, and advanced into Lagash territory. Ush proceeded with his forces to seize the fertile fields of Guedena.
Ush was later defeated from any further advance by an unknown Lagash king.
The Sumerian inscriptions state that, “Ningirsu, the hero of Enlil, by his just command, made war upon Umma. At the command of Enlil, his great net ensnared them. He erected their burial mound on the plain in that place.”
Rather than to the unknown king, the victory was granted to the patron god of the city of Lagash.
The reason there is no mention of the Lagash king is that Enmetena, the great-grandson of Ur-Nanshe
, wrote the story. Ur-Nashe was the founder of the dynasty from which Enmetena came. The man who defeated Ush has to be none other than Lugal-sha-engur
, the predecessor of King Ur-Nanshe.
So why would Enmetena not mention Lugal-sha-engur’s victory over Ush? Simple… Enmetena was not interested in giving thanks or glory to a dynasty that was not his own.
Check back Next week to see how Eannatum takes over the Sumerian valley and creates the first empire!
Read Part One: Lagash and the Too Fertile Valley Here: https://classicalwisdom.com/lagash/
“A War for Water – The tale of two City-States” was written by Cam Rea<|endoftext|>
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# How to Calculate IRR (Internal Rate of Return) in Excel (8 Ways)
## What Is Net Present Value?
If you have \$1000 today, you can invest it in a business or buy a product today and sell it later, or at least you can deposit it with a bank that gives you interest for your deposits.
Say you have a trustable bank nearby, and they give 12% interest on deposits. You go to the bank and deposit \$1000 with the bank at a 12% interest rate per year for the next 3 years.
Assume that you’re not withdrawing any money from the bank.
After 1 year, you will get:
`\$1000+ \$1000 x 12% = \$1000 (1+12%) = \$1120`
So, \$1120 after 1 year is the same as \$1000 today.
After 2 years, you will get:
`\$1000 (1+12%) + \$1000(1+12%)(12%) = \$1000(1+12%)(1+12%) = \$1000(1+12%)^2 = \$1254.40`
So, \$1254.40 after 2 years is the same as \$1000 today.
In the same way, we can conclude that after 3 years, we shall get:
`\$1000(1+12%)^3 = \$1404.93`
So, \$1404.93 after 3 years is the same as \$1000 today.
We can conclude an equation from the above discussion:
Future value of the money,
`FV = PV((1+r)^n)`
Here,
• PV = Present value
• r = Interest rate/year
• n = Number of years
Reversely, we can calculate the present value of the money with this equation:
`PV = FV/((1+r)^n)`
## What Is the Internal Rate of Return (IRR)?
IRR is the interest rate that balances your initial investment and future cash flows.
Let’s start with an investment proposal you got:
Sam is your friend who comes to you with an investment proposal. You are proposing to invest \$1000, and these are the cash flows you can expect from the investment over the next 5 years.
There is a bank nearby your place, and you trust this bank. This bank provides 12% yearly interest on a deposit. Or you can invest your money in a government savings bond with a 10% interest rate/per year. A government bond is more secure than depositing with a bank. So, you have three options to invest your \$1000:
• You can invest in your friend Sam’s project
• You can deposit the amount in your preferred bank that provides a 12% interest rate per year
• Or you can buy a government bond and enjoy a more secure income
What will you do? To make a better decision, you must first know the internal rate of return on your investment with Sam’s project.
Let’s calculate the IRR of these cash flows. There are several cash flow scenarios (regular, discrete, monthly, etc.), so your approach will also be different when you calculate IRR.
### Method 1 – Using the IRR Function to Calculate IRR
Steps:
• Select cell C12 and insert the following formula:
`=IRR(C5:C10)`
As the values argument, we have passed this cell range, C5:C10. The values must contain at least one negative value and one positive value. Otherwise, the IRR function will provide an #NUM! error value.
• Press ENTER to get the value of IRR.
### Method 2 – Applying VBA to Calculate IRR in Excel
Steps:
• Go to the Developer tab >> click on Visual Basic.
• The Microsoft Visual Basic for Application box will open.
• Click on Insert >> Select Module.
• Enter the following code in your Module:
``````Sub Calculate_IRR()
Static CashFlow(6) As Double
CashFlow(0) = -1000
CashFlow(1) = 200
CashFlow(2) = 500
CashFlow(3) = 200
CashFlow(4) = 150
CashFlow(5) = 500
Cells(12, 3) = IRR(CashFlow())
End Sub``````
Code Breakdown
• Firstly, I created a subprocedure called Calculate_IRR().
• Then, I declared a Static array and named it CashFlow, where the size and type are Double.
• After that, I inserted 6 values of CashFlow as -1000, 200, 500, 200, 150, and 500, as given in our dataset.
• Finally, I used the VBA IRR function, inserting these CashFlow array values into the function and assigning this value to Cell (12,3).
• Click on the Save button and go back to your worksheet.
• Go to the Developer tab >> click on Macros.
Now, the Macros box will appear.
• Select Calculate_IRR.
• Click on Run.
• You will get your IRR value in decimal.
• To change the number format, go to the Home tab >> click on Number Format >> select Percentage.
• You will get the value of IRR in percentage using VBA.
### Method 3 – Manually Calculating IRR from Net Present Value in Excel
Steps:
• Insert any value of your choice in Cell C13 as IRR. Here, I will insert 10%.
• Select cell D6 and insert the following formula:
`=C6/(1+\$C\$13)^B6`
• Press ENTER and drag down the Fill Handle tool to AutoFill the formula for the rest of the cells.
• You will get all the values of the Present Value.
• Select cell D11 and insert the following formula:
`=SUM(D6:D10)`
Here, in the SUM Function, we added all the Present Values in Cell range D6:D10.
• Press ENTER to get the total present value.
• In cell C13, you have to change the value manually to get a nearby value that makes the total present values equal to 1000.
• We change the value to 12%. Now, you see the total present value is \$1098.57.
• Change the value to, say, 16%. The value in cell C8 is \$993.03.
This is the manual way. It is not tough, but getting a very close value from this method will take time. You can also try this method on paper, but it might take a whole day to find the internal rate of return for some future cash flows.
### Method 4 – Applying the Goal Seek Feature to Calculate IRR in Excel
Steps:
• Go through the steps shown in Method 3Â to prepare the dataset to use the Goal Seek feature.
• Go to the Data tab >> click on Forecast >> click on What-If Analysis >> select Goal Seek.
• The Goal Seek box will appear.
• Insert in cell D11 (sum of all future cash flows) as Set cell, 1000 (equal to initial investment) as To value, and Cell C13 (Internal Rate of Return) as By changing cell.
• Press OK.
• The Excel Goal Seek feature does the iterations and comes up with a value that meets all the criteria. Here, you will get 15.715% as an IRR value that meets all the criteria and get the same internal rate of return as we got using Excel’s IRR function.
### Method 5 – Using the XIRR Function to Calculate IRR for Uneven Cash Flow
Steps:
• Select cell C19 and insert the following formula:
`=XIRR(C5:C17,B5:B17)`
In the XIRR function, I inserted Cell range C5:C17 as values and Cell range B5:B17 as dates.
• Press ENTER to get the value of IRR.
### Method 6 – Calculating IRR for Monthly Cash Flow in Excel
Steps:
• Select cell D19 and insert the following formula:
`=IRR(D5:D17)*12`
In the IRR function, I inserted Cell range D5:D17 as values and multiplied this by 12.
• Press ENTER to get the value of IRR using the IRR function.
• Select cell D20 and insert the following formula:
`=XIRR(D5:D17,C5:C17)`
In the XIRR function, I inserted Cell range D5:D17 as values and Cell range C5:C17 as dates.
• Press ENTER to get the value of IRR using the XIRR function.
You see, there is a difference between the values.
Note: If you use the IRR function to calculate the internal rate of return for monthly cash flows, you need to multiply the IRR value by 12, as IRR calculates the monthly rate of return, not yearly.
### Method 7 – Using the MIRR Function to Calculate Modified IRR in Excel
Steps:
• Select cell C15 and insert the following formula:
`=MIRR(C5:C10,C12,C13)`
In the MIRR function, I inserted values for cell range C5:C10, cell C12 as finance_rate, and cell C13 as reinvest_rate.
• Press ENTER to get the value of IRR.
### Method 8 – Calculating IRR for Real Estate in Excel
Steps:
• Select cell C8 and insert the following formula:
`=SUM(C5:C7)`
• Press ENTER and drag right the Fill Handle tool to AutoFill the formula for the rest of the cells.
In the SUM function, I have added all the values of the cell range C5:C7.
• Select cell C10 and insert the following formula:
`=IRR(C8:H8)`
In the IRR function, I inserted cell range C8:H8 as values.
• Press ENTER to get the value of IRR for this real estate company.
## What Is Good IRR?
It is a very tough question to answer. Usually, the IRR value is compared with a fixed value, such as a bank interest rate or a government bond rate, as you can earn these incomes idly without doing anything and risking anything. IRR has also compared the project to the project basis.
The following image shows data from two projects and their expected revenue over the next five years.
After calculating the IRR of these projects, it seems that Project B would have more potential for the company.
## Practice Section
In this section, we provide you with the dataset to practice these methods on your own.
<< Go Back to Excel Functions | Learn Excel
Get FREE Advanced Excel Exercises with Solutions!
Kawser Ahmed
Kawser Ahmed is a Microsoft Excel Expert, Udemy Course Instructor, Data Analyst, Finance professional, and Chief Editor of ExcelDemy. He is the founder and CEO of SOFTEKO (a software and content marketing company). He has a B.Sc in Electrical and Electronics Engineering. As a Udemy instructor, he offers 8 acclaimed Excel courses, one selected for Udemy Business. A devoted MS Excel enthusiast, Kawser has contributed over 200 articles and reviewed thousands more. His expertise extends to Data Analysis,... Read Full Bio
1. Kawser
Very understandable and effective article regarding Internal Rate of Return ( IRR). I have one point to complain on comparison IRR and XIRR. It seems that as IRR is a monthly rate and to compute it annually we must use a compound form. as (IRR ^12) and not multiplied. Doing so we have IRR = 0,542% monthly – and ( 1,00542^12) we get 6,70 % – quite the same om XIRR ( 6.72%).
Looking for a prompt reply.
Regards
• Hi Colvis.
We are very sorry for not reaching out to you promptly. But the annual IRR is calculated by multiplying the monthly IRR by 12. In that case, we get an annual IRR value of 6.504%. The value of IRR from the IRR formula and the XIRR formula should not be the same for irregular cash flows. And these values reflect that.
2. Hi – I am always confused with IRR as I say to myself (ok, it is used to give us an indication that Present Investment = Future Cash Flows) but how do we actually apply this in the “real world”. Can you please give me further insight with regard to your last example comparing Project A to Project B. Finally, is IRR of A (3%) and B (5%) of any relevance to the comparison of the “profitability” of these projects? In order to do so, wouldn’t we need to introduce WACC?
Thank you in advance for coming back to me on this.
Regards,
Romesh
• Hi Romesh.
IRR is not the sole measure of comparing profitability. It is just one of the many parameters you can use to compare two paths of investment.
3. This is a wonderful articles . Great many thanks
Advanced Excel Exercises with Solutions PDF<|endoftext|>
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# What are the 3 laws of motion according to Newton?
## What are the 3 laws of motion according to Newton?
The laws are: (1) Every object moves in a straight line unless acted upon by a force. (2) The acceleration of an object is directly proportional to the net force exerted and inversely proportional to the object’s mass. (3) For every action, there is an equal and opposite reaction.
## What is Newton’s law of motion in physics?
What are Newton’s Laws of Motion? An object at rest remains at rest, and an object in motion remains in motion at constant speed and in a straight line unless acted on by an unbalanced force. The acceleration of an object depends on the mass of the object and the amount of force applied.
What does Newton’s 2nd law say?
It states that the time rate of change of the momentum of a body is equal in both magnitude and direction to the force imposed on it. The momentum of a body is equal to the product of its mass and its velocity.
What is Newton’s third law formula?
This is an example of Newton’s third law. The rope is the medium that transmits forces of equal magnitude between the two objects but that act in opposite directions. T = W = m g.
### What is Newton’s law of Class 10?
In the first law, we understand that an object will not change its motion unless a force acts on it. The second law states that the force on an object is equal to its mass times its acceleration. And finally, the third law states that there is an equal and opposite reaction for every action.
### What is Newton’s second law easy definition?
Newton’s Second Law of Motion says that acceleration (gaining speed) happens when a force acts on a mass (object). Riding your bicycle is a good example of this law of motion at work. Your bicycle is the mass. Your leg muscles pushing pushing on the pedals of your bicycle is the force.
What is the 2nd law of motion called?
According to Newton s Second Law of Motion, also known as the Law of Force and Acceleration, a force upon an object causes it to accelerate according to the formula net force = mass x acceleration.
What is the 1st law of motion called?
The property of a body to remain at rest or to remain in motion with constant velocity is called inertia. Newton’s first law is often called the law of inertia.
## What are 5 examples of law of acceleration?
Examples
• An object was moving north at 10 meters per second.
• An apple is falling down.
• Jane is walking east at 3 kilometers per hour.
• Tom was walking east at 3 kilometers per hour.
• Sally was walking east at 3 kilometers per hour.
• Acceleration due to gravity.
## What is Vt graph?
A velocity-time graph shows the speed and direction an object travels over a specific period of time. Velocity-time graphs are also called speed-time graphs. The vertical axis of a velocity-time graph is the velocity of the object. The horizontal axis is the time from the start.<|endoftext|>
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Pencil & Paper Square Roots the Easy Way
An Inside-Out Approach
This method for extracting roots is unorthodox, in that it produces results in fractional form. If a decimal value is desired, then a subsequent long division would have to be performed.
The special properties of this continued fraction (CF) lend themselves to manipulation by a calculator — or by hand:
Example: for sqrt(27), an expansion of CF looks like this:
Since the original estimate never is changed, and the iterations always use the same values, this construct lends itself to an "inside-out" approach to the calculation; that is, choosing an arbitrary starting point as the "bottom" of the sequence, and working up. A sample procedure is fairly simple:
2. Add 10, then divide into 2.
3. repeat step #2 until desired decimal accuracy is achieved.
It sounds like a kitchen recipe, doesn't it? In any case, the first step is trivial; it is simply 2/10. Here is the second iteration:
Similarly, loop #3:
For ease of calculation, the idea is to keep the numbers as small as possible, by simplifying each fraction to two integers and by reducing to lowest terms when possible. If CF itself is in lowest terms, then no factoring will be available; otherwise, some quotients can be reduced to lower terms. In the case of 2 and 10, where both are divisible by 2, each quotient can be reduced by a factor of 2.
Warning: the original continued fraction itself must never be changed. For example, reducing a CF of 2/10 to 1/5 would effectively represent the square root of !
Here is a chart of iterations of sqrt(27):
Loops Fraction Decimal value 1 2/10 0.2 2 10/51 0.1960784 3 51/260 0.1961538 4 520/2651 0.1961524 Actual value, sqrt(27) 5.1961524
In this case, just three iterations were sufficient to achieve 5-decimal accuracy.
Now let's try a rather less friendly example — sqrt(47), using an estimate of 6:
The numbers have become uncomfortably large in a hurry. At this stage, you somehow know that the prior total isn't going to be very accurate, and you are right. You probably are asking yourself why you don't just borrow someone's calculator and be done with it. The problem here is that a CF of 11/12 is too large; it converges slowly, and the size of the numerator escalates the size of all subsequent numbers.
Is there an alternative? Yes! It is not necessary that the estimate be less than the actual square root; but ideally it is as close as possible47 is much closer to 49 than to 36, so let's revise our setup for sqrt(47), using an estimate of 7. The CF numerator will be negative; but that's a minor detail:
Adding the last result to 7 leaves 6.8556548, which is sqrt(47) accurate to 6 decimal places! Had we stopped after just two iterations, we still would have achieved 4-decimal accuracy with minimal calculation.
Manipulating fractions in this fashion is straightforward enough, but there is a more concise method that, with a little practice, makes the whole procedure a snap!
Plugging the numbers into this equation simplifies the process by obviating the fractions within fractions. Calculating three iterations should be fairly easy irrespective of the parameters. Let's try sqrt(21) with an estimate of 4:
That's good for 3-decimal accuracy, despite a large CF: 4.5828 compared to the actual 4.5826
Note: the smaller the continued fraction, the more quickly it converges upon the root, and the easier will be any manual calculations. For maximum accuracy with the minimum of effort, it is incumbent upon you to select the closest original estimate.
Ideally, e would be the greatest integer less than the root; then the second term of the equation would calculate only the fractional, or decimal portion of the root. The working fraction also will be as small as possible. Any estimate will work, however.
So how many iterations should be performed? It depends upon the desired digital accuracy. If the CF is relatively small, two iterations will yield sufficient accuracy for most real-world applications, and three iterations should satisfy anyone. If CF is larger (over ½), then three or more iterations might be required for meaningful results.<|endoftext|>
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What is Friction?
Friction is the action of one surface or object rubbing against another. There are two primary types of friction: dynamic friction and static friction. Dynamic Friction aka Kinetic Friction is the friction between surfaces or objects which move relative to one another. Static friction is the friction between stationary surfaces or objects.
Coefficient of Friction (C.O.F.) is defined as follows: A value that illustrates the relationship between the force of friction amid two objects and the normal friction among the objects involved.
The higher the number, the lower the probability that a slip occurs.
Per NFSI (B101.1): C.O.F. > .60 is considered "High Traction".
All of our products fall into this tier of High Traction.
ANSI (American National Standard Institute) was founded in 1918 with the mission "to enhance both the global competitiveness of U.S. business and the U.S. quality of life by promoting and facilitating voluntary consensus standards and conformity assessment systems, and safeguarding their integrity." ANSI oversees the creation, promulgation and use of thousands of norms and guidelines that directly impact businesses in nearly every sector: from acoustical devices to construction equipment, from dairy and livestock production to energy distribution, and many more.
The NFSI (National Floor Safety Institute) is a 501(c) (3) not-for-profit organization. NFSI was founded in 1997 with the intention of preventing slip and fall accidents by educating the public and businesses through informative programming, research, training and product certification. NFSI certifies flooring materials, coatings, chemical floor-cleaning products, and treatments.
ANSI and NFSI have devised various detailed standards for measuring the Coefficient of Friction (COF) of a surface.
B101.1 ANSI/NFSI Standard: Test Method for Measuring Wet SCOF of Common Hard-Surface Floor Materials.
The American with Disabilities Act (ADA) was enacted in 1990 to guarantee equal opportunities for individuals with disabilities. The ADA made it illegal to discriminate on the basis of disability in employment, transportation, telecommunications, state and local government and public accommodations. In 2003 the ADA advisory on surface conditions issued "Bulletin 4" which recommended a static coefficient of friction (SCOF) value of 0.6 for level surfaces and .8 for ramps and inclined surfaces.<|endoftext|>
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In the past year, native Californians have adjusted to the fact that wildfires are just a part of California. Although many Californians have gotten used to hearing about a new wildfire on their televisions, many do not understand how wildfires start and how vast the consequences can be.
For starters, when many people hear the word “wildfire”, they automatically assume that the causes are natural simply because the word “wild” is a part of the phrase. However, as much as 90% of wildfires are started by humans (4). This can be anywhere from unattended campfires, cigarettes, and even arson. The other 10% are natural causes like lighting or excessive heat from the sun.
In order for wildfires to survive for long periods of time, they require what scientist refer to as the “fire triangle.” The fire triangle consists of heat, fuel, and oxygen (2). Especially in states like California, where the fuel is plentiful because of drought and there is lots of wind in the summer, wildfires are bound to start. The air we breath also contains about 21% oxygen and fire requires about 16% to continue burning, therefore fire burns very easy.
Once a fire has ignited, the three major factors that determine its longevity: weather, fuel, and topography (1). The fuel load, or amount of fuel available to burn, will affect how the fire burns. When there is a large fuel load, the fire will burn quickly and large, whereas a smaller fuel load will be a slower, smaller burn. The weather can also affect the fire’s spread. If it is windy, the wind can carry embers and cause the fire to spread. Air moisture can also help to control fires, the wetter the air, the better it is for eliminating fires. Topography is also a huge factor in the lifespan of a fire. Unlike humans, it is easier for fires to travel uphill than downhill.
Once wildfires have run their course and have been subdued by our hardworking firefighters, the wake of destruction they leave behind impacts everyone and everything. Not only are entire ecosystems destroyed and many wild animals are left unprotected, the aftermath of wildfires has huge effects on human health. The largest threat of wildfires to human health is the smoke from the wildfires (3). Although there are rarely long-term effects of smoke inhalation in healthy individuals, short-term effects include trouble breathing, rapid heartbeat, and symptoms similar to a sinus infection like fatigue and headaches. Individuals with asthma or related diseases should consult their doctors when air quality is not ideal.
Hopefully, with this information, you can be more informed the next time you hear the word “wildfire” come out from your tv or radio.
Bonsor, Kevin. “How Wildfires Work.” HowStuffWorks Science, HowStuffWorks, 8 Mar. 2018, science.howstuffworks.com/nature/natural-disasters/wildfire4.htm.
“Elements of Fire.” Smokey Bear, smokeybear.com/en/about-wildland-fire/fire-science/elements-of-fire.
Strickland, Ashley. “Do You Need to Worry about Wildfire Smoke?” CNN, Cable News Network, 30 July 2018, www.cnn.com/2016/11/15/health/wildfire-smoke-air-quality-health/index.html.
“Wildfire Causes and Evaluations (U.S. National Park Service).” National Parks Service, U.S. Department of the Interior, www.nps.gov/articles/wildfire-causes-and-evaluations.htm.<|endoftext|>
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# Pair of Linear Equation in Two Variables
An equation in the form of px + qy + r, where p, q and r are real numbers and the variables p and q are not equivalent to zero, is known as a linear equation in two variables.
The two linear equations with similar two variables are considered as the pair of linear equations in two variables. There are two methods to solve a pair of linear equations in two variables. These two methods are:
1. Graphical method
2. Algebraic Method
The general form of pair of linear equation in two variables is given as:
p1x + q1y+ r1 = 0 p2x + q2y + r2 = 0
Here, p1, q1, r1, p2, q2, r2 are real numbers
P12 + q12 ≠ 0
p22 + q22 ≠ 0
### Pair of Linear Equation in Two variables Introduction
We have studied linear equations in one variable in our previous classes and we know how to solve the linear equation in one variable. Solving the linear equation in one variable is simple and can be calculated easily. In order to solve the linear equations in two variables, two different sets are needed which will be helpful to determine the values of two unknown variables say x and y. If one equation is given and two variables are asked to be solved, we will not be able to get a particular solution for a given equation.
For example
2x + 5y = 6 and 5x + y = 2
We can get a particular solution for the above two equations. But , on the other side
We cannot get a particular solution for the equation 3x + 6y = 5 as only one equation is given and we are asked to find the values of two unknown variables. Hence, we can express the equation 3x + 6y = 5 as, y = (5-3x)/6
The values of y will change accordingly on the basis of the values of variable x, Hence, the particular solution for a given equation is not possible.
It can be said that two distinct sets of independent conditions are required to obtain a particular solution of systems of linear equations in two variables
### Expression of Pair of Linear Equation in Two Variables
There are generally two methods to solve and express the pair of linear equations in two variables. These two methods are:
• Graphical Method
• Algebraic Method
The general representation of pair of linear equation for two variables say x and y is given by:
p₁x + q₁y + r₁ = 0………….(1)
p₂x + q₂y + r₂ = 0 …………(2)
Where , p₁ q₁, r₁, p₂, q₂, r₂ are real numbers and p₁² + q₁² ≠ 0 and p₂²+ q₂² ≠ 0.
If the pair of linear equations are given in the form of p₁x + q₁y + r₁ = 0 and p₂x + q₂y + r₂ = 0, then the following situation takes place.
1. If the given pair of linear equations is consistent then p₁/p₂ ≠ q₁/q₂
2. If the given pair of linear equations is inconsistent then p₁/p₂ = q₁/q₂ ≠ r₁/r₂
3. If the given pair of linear equations is dependent and consistent then p₁/p₂ = q₁/q₂ = r₁/r₂.
For example, the two equations such as 2x- y = -1 and 3x + 2y = 9 are pairs of linear equations with two variables x and y. The diagram given below represents the solutions for both the equations by substituting the value of x to determine the value of y.
Similarly, we can find the solutions for other pairs of linear equations in two variables.
### Graphical Expression of Pair of Linear Equation in Two Variables
As we know, the graphical representation of linear equations in two variables always forms a straight line. Therefore, a pair of linear equations in two variables forms a straight line which are observed together. There are three possibilities when two lines are drawn in a plane. These three possibilities are :
• Two lines intersect with each other at one point.
• Two lines which are drawn are parallel to each other.
• The two lines will coincide with each other.
Now, let us take a pair of linear equations such as x + y + 5 and 2x + 2y = 10. To represent these pair of linear quotations in a graph, we need to determine the solutions.
We will get two different values of x and y after determining the solutions. Hence it can expressed as below:
### Algebraic Method of Solving a Pair of Linear Equations of Two Variables.
Pair of linear equations are found in every situation. Let us learn the algebraic method of solving a pair of linear equations of two variables by considering practical application of daily life. For example, suppose you went to the fruits market. There were two types of oranges available. The fruit seller said that the small oranges are 3 times smaller than the large oranges. The total money the lady was having with her is Rs.100. Can you determine how much the lady spent to purchase two types of oranges?
Let us understand the above situation mathematically,
Let us consider the price of the smaller oranges = x
And ,the price of the bigger oranges = y.
As per the first situation in the questions x = 4y…………(1)
And as per the second situation in the questions, x + y = 100……………(2)
For determining the solution, we are required to find the value of both x and y and solve both the equations. The coordinates (x,y) can be represented easily in a graph. But the graphical method is the most easier way in the situations when the point representing the solution of the linear equations includes non-integral coordinates like ( √2, 5√6 ) , (–1.42, 5.3), (6/19, 1/17), etc. Hence, the algebraic methods are used to solve such situations.
The two algebraic methods to solve a pair of linear equations:
1. Substitution Method
2. Elimination Method
3. Cross-Multiplication Method
### Solved Examples
1. The sum of 4 times a larger integer and 5 times a smaller integer is 7. When twice the smaller integer is subtracted from 3 times the larger, the result is 11.Find the integers.
Solution:
We will first assign the variables to the larger and smaller integer.
Let us consider x as the smaller integer and y as the larger integer.
When Using two variables, we are required to form two equations. The first sentence explains addition and the second statement explains the difference.
This gives the following situation
This gives us the following two equations.
4x + 5y = 7
3x- 2y = 11
We will solve the above equation through the substitution method.
In the substitution method, we will multiply the equation 4x + 5y = 7 by -2 and the equation 3x – 2y = 11 by 5
Hence, two equations which we got after multiplication are given below:
-8x -10y = -14……(1)
15x – 10y = 55…..(2)
Now, we will subtract the equation (2) from the equation (1)
(15x – 10y = 55) – ( -8x -10y = -14)
23x = 69
x = 69/23
x = 3
Now, we will substitute the value of x to find the value of y
4x + 5y = 7
4(3) + 5y = 7
12 + 5y = 7
5y = -5
y = -1
Hence, the larger integer is 3 and the smaller integer is -1
2. Calculate the value of variables which meets the following equation
2x + 5y = 20 and 3x+6y =12.
Solution:
We will solve the pair of linear equations by substitution method .
Two equations are given below:
2x + 5y = 20…………………….(1)
3x + 6y =12……………………..(ii)
Now, we will multiply the equation (i) by 3 and equation (ii) by 2,
After multiplication, we get following two equations
6x + 15y = 60…………………….(iii)
6x+12y = 24……………………..(iv)
Now , we will Subtract the equation (iv) from (iii)
(6x+12y = 24) – (6x + 15y = 60) = 0
-3y = – 36
y = 12
By substituting the value of y in any of the equation (i) or (ii), we get the value of x
2x + 5(12) = 20
x = −20
Hence x=-20 and y =12 are two point where the given equations intersect
1. What are the different steps to write linear equations in two variables?
Ans. The different steps to write linear equations in two variable are as follows:
1. The first step is to define the variables.
2. The second step is to write any two equations that include any two variables.
3. The third step is to state the equation in slope intercept form i.e y = ax + b
4. The fourth step is to graph the equation with the help of the y-intercept and slope.
5. The fourth step is to recognise the variables by observing the points where the intercepts meet.
6. The sixth step is to mention what is meant by points with reference to the problems.
2. What is the difference between Linear equations and Nonlinear equations?
Ans. The difference between linear equation and non-linear equation is stated below in tabulated from:
Linear Equation Non Linear Equation Linear equation forms a straight line or represent the equation for straight line Nonlinear equation does not form straight line but forms curve Linear equation has only degree 1 A nonlinear equation has degree 2 or more than degree 2 but not less than degree 2. Linear equations forms a straight line in XY-plane and can be lengthen in any direction but in straight form Nonlinear equation forms curve and if the curve is been increased, the curvature of the graph increases Linear equation is expressed in the form of y = ax + b Here, x and y represent the variables, a is the slope of line and c is a constant value. Nonlinear equation is expressed in the form of ax² + by + c Here, x and y represent the variables, a and c is the slope of the line and constant value respectively. Example 9x =1 3x + y + 2 =0 3y = 2x Examples x² + y² = 2 3x² + x + 2= 5<|endoftext|>
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- Take notes in a spiral notebook, using a separate notebook for each course or in a binder where pages can be inserted and rearranged. Write your name, address, and phone number in each notebook. Write your lecture notes leaving a wide margin on the left for writing probable test questions, key words, or additional notes.
- Label, number, and date all notes. Develop the habit of labeling and dating to integrate reading and lecture notes.
- Use blank space. Notes tightly crammed into every corner of the page are hard to read and difficult to use for review. Give your eyes a break by leaving plenty of space.
- Write down the major ideas and statements in the lecture. Don't try to write down every word; instead, use key phrases and ideas. Underline ideas that your instructor emphasizes.
- Use a "lost" signal. No matter how attentive and alert you are, you might get lost and confused in a lecture. If it is inappropriate to ask a question, record in your notes that you were lost.
- Use standard abbreviations. Be consistent with your abbreviations. If you devise your own abbreviations or symbols, write a key explaining them in your notes.
- After the lecture, review notes as soon as possible and fill in missing ideas, key words, and phrases. Underline headings that are of major significance. You may also wish to compare your notes with a friend's to see what you may have missed. The sooner and more frequently you review notes after lecture, the more you retain.
- After each lecture, take several minutes to turn your notes into questions, focusing on the main theme and sub topics. Each lecture will usually supply three to seven good exam questions. The questions should be written in the left hand margin.
- At least once a week, review the questions written in your notebook. Pretend you are taking a test, give yourself an oral quiz, or even better, practice by taking a written quiz. Compare your answers with those given in your notes or textbook.<|endoftext|>
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# lec0116 - Answer to the Lecture Question from Last Time...
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Answer to the Lecture Question from Last Time Total number of ways to choose 7 questions: 10 C 7 Total number of ways to choose 7 questions by choosing less than 4 of the first 6: 6 C 3 . Now, the second answer does not seem obvious. It should not be. If you are answering 7 of 10 questions, then you can leave at MOST 3 of them blank. Thus, if MUST choose at LEAST 3 questions of the first 6, no matter what. So, the only way to choose 7 out of 10 questions, WITHOUT choosing at least 4 of these first 6 questions is to choose exactly 3 of them. The answer to our question is simply the difference of these two values: 10 C 7 - 6 C 3 = 100. Another way to view this question is the following: We can choose our questions in the following 3 ways: 1) 4 of the first 6, followed by 3 of the last 4 2) 5 of the first 6, followed by 2 of the last 4 3) 6 of the first 6, followed by 1 of the last 4. We can do #1 in ( 6 C 4 ) ( 4 C 3 ) = 60 We can do #2 in ( 6 C 5 ) ( 4 C 2 ) = 36 We can do #3 in ( 6 C 6 ) ( 4 C 1 ) = 4 We used the multiplication principle for each of these parts and will use the sum rule to deduce that there are a total of 100 possible test question choices.
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A common mistake made on this problem is the following: 1) Choose 4 of the first 6 questions. 2) Then, you have 3 choices to make of the remaining 6 questions. It seems as if we can do this in ( 6 C 4 ) ( 6 C 3 ) = 300 ways. But, as we can see, this answer is incorrect, by quite a bit. We are actually counting some of the combinations of questions more than once, that is why this answer is incorrect. Consider the combination 1, 2, 3, 4, 5, 7, and 8. We can arrive at this combination in a couple ways: 1) Pick 1,2,3,4 of the first 6, then 5,7,8 of the remaining Qs. 2) Pick 1,2,3,5 of the first 6, then 4,7,8 of the remaining Qs. We count these two possibilities separately in the count above. But, we shouldn’t. Thus, we are over counting and end up with an incorrect answer. Combinations with Repetition
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# AP Statistics Curriculum 2007 Estim Var
(Difference between revisions)
Revision as of 17:06, 2 June 2010 (view source)IvoDinov (Talk | contribs)m (→More Examples: minor)← Older edit Revision as of 23:14, 8 June 2010 (view source)IvoDinov (Talk | contribs) m (→Interval Estimates of Population Variance and Standard Deviation: typo)Newer edit → Line 20: Line 20: Notice that the Chi-Square Distribution is '''not''' symmetric (positively skewed) and therefore, there are two critical values for each level of confidence. The value $\chi_L^2$ represents the left-tail critical value and $\chi_R^2$ represents the right-tail critical value. For various degrees of freedom and areas, you can compute all critical values either using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions] or using the [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR Chi-square Distribution Calculator]. Notice that the Chi-Square Distribution is '''not''' symmetric (positively skewed) and therefore, there are two critical values for each level of confidence. The value $\chi_L^2$ represents the left-tail critical value and $\chi_R^2$ represents the right-tail critical value. For various degrees of freedom and areas, you can compute all critical values either using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions] or using the [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR Chi-square Distribution Calculator]. - * Example: Find the critical values, $\chi_L^2$ and $\chi_R^2$, for a 95% confidence interval when the sample size is 25. Use the following Protocol: + * Example: Find the critical values, $\chi_L^2$ and $\chi_R^2$, for a 90% confidence interval when the sample size is 25. Use the following Protocol: - ** Identify the degrees of freedom ($df=n-1=24$) and the level of confidence (${\alpha\over 2}=0.025$). + ** Identify the degrees of freedom ($df=n-1=24$) and the level of confidence (${\alpha\over 2}=0.05$). ** Find the left and right critical values, $\chi_L^2=13.848$ and $\chi_R^2=36.415$, as in the image below. ** Find the left and right critical values, $\chi_L^2=13.848$ and $\chi_R^2=36.415$, as in the image below.
[[Image:SOCR_EBook_Dinov_Estim_Var_020408_Fig2.jpg|500px]]
[[Image:SOCR_EBook_Dinov_Estim_Var_020408_Fig2.jpg|500px]]
## General Advance-Placement (AP) Statistics Curriculum - Estimating Population Variance
In manufacturing, and many other fields, controlling the amount of variance in producing machinery parts is very important. It is important that the parts vary little or not at all.
### Point Estimates of Population Variance and Standard Deviation
The most unbiased point estimate for the population variance σ2 is the Sample-Variance (s2) and the point estimate for the population standard deviation σ is the Sample Standard Deviation (s).
We use a Chi-Square Distribution to construct confidence intervals for the variance and standard distribution. If the process or phenomenon we study generates a Normal random variable, then computing the following random variable (for a sample of size n > 1) has a Chi-Square Distribution
$\chi_o^2 = {(n-1)s^2 \over \sigma^2}$
### Chi-Square Distribution Properties
• All chi-squares values $\chi_o^2 \geq 0$.
• The chi-square distribution is a family of curves, each determined by the degrees of freedom (n-1). See the interactive Chi-Square distribution.
• To form a confidence interval for the variance (σ2), use the χ2(df = n − 1) distribution with degrees of freedom equal to one less than the sample size.
• The area under each curve of the Chi-Square Distribution equals one.
• All Chi-Square Distributions are positively skewed.
### Interval Estimates of Population Variance and Standard Deviation
Notice that the Chi-Square Distribution is not symmetric (positively skewed) and therefore, there are two critical values for each level of confidence. The value $\chi_L^2$ represents the left-tail critical value and $\chi_R^2$ represents the right-tail critical value. For various degrees of freedom and areas, you can compute all critical values either using the SOCR Distributions or using the SOCR Chi-square Distribution Calculator.
• Example: Find the critical values, $\chi_L^2$ and $\chi_R^2$, for a 90% confidence interval when the sample size is 25. Use the following Protocol:
• Identify the degrees of freedom (df = n − 1 = 24) and the level of confidence (${\alpha\over 2}=0.05$).
• Find the left and right critical values, $\chi_L^2=13.848$ and $\chi_R^2=36.415$, as in the image below.
#### Confidence Interval for σ2
${(n-1)s^2 \over \chi_R^2} \leq \sigma^2 \leq {(n-1)s^2 \over \chi_L^2}$
#### Confidence Interval for σ
$\sqrt{(n-1)s^2 \over \chi_R^2} \leq \sigma \leq \sqrt{(n-1)s^2 \over \chi_L^2}$
### Hands-on Activities
• Construct the confidence intervals for σ2 and σ assuming the observations below represent a random sample from the liquid content (in fluid ounces) of 16 beverage cans and can be considered as Normally distributed. Use a 90% level of confidence.
14.816 14.863 14.814 14.998 14.965 14.824 14.884 14.838 14.916 15.021 14.874 14.856 14.86 14.772 14.98 14.919
• Get the sample statistics from SOCR Charts (e.g., Index Plot); Sample-Mean=14.8875; Sample-SD=0.072700298, Sample-Var=0.005285333.
• Identify the degrees of freedom (df = n − 1 = 15) and the level of confidence (α / 2 = 0.05), as we are looking for a (1 − α)100%CI2).
• Find the left and right critical values, $\chi_L^2=7.261$ and $\chi_R^2=24.9958$ using SOCR Chi-Square Distribution, as in the image below.
• CI(σ2)
$0.00318={15\times 0.0053 \over 24.9958} \leq \sigma^2 \leq {15\times 0.0053 \over 7.261}=0.01095$
• CI(σ)
$0.0564=\sqrt{15\times 0.0053 \over 24.9958} \leq \sigma \leq \sqrt{15\times 0.0053 \over 7.261}=0.10464$
### More Examples
• You randomly select and measure the contents of 15 bottles of cough syrup. The results (in fluid ounces) are shown. Use a 95% level of confidence to construct a confidence interval for the standard deviation (σ) assuming the contents of these cough syrup bottles is Normally distributed. Does this CI(σ) suggest that the variation in the bottles is at an acceptable level if the population standard deviation of the bottle’s contents should be less than 0.025 fluid ounce?
4.211 4.246 4.269 4.241 4.26 4.293 4.189 4.248 4.22 4.239 4.253 4.209 4.3 4.256 4.29
• The gray whale has the longest annual migration distance of any mammal. Gray whales leave Baja, California, and western Mexico in the spring, migrating to the Bering and Chukchi seas for the summer months. Tracking a sample of 50 whales for a year provided a mean migration distance of 11,064 miles with a standard deviation of 860 miles. Construct a 90% confidence interval for the variance for the migrating whales. Assume that the population of migration distances is Normally distributed.
• For the hot-dogs dataset construct 97% CI for the population standard deviation of the calorie and sodium contents, separately.<|endoftext|>
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How Do Ants Walk?
Ants are incredible creatures. They are excellent at working in groups to forage food. Ants are also not as lacking in intelligence like you may think. For example, an old study demonstrated that a single ant could find its way home after traveling long distances.
This may not sound impressive, but for an organism with a brain as large as a speck of dust, this is intelligent behavior. Since the study showed that ants could find their way back to their homes, the scientists naturally concluded that ants are capable of surveying their environment and storing memories that help them find their way home. However, a strange type of ant is challenging this old conclusion by showing that they can walk backwards and still know where they were going, as though they have eyes on the backs of their heads.
After observing ants walking backwards, scientists are now under the impression ants do not use visual-spatial reasoning or their memory to find their way back to their nests. Instead, researchers are now focused on the results of a new study that demonstrates ants use the landscape and sky to navigate their way back to their homes.
Ants Walk Backwards For a Reason
Foraging desert ants typically crawl in a forward fashion. It is only when an ant is carrying something very heavy relative in size to its body that it walks backwards. In these scenarios, walking backwards gives these ants a greater advantage, and they don’t need to expend as much energy as they would if they had to turn their bodies around.
The researchers split these ants into two groups. One group had a light object to carry while the second group had a heavier object. To the amazement of the researchers, the ants that walked backwards made it back to their homes just as quickly and accurately as the ants walking forwards.
Have you ever wondered why you sometimes see ants crawling backwards?<|endoftext|>
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NCERT Class XII Sociology: Chapter 6 – The Challenges of Cultural Diversity
National Council of Educational Research and Training (NCERT) Book for Class XII
Subject: Sociology (Indian Society)
Chapter: Chapter 6 – The Challenges of Cultural Diversity
Class XII NCERT Sociology (Indian Society) Text Book Chapter 6 The Challenges of Cultural Diversity is given below
Different kinds of social institutions, ranging from the family to the market,can bring people together, create strong collective identities and strengthensocial cohesion, as you learnt in Chapters 3 and 4. But, on the other hand, asChapters 4 and 5 showed, the very same institutions can also be sources ofinequality and exclusion. In this chapter, you will learn about some of thetensions and difficulties associated with cultural diversity. What precisely does‘cultural diversity’ mean, and why is it seen as a challenge?
The term ‘diversity’ emphasises differences rather than inequalities. Whenwe say that India is a nation of great cultural diversity, we mean that there aremany different types of social groups and communities living here. These arecommunities defined by cultural markers such as language, religion, sect, raceor caste. When these diverse communities are also part of a larger entity like anation, then difficulties may be created by competition or conflict between them.
This is why cultural diversity can present tough challenges. The difficultiesarise from the fact that cultural identities are very powerful – they can arouseintense passions and are often able to moblise large numbers of people.Sometimes cultural differences are accompanied by economic and socialinequalities, and this further complicates things. Measures to address theinequalities or injustices suffered by one community can provoke oppositionfrom other communities. The situation is made worse when scarce resources –like river waters, jobs or government funds – have to be shared.
If you read the newspapers regularly, or watch the news on television, youmay often have had the depressing feeling that India has no future. Thereseem to be so many divisive forces hard at work tearing apart the unity andintegrity of our country – communal riots, demands for regional autonomy,caste wars… You might have even felt upset that large sections of our populationare not being patriotic and don’t seem to feel as intensely for India as you andyour classmates do. But if you look at any book dealing with the history ofmodern India, or books dealing specifically with issues like communalism orregionalism (for example, Brass 1974), you will realise that these problems arenot new ones. Almost all the major ‘divisive’ problems of today have been thereever since Independence, or even earlier. But in spite of them India has notonly survived as a nation, but is a stronger nation-state today.
As you prepare to read on, remember that this chapter deals with difficultissues for which there are no easy answers. But some answers are better thanothers, and it is our duty as citizens to try our utmost to produce the bestanswers that are possible within the limitations of our historical and socialcontext. Remember also that, given the immense challenges presented by avast and extremely diverse collection of peoples and cultures, India has on thewhole done fairly well compared to most other nations. On the other hand, wealso have some significant shortcomings. There is a lot of room for improvementand much work needs to be done in order to face the challenges of the future …
CULTURAL COMMUNITIES AND THE NATION-STATE
Before discussing the major challenges that diversity poses in India – issuessuch as regionalism, communalism and casteism – we need to understand therelationship between nation-states and cultural communities. Why is it soimportant for people to belong to communities based on cultural identities likea caste, ethnic group, region, or religion? Why is so much passion arousedwhen there is a perceived threat, insult, or injustice to one’s community? Whydo these passions pose problems for the nation-state?
THE IMPORTANCE OF COMMUNITY IDENTITY
Every human being needs a sense of stable identity to operate in this world.Questions like — Who am I? How am I different from others? How do othersunderstand and comprehend me? What goals and aspirations should I have?– constantly crop up in our life right from childhood. We are able to answermany of these questions because of the way in which we are socialised, ortaught how to live in society by our immediate families and our community invarious senses. (Recall the discussion of socialisation in your Class XI textbooks.)The socialisation process involves a continuous dialogue, negotiation and evenstruggle against significant others (those directly involved in our lives) like ourparents, family, kin group and our community. Our community provides usthe language (our mother tongue) and the cultural values through which wecomprehend the world. It also anchors our self-identity.
Community identity is based on birth and ‘belonging’ rather than on someform of acquired qualifications or ‘accomplishment’. It is what we ‘are’ ratherthan what we have ‘become’. We don’t have to do anything to be born into acommunity – in fact, no one has any choice about which family or communityor country they are born into. These kinds of identities are called ‘ascriptive’ –that is, they are determined by the accidents of birth and do not involve anychoice on the part of the individuals concerned. It is an odd fact of social lifethat people feel a deep sense of security and satisfaction in belonging tocommunities in which their membership is entirely accidental. We often identifyso strongly with communities we have done nothing to ‘deserve’ – passed noexam, demonstrated no skill or competence… This is very unlike belonging to,say, a profession or team. Doctors or architects have to pass exams anddemonstrate their competence. Even in sports, a certain level of skill andperformance are a necessary pre-condition for membership in a team. But ourmembership in our families or religious or regional communities is withoutpreconditions, and yet it is total. In fact, most ascriptive identities are very hardto shake off; even if we choose to disown them, others may continue to identifyus by those very markers of belonging.
Perhaps it is because of this accidental, unconditional and yet almostinescapable belonging that we can often be so emotionally attached to our community identity. Expanding and overlapping circles of community ties(family, kinship, caste, ethnicity, language, region or religion) give meaning toour world and give us a sense of identity, of who we are. That is why peopleoften react emotionally or even violently whenever there is a perceived threat totheir community identity.
To get a clearer understanding of the expanding circles of community ties which shape oursense of identity, you can do a small survey designed as a game. Interview your schoolmates or other friends: each interviewee gets four chances to answer each of two questions:‘Who am I?’ and ‘Who do others think I am?’. But the answers must be in a single word orshort phrase; they cannot include any names (your own or your parents’/guardians’ names;cannot include your class/school, etc.). Interviews must be done singly and in private,i.e., other potential interviewees should not be able to hear what is said. Each person shouldonly be interviewed once (i.e., different interviewers cannot interview the same person).You can record the answers and analyse them later. Which types of identities predominated?What was the most common first choice? Which was often the last choice? Were thereany patterns to the answers? Did the answers for ‘who am I’ differ greatly, somewhat, or notat all from answers to ‘who do others think I am’?
A second feature of ascriptive identities and community feeling is that theyare universal. Everyone has a motherland, a mother tongue, a family, a faith…This may not necessarily be strictly true of every individual, but it is true in ageneral sense. And we are all equally committed and loyal to our respectiveidentities. Once again it is possible to come across people who may not beparticularly committed to one or the other aspect of their identity. But thepossibility of this commitment is potentially available to most people. Becauseof this, conflicts that involve our communities (whether of nation, language,religion, caste or region) are very hard to deal with. Each side in the conflictthinks of the other side as a hated enemy, and there is a tendency to exaggeratethe virtues of one’s own side as well as the vices of the other side. Thus, whentwo nations are at war, patriots in each nation see the other as the enemyaggressor; each side believes that God and truth are on their side. In the heatof the moment, it is very hard for people on either side to see that they areconstructing matching but reversed mirror images of each other.
It is a social fact that no country or group ever mobilises its members tostruggle for untruth, injustice or inequality – everyone is always fighting for truth,justice, equality… This does not mean that both sides are right in every conflict,or that there is no right and wrong, no truth. Sometimes both sides are indeedequally wrong or right; at other times history may judge one side to be the aggressorand the other to be the victim. But this can only happen long after the heat of theconflict has cooled down. Some notion of mutually agreed upon truth is veryhard to establish in situations of identity conflict; it usually takes decades,sometimes centuries for one side to accept that it was wrong (See Box 6.1).
When ‘Victors’ Apologise
It is not uncommon for the losing side in a war to be forced to apologise for the badthings that it did. It is only rarely that the winners accept that they were guilty ofwrong doing. However, in recent times there have been many such examples from aroundthe world. Nations or communities that were on the ‘winning’ side, or that are still in a dominantposition, are beginning to accept that they have been responsible for grave injustices in thepast and are seeking to apologise to the affected communities.
In Australia, there has been a long debate on an official apology from the Australian nation(where the majority of the population today is of white-European origin) to the descendants ofthe native peoples who were the original inhabitants of the forcibly colonised land. Most stategovernments in Australia have passed some variant of the following apology resolution:
We, the peoples of Australia, of many origins as we are, make a commitment to goon together in a spirit of reconciliation. We value the unique status of Aboriginaland Torres Strait Islander peoples as the original owners and custodians of landsand waters.
We recognise this land and its waters were settled as colonies without treaty orconsent. […] Our nation must have the courage to own the truth, to heal the woundsof its past so that we can move on together at peace with ourselves. As we walkthe journey of healing, one part of the nation apologises and expresses its sorrowand sincere regret for the injustices of the past, so the other part accepts theapologies and forgives. […] And so, we pledge ourselves to stop injustice, overcomedisadvantage, and respect that Aboriginal and Torres Strait Islander peoples havethe right to self-determination within the life of the nation.
In the United States of America there has been a longstanding debate about apologies tothe Native American community (the original inhabitants of the land driven out by war) andto the Black community (brought as slaves from Africa). No consensus has been reached yet.
In Japan, official policy has long recognised the need to apologise for the atrocities of warand colonisation during the periods when Japan occupied parts of East Asia including Koreaand parts of China. The most recent apology is from a 15th August 2005 speech by PrimeMinister Junichiro Koizumi:
In the past, Japan, through its colonial rule and aggression, caused tremendousdamage and suffering to the people of many countries, particularly to those ofAsian nations. Sincerely facing these facts of history, I once again express my feelingsof deep remorse and heartfelt apology, and also express the feelings of mourningfor all victims, both at home and abroad, in the war. I am determined not to allowthe lessons of that horrible war to erode, and to contribute to the peace andprosperity of the world without ever again waging a war.
Similar debates have gone on in South Africa, where a white minority was in power and brutallyoppressed the black majority consisting of the native population. In Britain as well, there hasbeen public discussion on whether the nation should apologise for its role in colonialism, or inpromoting slavery. Interestingly, the latter issue has also been taken up by cities – for example,the port city of Bristol debated whether the city council should pass a resolution apologisingfor the role that Bristol played in the slave trade.
Read Box 6.1 carefully. What purpose do you think such apologies serve? After all, theactual victims and the actual exploiters or oppressors may be long dead – they cannot becompensated or punished. Then for whom and for what reason are such apologies offeredor debated?
Can you think of other examples where anonymous ordinary people (i.e., people who arenot famous or powerful) who are no longer living are remembered, celebrated or honouredin a public way? What purpose is served by memorials and monuments like, for example,the India Gate monument in Delhi? (To whom is this monument dedicated? If you don’tknow, try to find out.)
Think about the kind of apology mentioned in Box 6.1 in the Indian context. If you wereasked to propose such a thing, which groups or communities do you think we as a nationshould ‘apologise’ to? Discuss this in class and try to reach a consensus. What are thearguments and counter-arguments given for various candidate groups? Did your opinion on such ‘apologies’ change after the class discussion?
COMMUNITIES, NATIONS AND NATION-STATES
At the simplest level, a nation is a sort of large-scale community – it is acommunity of communities. Members of a nation share the desire to be part ofthe same political collectivity. This desire for political unity usually expressesitself as the aspiration to form a state. In its most general sense, the term staterefers to an abstract entity consisting of a set of political-legal institutionsclaiming control over a particular geographical territory and the people living init. In Max Weber’s well-known definition, a state is a “body that successfullyclaims a monopoly of legitimate force in a particular territory” (Weber 1970:78).
A nation is a peculiar sort of community that is easy to describe but hard todefine. We know and can describe many specific nations founded on the basisof common cultural, historical and political institutions like a shared religion,language, ethnicity, history or regional culture. But it is hard to come up withany defining features, any characteristics that a nation must possess. For everypossible criterion there are exceptions and counter-examples. For example,there are many nations that do not share a single common language, religion,ethnicity and so on. On the other hand, there are many languages, religions orethnicities that are shared across nations. But this does not lead to the formationof a single unified nation of, say, all English speakers or of all Buddhists.
How, then, can we distinguish a nation from other kinds of communities,such as an ethnic group (based on common descent in addition to othercommonalities of language or culture), a religious community, or a regionally-defined community? Conceptually, there seems to be no hard distinction – anyof the other types of community can one day form a nation. Conversely, noparticular kind of community can be guaranteed to form a nation.
aspiringIs it really true that there is no characteristic that is common to each and every nation?Discuss this in class. Try to make a list of possible criteria or characteristics that could definea nation. For each such criterion, make a list of examples of nations that meet the criterion,and also a list of nations that violate it.In case you came up with the criterion that every nation must possess a territory in the formof a continuous geographical area, consider the cases mentioned below. [Locate eachcountry or region on a world map; you will also need to do a little bit of prior research oneach case… ]
- Alaska and the United States of America
- Pakistan before 1971 (West Pakistan + East Pakistan)
- Malvinas/Falkland Islands and the United Kingdom
- Austria and Germany
- Ecuador, Colombia, Venezuela
- Yemen, Saudi Arabia, Kuwait, United Arab Emirates
[Hint: The first three cases are examples of geographically distant territories belonging tothe same nation; the last three cases are examples of countries with contiguous territory,shared language and culture but separate nation-states.]Can you add to this list of examples?
The criterion that comes closest to distinguishing a nation is the state. Unlikethe other kinds of communities mentioned before, nations are communitiesthat have a state of their own. That is why the two are joined with a hyphen toform the term nation-state. Generally speaking, in recent times there has been aone-to-one bond between nation and state (one nation, one state; one state, onenation). But this is a new development. It was not true in the past that a singlestate could represent only one nation, or that every nation must have its ownstate. For example, when it was in existence, the Soviet Union explicitly recognisedthat the peoples it governed were of different ‘nations’ and more than one hundredsuch internal nationalities were recognised. Similarly, people constituting a nationmay actually be citizens or residents of different states. For example, there aremore Jamaicans living outside Jamaica than in Jamaica – that is, the populationof ‘non-resident’ Jamaicans exceeds that of ‘resident’ Jamaicans. A differentexample is provided by ‘dual citizenship’ laws. These laws allow citizens of aparticular state to also – simultaneously – be citizens of another state. Thus, tocite one instance, Jewish Americans may be citizens of Israel as well as the USA;they can even serve in the armed forces of one country without losing theircitizenship in the other country.
In short, today it is hard to define a nation in any way other than to say thatit is a community that has succeeded in acquiring a state of its own. Interestingly,the opposite has also become increasingly true. Just as would-be or aspiringIs
nationalities are now more andmore likely to work towards forminga state, existing states are alsofinding it more and more necessaryto claim that they represent anation. One of the characteristicfeatures of the modern era (recallthe discussion of modernity fromChapter 4 of your Class XI textbook,Understanding Society) is theestablishment of democracy andnationalism as dominant sources ofpolitical legitimacy. This meansthat, today, ‘the nation’ is the mostaccepted or proper justification fora state, while ‘the people’ are theultimate source of legitimacy of the nation. In other words, states ‘need’ thenation as much or even more than nations need states.
But as we have seen in the preceding paragraphs, there is no historicallyfixed or logically necessary relationship between a nation-state and the variedforms of community that it could be based on. This means that there is nopre-determined answer to the question: How should the ‘state’ part of thenation-state treat the different kinds of community that make up the ‘nation’part? As is shown in Box 6.2 (which is based on the United Nations DevelopmentProgram (UNDP) report of 2004 on Culture and Democracy), most states havegenerally been suspicious of cultural diversity and have tried to reduce oreliminate it. However, there are many successful examples – including India –which show that it is perfectly possible to have a strong nation-state without havingto ‘homogenise’ different types of community identities into one standard type.
Threatened by community identities, states try to eliminatecultural diversity
Historically, states have tried to establish and enhance their political legitimacy throughnation-building strategies. They sought to secure … the loyalty and obedience oftheir citizens through policies of assimilation or integration. Attaining these objectiveswas not easy, especially in a context of cultural diversity where citizens, in addition totheir identifications with their country, might also feel a strong sense of identity withtheir community – ethnic, religious, linguistic and so on.Most states feared that the recognition of such difference would lead to socialfragmentation and prevent the creation of a harmonious society. In short, suchidentity politics was considered a threat to state unity. In addition, accommodatingthese differences is politically challenging, so many states have resorted to eithersuppressing these diverse identities or ignoring them on the political domain.
Policies of assimilation – often involving outright suppression of the identities of ethnic,religious or linguistic groups – try to erode the cultural differences between groups.Policies of integration seek to assert a single national identity by attempting to eliminateethno-national and cultural differences from the public and political arena, whileallowing them in the private domain. Both sets of policies assume a singular nationalidentity.
Assimilationist and integrationist strategies try to establish singular national identitiesthrough various interventions like:
- Centralising all power to forums where the dominant group constitutes a majority,and eliminating the autonomy of local or minority groups;
- Imposing a unified legal and judicial system based on the dominant group’straditions and abolishing alternative systems used by other groups;
- Adopting the dominant group’s language as the only official ‘national’ languageand making its use mandatory in all public institutions;
- Promotion of the dominant group’s language and culture through nationalinstitutions including state-controlled media and educational institutions;
- Adoption of state symbols celebrating the dominant group’s history, heroes andculture, reflected in such things as choice of national holidays or naming of streetsetc.;
- Seizure of lands, forests and fisheries from minority groups and indigenous peopleand declaring them ‘national resources’…
Box 6.2 speaks of ‘assimilationist’ and ‘integrationist’ policies. Policies thatpromote assimilation are aimed at persuading, encouraging or forcing all citizensto adopt a uniform set of cultural values and norms. These values and normsare usually entirely or largely those of the dominant social group. Other,non-dominant or subordinated groups in society are expected or required to giveup their own cultural values and adopt the prescribed ones. Policies promotingintegration are different in style but not in overall objective: they insist that thepublic culture be restricted to a common national pattern, while all ‘non-national’cultures are to be relegated to the private sphere. In this case too, there is thedanger of the dominant group’s culture being treated as ‘national’ culture.
You can probably see what the problem is by now. There is no necessaryrelationship between any specific form of community and the modern form ofthe state. Any of the many bases of community identity (like language, religion,ethnicity and so on) may or may not lead to nation formation – there are noguarantees. But because community identities can act as the basis fornation-formation, already existing states see all forms of community identityas dangerous rivals. That is why states generally tend to favour a single,homogenous national identity, in the hope of being able to control and manageit. However, suppressing cultural diversity can be very costly in terms of thePolicies alienation of the minority or subordinated communities whose culture is treatedas ‘non-national’. Moreover, the very act of suppression can provoke theopposite effect of intensifying community identity. So encouraging, or at leastallowing, cultural diversity is good policy from both the practical and theprincipled point of view.
CULTURAL DIVERSITY AND THE INDIAN NATION-STATE – AN OVERVIEW
The Indian nation-state is socially and culturally one of the most diversecountries of the world. It has a population of about 1029 million people,currently the second largest – and soon to become the largest – nationalpopulation in the world. These billion-plus people speak about 1,632 differentlanguages and dialects. As many as eighteeen of these languages have beenofficially recognised and placed under the 8th Schedule of the Constitution,thus guaranteeing their legal status. In terms of religion, about 80.5% of thepopulation are Hindus, who in turn are regionally specific, plural in beliefsand practices, and divided by castes and languages. About 13.4% of thepopulation are Muslims, which makes India the world’s third largest Muslimcountry after Indonesia and Pakistan. The other major religious communitiesare Christians (2.3%), Sikhs (1.9%), Buddhists (0.8%) and Jains (0.4%).Because of India’s huge population, these small percentages can also add upto large absolute numbers.
In terms of the nation-state’s relationship with community identities, theIndian case fits neither the assimilationist nor the integrationist model describedin Box 6.2. From its very beginning the independent Indian state has ruledout an assimilationist model. However, the demand for such a model hasbeen expressed by some sections of the dominant Hindu community. Although‘national integration’ is a constant theme in state policy, India has not been‘integrationist’ in the way that Box 6.2 describes. The Constitution declaresthe state to be a secular state, but religion, language and other such factorsare not banished from the public sphere. In fact these communities have beenexplicitly recognised by the state. By international standards, very strongconstitutional protection is offered to minority religions. In general, India’sproblems have been more in the sphere of implementation and practice ratherthan laws or principles. But on the whole, India can be considered a goodexample of a ‘state-nation’ though it is not entirely free from the problemscommon to nation-states.
National unity with cultural diversity – Building a democratic “state-nation’’
An alternative to the nation-state, then, is the “state nation”, where various“nations”— be they ethnic, religious, linguistic or indigenous identities— can co-exist peacefully and cooperatively in a single state polity. Case studies and analyses demonstrate that enduring democracies can beestablished in polities that are multicultural. Explicit efforts are required to end theBOX
cultural exclusion of diverse groups … and to build multiple and complementaryidentities. Such responsive policies provide incentives to build a feeling of unity in diversity— a “we” feeling. Citizens can find the institutional and political space to identify withboth their country and their other cultural identities, to build their trust in commoninstitutions and to participate in and support democratic politics. All of these are keyfactors in consolidating and deepening democracies and building enduring “state-nations”.
India’s constitution incorporates this notion. Although India is culturally diverse,comparative surveys of long-standing democracies including India show that it hasbeen very cohesive, despite its diversity. But modern India is facing a grave challengeto its constitutional commitment to multiple and complementary identities with the riseof groups that seek to impose a singular Hindu identity on the country. These threatsundermine the sense of inclusion and violate the rights of minorities in India today.Recent communal violence raises serious concerns for the prospects for social harmonyand threatens to undermine the country’s earlier achievements.
And these achievements have been considerable. Historically, India’s constitutionaldesign recognised and responded to distinct group claims and enabled the polity tohold together despite enormous regional, linguistic and cultural diversity. As evidentfrom India’s performance on indicators of identification, trust and support (Chart 1), itscitizens are deeply committed to the country and to democracy, despite the country’sdiverse and highly stratified society. This performance is particularly impressive whencompared with that of other long-standing—and wealthier—democracies.
The challenge is in reinvigorating India’s commitment to practices of pluralism,institutional accommodation and conflict resolution through democratic means. Criticalfor building a multicultural democracy is a recognition of the shortcomings of historicalnation-building exercises and of the benefits of multiple and complementary identities.Also important are efforts to build the loyalties of all groups in society throughidentification, trust and support. National cohesion does not require the imposition of asingle identity and the denunciation of diversity. Successful strategies to build “state-nations” can and do accommodate diversity constructively by crafting responsivepolicies of cultural recognition. They are effective solutions for ensuring the longer termsobjectives of political stability and social harmony.
REGIONALISM IN THE INDIAN CONTEXT
Regionalism in India is rooted in India’s diversity of languages, cultures, tribes,and religions. It is also encouraged by the geographical concentration of theseidentity markers in particular regions, and fuelled by a sense of regionaldeprivation. Indian federalism has been a means of accommodating theseregional sentiments. (Bhattacharyya 2005).
After Independence, initially the Indian state continued with theBritish-Indian arrangement dividing India into large provinces, also called‘presidencies’. (Madras, Bombay, and Calcutta were the three major presidencies;incidentally, all three cities after which the presidencies were named have changedtheir names recently). These were large multi-ethnic and multilingual provincialstates constituting the major political-administrative units of a semi-federal state
called the Union of India. For example, the old Bombay State (continuation ofthe Bombay Presidency) was a multilingual state of Marathi, Gujarati, Kannadaand Konkani speaking people. Similarly, the Madras State was constituted byTamil, Telugu, Kannada and Malayalam speaking people. In addition to thepresidencies and provinces directly administered by the British Indiangovernment, there were also a large number of princely states and principalitiesall over India. The larger princely states included Mysore, Kashmir, and Baroda.But soon after the adoption of the Constitution, all these units of the colonialera had to be reorganised into ethno-linguistic States within the Indian unionin response to strong popular agitations. (See Box 6.4 on the next page).
Linguistic States Helped Strengthen Indian Unity
The Report of the States Reorganisation Commission (SRC) which wasimplemented on November 1, 1956, has helped transform the political andinstitutional life of the nation.
The background to the SRC is as follows. In the 1920s, the Indian National Congresswas reconstituted on lingusitic lines. Its provincial units now followed the logic oflanguage – one for Marathi speakers, another for Oriya speakers, etc. At the sametime, Gandhi and other leaders promised their followers that when freedom came,the new nation would be based on a new set of provinces based on the principle oflanguage.
However, when India was finally freed in 1947, it was also divided. Now, when theproponents of linguistic states asked for this promise to be redeemed, the Congresshesitated. Partition was the consequence of intense attachment to one’s faith; howmany more partitions would that other intense loyalty, language, lead to? So ran thethinking of the top Congress bosses including Nehru, Patel and Rajaji.
On the other side, the rank and file Congressmen were all for the redrawing of themap of India on the lines of language. Vigorous movements arose among Marathiand Kannada speakers, who were then spread across several different political regimes– the erstwhile Bombay and Madras presidencies, and former princely states such asMysore and Hyderabad. However, the most militant protests ensued from the verylarge community of Telugu speakers. In October 1953, Potti Sriramulu, a formerGandhian, died seven weeks after beginning a fast unto death. Potti Sriramulu’smartyrdom provoked violent protests and led to the creation of the state of AndhraPradesh. It also led to the formation of the SRC, which in 1956 put the formal, final sealof approval on the principle of linguistic states.
In the early 1950s, many including Prime Minister Jawaharlal Nehru feared thatstates based on language might hasten a further subdivision of India. In fact, somethinglike the reverse has happened. Far from undermining Indian unity, linguistic stateshave helped strengthen it. It has proved to be perfectly consistent to be Kannadigaand Indian, Bengali and Indian, Tamil and Indian, Gujarati and Indian…
To be sure, these states based on language sometimes quarrel with each other. Whilethese disputes are not pretty, they could in fact have been far worse. In the sameyear, 1956, that the SRC mandated the redrawing of the map of India on linguisticlines, the Parliament of Ceylon (as Sri Lanka was then known) proclaimed Sinhala thecountry’s sole official language despite protests from the Tamils of the north. Oneleft-wing Sinhala MP issued a prophetic warning to the chauvinists. “One language,two nations”, he said, adding: “Two languages, one nation”.
The civil war that has raged in Sri Lanka since 1983 is partly based on the denial by themajority linguistic group of the rights of the minority. Another of India’s neighbours,Pakistan, was divided in 1971 because the Punjabi and Urdu speakers of its westernwing would not respect the sentiments of the Bengalis in the east.
It is the formation of linguistic states that has allowed India to escape an even worsefate. If the aspirations of the Indian language communities had been ignored, whatwe might have had here was – “One language, fourteen or fifteen nations.”
Language coupled with regional and tribal identity – and not religion – hastherefore provided the most powerful instrument for the formation of ethno-national identity in India. However, this does not mean that all linguisticcommunities have got statehood. For instance, in the creation of three newstates in 2000, namely Chhatisgarh, Uttaranchal and Jharkhand, languagedid not play a prominent role. Rather, a combination of ethnicity based ontribal identity, language, regional deprivation and ecology provided the basisfor intense regionalism resulting in statehood. Currently there are 28 States(federal units) and 7 Union territories (centrally administered) within the Indiannation-state.
NOTE: In this chapter, the word “State” has a capital S when it is used to denotethe federal units within the Indian nation-state; the lower case ‘state’ is usedfor the broader conceptual category described above.
Find out about the origins ofyour own State. When was itformed? What were themain criteria used to defineit? – Was it language, ethnic/tribal identity, regionaldeprivation, ecologicaldifference or other criterion?How does this compare withother States within the Indiannation-state?Try to classify all the States ofIndia in terms of the criteriafor their formation.Are you aware of anycurrent social movementsthat are demanding thecreation of a State? Try tofind out the criteria beingused by these movements.[Hint: Check the Telenganaand Vidarbha movements,and others in your region…]
Respecting regional sentiments is not just a matter ofcreating States: this has to be backed up with aninstitutional structure that ensures their viability asrelatively autonomous units within a larger federalstructure. In India this is done by Constitutionalprovisions defining the powers of the States and the Centre.There are lists of ‘subjects’ or areas of governance whichare the exclusive responsibility of either State or Centre,along with a ‘Concurrent List’ of areas where both areallowed to operate. The State legislatures determine thecomposition of the upper house of Parliament, the RajyaSabha. In addition there are periodic committees andcommissions that decide on Centre-State relations. Anexample is the Finance Commission which is set up everyten years to decide on sharing of tax revenues betweenCentre and States. Each Five Year Plan also involvesdetailed State Plans prepared by the State PlanningCommissions of each state.
On the whole the federal system has worked fairly well,though there remain many contentious issues. Since theera of liberalisation (i.e., since the 1990s) there is concernamong policy makers, politicians and scholars aboutincreasing inter-regional economic and infrastructuralinequalities. As private investment (both foreign and Indian)is given a greater role in economic development,considerations of regional equity get diluted. This happensbecause private investors generally want to invest in alreadydeveloped States where the infrastructure and other facilities are better. Unlikeprivate industry, the government can give some consideration to regional equity(and other social goals) rather than just seek to maximise profits. So left to itself,the market economy tends to increase the gap between developed and backwardregions. Fresh public initiatives will be needed to reverse current trends.
THE NATION-STATE AND RELIGION-RELATEDISSUES AND IDENTITIES
Perhaps the most contentious of all aspects of cultural diversity are issuesrelating to religious communities and religion-based identities. These issuesmay be broadly divided into two related groups – the secularism–communalismset and the minority–majority set. Questions of secularism and communalismare about the state’s relationship to religion and to political groupings thatinvoke religion as their primary identity. Questions about minorities andmajorities involve decisions on how the state is to treat different religious, ethnic
or other communities that are unequal in terms of numbers and/or power(including social, economic and political power).
MINORITY RIGHTS AND NATION BUILDING
In Indian nationalism, the dominant trend was marked by an inclusive anddemocratic vision. Inclusive because it recognised diversity and plurality.Democratic because it sought to do away with discrimination and exclusionand bring forth a just and equitable society. The term ‘people’ has not beenseen in exclusive terms, as referring to any specific group defined by religion,ethnicity, race or caste. Ideas of humanism influenced Indian nationalists andthe ugly aspects of exclusive nationalism were extensively commented upon byleading figures like Mahatma Gandhi and Rabindranath Tagore.
Rabindranath Tagore on the evils of exclusive nationalism
where the spirit of the Western nationalism prevails, the whole people isbeing taught from boyhood to foster hatreds and ambitions by all kinds ofmeans — by the manufacture of half-truths and untruths in history, by persistentmisrepresentation of other races and the culture of unfavourable sentiments towardsthem…Never think for a moment that the hurt you inflict upon other races will notinfect you, or that the enemities you sow around your homes will be a wall of protectionto you for all time to come? To imbue the minds of a whole people with an abnormalvanity of its own superiority, to teach it to take pride in its moral callousness and ill-begotten wealth, to perpetuate humiliation of defeated nations by exhibiting trophieswon from war, and using these schools in order to breed in children’s minds contemptfor others, is imitating the West where she has a festering sore…
To be effective, the ideas of inclusive nationalism had to be built into theConstitution. For, as already discussed (in section 6.1), there is a very strongtendency for the dominant group to assume that their culture or language orreligion is synonymous with the nation state. However, for a strong anddemocratic nation, special constitutional provisions are required to ensure therights of all groups and those of minority groups in particular. A brief discussionon the definition of minorities will enable us to appreciate the importance ofsafeguarding minority rights for a strong, united and democratic nation.
The notion of minority groups is widely used in sociology and is more thana merely numerical distinction – it usually involves some sense of relativedisadvantage. Thus, privileged minorities such as extremely wealthy peopleare not usually referred to as minorities; if they are, the term is qualified insome way, as in the phrase ‘privileged minority’. When minority is used withoutqualification, it generally implies a relatively small but also disadvantaged group.
The sociological sense of minority alsoimplies that the members of the minorityform a collectivity – that is, they have astrong sense of group solidarity, a feeling oftogetherness and belonging. This is linkedto disadvantage because the experience ofbeing subjected to prejudice anddiscrimination usually heightens feelings ofintra-group loyalty and interests (Giddens2001:248). Thus, groups that may beminorities in a statistical sense, such aspeople who are left-handed or people bornon 29th February, are not minorities in the sociological sensebecause they do not form a collectivity.
However, it is possible to have anomalous instances where aminority group is disadvantaged in one sense but not in another.Thus, for example, religious minorities like the Parsis or Sikhsmay be relatively well-off economically. But they may still bedisadvantaged in a cultural sense because of their small numbersrelative to the overwhelming majority of Hindus. Religious orcultural minorities need special protection because of thedemographic dominance of the majority. In democratic politics, itis always possible to convert a numerical majority into politicalpower through elections. This means that religious or culturalminorities – regardless of their economic or social position – arepolitically vulnerable. They must face the risk that the majoritycommunity will capture political power and use the state machineryto suppress their religious or cultural institutions, ultimately forcingthem to abandon their distinctive identity.
Relative size and distribution of religious minorities
As is well known, Hindus constitute an overwhelming majority in India: theynumber about 828 millions and account for 80.5% of the total populationaccording to the 2001 Census. The Hindu population is four times larger than the combinedpopulation of all other minority religions, and about six times larger than the largest minoritygroup, the Muslims.However, this can also be misleading because Hindus are not a homogenous group andare divided by caste – as indeed are all the other major religions, albeit to different extents.The Muslims are by far the largest religious minority in India – they numbered 138 millionsand were 13.4% of the population in 2001. They are scattered all over the country,constitute a majority in Jammu and Kashmir and have sizeable pockets in West Bengal,Uttar Pradesh, Kerala, Andhra Pradesh, Karnataka and Rajasthan.Christians constitute around 2.3% of the population (24 million) and are scattered allover, with sizeable pockets in the north eastern and southern states. The three Christian-majority states are all in the North East – Nagaland (90%), Mizoram (87%) andMeghalaya (70%). Sizeable proportions of Christians are also found in Goa (27%) andKerala (19%).The Sikhs constitute 1.9% of the population (19 million) and although they are foundscattered across the country, they are concentrated in Punjab where they are in amajority (60%).There are also several other small religious groups – Buddhists (8 million, 0.8%), Jains (4 million,0.4%), and ‘Other Religions and Persuasions’ (under 7 million, 0.7%). The highest proportionof Buddhists is found in Sikkim (28%) and Arunachal Pradesh (13%), while among the largerstates Maharashtra has the highest share of Buddhists at 6%. The highest concentrations ofJains are found in Maharashtra (1.3%), Rajasthan (1.2%) and Gujarat (1%).
In the long years of struggle against British colonialism, Indian nationalistsunderstood the imperative need to recognise and respect India’s diversity.Indeed ‘unity in diversity’ became a short hand to capture the plural and diversenature of Indian society. Discussions on minority and cultural rights markmany of the deliberations of the Indian National Congress and find finalexpression in the Indian Constitution (Zaidi 1984).
Dr. Ambedkar on protection of minorities
To diehards who have developed a kind of fanaticism against minorityprotection I would like to say two things. One is that minorities are an explosiveforce which, if it erupts, can blow up the whole fabric of the state. The history of Europebears ample and appalling testimony to this fact. The other is that the minorities inIndia have agreed to place their existence in the hands of the majority. In the historyof negotiations for preventing the partition of Ireland, Redmond said to Carson “Askfor any safeguard you like for the Protestant minority but let us have a United Ireland.”Carson’s reply was “Damn your safeguards, we don’t want to be ruled by you.” Nominority in India has taken this stand. [John Redmond, catholic majority leader; Sir Edward Carson, protestant minority leader]
The makers of the Indian Constitution were aware that astrong and united nation could be built only when all sectionsof people had the freedom to practice their religion, and todevelop their culture and language. Dr. B.R. Ambedkar, thechief architect of the Constitution, made this point clear inthe Constituent Assembly, as shown in Box 6.7.
In the last three decades we have witnessed hownon-recognition of the rights of different groups of peoplein a country can have grave implications for national unity.One of key issues that led to the formation of Bangladeshwas the unwillingness of the Pakistani state to recognisethe cultural and linguistic rights of the people of Bangladesh.
The Indian Constitution on minorities and cultural diversityArticle 29:
(1) Any section of the citizens residing in the territory of India or any part thereofhaving a distinct language, script or culture of its own shall have the right to conservethe same.
(2) No citizen shall be denied admission into any educational institution maintainedby the State or received out of State funds on grounds only of religion, race, caste,language or any of them.
(1)All minorities, whether based on religion or language, shall have the right toestablish and administer educational institutions of their choice.
(2)The State shall not, in granting aid to educational institutions, discriminateagainst any educational institution on the ground that it is under themanagement of a minority, whether based on religion or language.
There are many instances ofa ‘majority’ in one contextbeing converted into a‘minority’ in another context(or the other way around).Find out about concreteexamples of this, and discussthe implications.Remember that thesociological concept of aminority involves not justrelative numbers but alsorelative power.[Suggestions: Whites in SouthAfrica before and after theend of apartheid; Hindus inKashmir; Muslims in Gujarat;Upper castes among Hindus;Tribals in North Easternstates;]
One of the many contentious issues that formed thebackdrop of the ethnic conflict in Sri Lanka was theimposition of Sinhalese as a national language. Likewiseany forcible imposition of a language or religion on anygroup of people in India weakens national unity which isbased upon a recognition of differences. Indiannationalism recognises this, and the Indian Constitutionaffirms this (Box 6.8).
Finally, it is useful to note that minorities existeverywhere, not just in India. In most nation-states, theretend to be a dominant social group whether cultural,ethnic, racial or religious. Nowhere in the world is therea nation-state consisting exclusively of a singlehomogenous cultural group. Even where this was almosttrue (as in countries like Iceland, Sweden or South Korea),modern capitalism, colonialism and large scale migrationhave brought in a plurality of groups. Even the smalleststate will have minorities, whether in religious, ethnic,linguistic or racial terms.
COMMUNALISM, SECULARISM ANDTHE NATION-STATE
In everyday language, the word ‘communalism’ refers to aggressive chauvinismbased on religious identity. Chauvinism itself is an attitude that sees one’sown group as the only legitimate or worthy group, with other groups being seen– by definition – as inferior, illegitimate and opposed. Thus, to simplify further,communalism is an aggressive political ideology linked to religion. This is apeculiarly Indian, or perhaps South Asian, meaning that is different from thesense of the ordinary English word. In the English language, “communal” meanssomething related to a community or collectivity as different from an individual.The English meaning is neutral, whereas the South Asian meaning is stronglycharged. The charge may be seen as positive – if one is sympathetic tocommunalism – or negative, if one is opposed to it.
It is important to emphasise that communalism is about politics,not about religion. Although communalists are intensely involved withreligion, there is in fact no necessary relationship between personalfaith and communalism. A communalist may or may not be a devoutperson, and devout believers may or may not be communalists. However,all communalists do believe in a political identity based on religion.The key factor is the attitude towards those who believe in other kindsof identities, including other religion-based identities. Communalistscultivate an aggressive political identity, and are prepared to condemnor attack everyone who does not share their identity.
One of the characteristic features of communalism is its claim thatreligious identity overrides everything else. Whether one is poor or rich,whatever one’s occupation, caste or political beliefs, it is religion alonethat counts. All Hindus are the same as are all Muslims, Sikhs and soon. This has the effect of constructing large and diverse groups assingular and homogenous. It is noteworthy that this is done for one’sown group as well as for others. This would obviously rule out thepossibility that Hindus, Muslims and Christians who belong to Kerala,for example, may have as much or more in common with each otherthan with their co-religionists from Kashmir, Gujarat or Nagaland. Italso denies the possibility that, for instance, landless agriculturallabourers (or industrialists) may have a lot in common even if theybelong to different religions and regions.
Communalism is an especially important issue in India because ithas been a recurrent source of tension and violence. During communalriots, people become faceless members of their respective communities.They are willing to kill, rape, and loot members of other communities inorder to redeem their pride, to protect their home turf. A commonlycited justification is to avenge the deaths or dishonour suffered by theirco-religionists elsewhere or even in the distant past. No region has beenwholly exempt from communal violence of one kind or another. Everyreligious community has faced this violence in greater or lesser degree, althoughthe proportionate impact is far more traumatic for minority communities. To theextent that governments can be held responsible for communal riots, nogovernment or ruling party can claim to be blameless in this regard. In fact, thetwo most traumatic contemporary instances of communal violence occurred undereach of the major political parties. The anti-Sikh riots of Delhi in 1984 took placeunder a Congress regime. The unprecedented scale and spread of anti-Muslimviolence in Gujarat in 2002 took place under a BJP government.
India has had a history of communal riots from pre-Independence times,often as a result of the divide-and-rule policy adopted by the colonial rulers.But colonialism did not invent inter-community conflicts – there is also a longhistory of pre-colonial conflicts – and it certainly cannot be blamed for post-Independence riots and killings. Indeed, if we wish to look for instances of
Kabir Das – A Lasting Symbol of Syncretic Traditions
The poems of Kabir, synthesising Hindu and Muslim devotion arecherished symbols of pluralism:Moko Kahan Dhundhe re Bande Where do you search for me?Mein To Tere Paas Mein I am with youNa Teerath Mein, Na Moorat Mein Not in pilgrimage, nor in iconsNa Ekant Niwas Mein Neither in solitudeNa Mandir Mein, Na Masjid Mein Not in temples, nor in mosquesNa Kabe Kailas Mein Neither in Kaaba nor in KailashMein To Tere Paas Mein Bande I am with you o manMein To Tere Paas Mein… I am with you …
6.9Talk to your parents and theelders in your family andcollect from them poems,songs, short stories whichhighlight issues such asreligious pluralism, syncretismor communal harmony.When you have collected allthis material and presentedthem in class, you may bepleasantly surprised to learnhow broad based ourtraditions of religious pluralismare, and how widely they areshared across differentlinguistic groups, regions andreligions.
religious, cultural, regional or ethnic conflict they can befound in almost every phase of our history. But we shouldnot forget that we also have a long tradition of religiouspluralism, ranging from peaceful co-existence to actualinter-mixing or syncretism. This syncretic heritage isclearly evident in the devotional songs and poetry of theBhakti and Sufi movements (Box 6.9). In short, historyprovides us with both good and bad examples; what wewish to learn from it is up to us.
As we have seen above, the meanings of the terms communaland communalism are more or less clear, despite the bittercontroversies between supporters and opponents. Bycontrast, the terms ‘secular’ and ‘secularism’ are very hardto define clearly, although they are also equally controversial.In fact, secularism is among the most complex terms insocial and political theory. In the western context the mainsense of these terms has to do with the separation of churchand state. The separation of religious and political authoritymarked a major turning point in the social history of thewest. This separation was related to the process of “secularisation”, or theprogressive retreat of religion from public life, as it was converted from a mandatoryobligation to a voluntary personal practice. Secularisation in turn was related tothe arrival of modernity and the rise of science and rationality as alternatives toreligious ways of understanding the world.
The Indian meanings of secular and secularism include the western sensebut also involve others. The most common use of secular in everyday languageis as the opposite of communal. So, a secular person or state is one that doesnot favour any particular religion over others. Secularism in this sense is theopposite of religious chauvinism and it need not necessarily imply hostility toreligion as such. In terms of the state-religion relationship, this sense ofKabir secularism implies equal respect for all religions, rather than separation ordistancing. For example, the secular Indian state declares public holidays tomark the festivals of all religions.
One kind of difficulty is created by the tension between the western sense ofthe state maintaining a distance from all religions and the Indian sense of thestate giving equal respect to all religions. Supporters of each sense are upsetby whatever the state does to uphold the other sense. Should a secular stateprovide subsidies for the Haj pilgrimage, or manage the Tirupati-Tirumala templecomplex, or support pilgrimages to Himalayan holy places? Should all religiousholidays be abolished, leaving only Independence Day, Republic Day, GandhiJayanti and Ambedkar Jayanti for example? Should a secular state ban cowslaughter because cows are holy for a particular religion? If it does so, shouldit also ban pig slaughter because another religion prohibits the eating of pork?If Sikh soldiers in the army are allowed to have long hair and wear turbans,should Hindu soldiers also be allowed to shave their heads or Muslim soldiersallowed to have long beards? Questions of this sort lead to passionatedisagreements that are hard to settle.
Another set of complications is created by the tension between the Indianstate’s simultaneous commitment to secularism as well as the protection ofminorities. The protection of minorities requires that they be given specialconsideration in a context where the normal working of the political systemplaces them at a disadvantage vis-à-vis the majority community. But providingsuch protection immediately invites the accusation of favouritism or‘appeasement’ of minorities. Opponents argue that secularism of this sort isonly an excuse to favour the minorities in return for their votes or other kindsof support. Supporters argue that without such special protection, secularismcan turn into an excuse for imposing the majority community’s values andnorms on the minorities.
These kind of controversies become harder to solve when political partiesand social movements develop a vested interest in keeping them alive. In recenttimes, communalists of all religions have contributed to the deadlock. Theresurgence and newly acquired political power of the Hindu communalists hasadded a further dimension of complexity. Clearly a lot needs to be done toimprove our understanding of secularism as a principle and our practice of itas a policy. But despite everything, it is still true that India’s Constitution andlegal structure has proved to be reasonably effective in handling the problemscreated by various kinds of communalism.
The first generation of leaders of independent India (who happened to beoverwhelmingly Hindu and upper caste) chose to have a liberal, secular stategoverned by a democratic constitution. Accordingly, the ‘state’ was conceivedin culturally neutral terms, and the ‘nation’ was also conceived as an inclusiveterritorial-political community of all citizens. Nation building was viewed mainlyas a state-driven process of economic development and social transformation.
The expectation was that the universalisation of citizenship rights and theinduction of cultural pluralities into the democratic process of open andcompetitive politics would evolve new, civic equations among ethnic communities,and between them and the state (Sheth:1999). These expectations may nothave materialised in the manner expected. But ever since Independence, thepeople of India, through their direct political participation and election verdictshave repeatedly asserted their support for a secular Constitution and state.Their voices should count.
STATE AND CIVIL SOCIETY
You may have noticed that much of this chapter has been concerned with thestate. The state is indeed a very crucial institution when it comes to themanagement of cultural diversity in a nation. Although it claims to representthe nation, the state can also become somewhat independent of the nation andits people. To the extent that the state structure – the legislature, bureaucracy,judiciary, armed forces, police and other arms of the state – becomes insulatedfrom the people, it also has the potential of turning authoritarian. Anauthoritarian state is the opposite of a democratic state. It is a state in whichthe people have no voice and those in power are not accountable to anyone.Authoritarian states often limit or abolish civil liberties like freedom of speech,freedom of the press, freedom of political activity, right to protection fromwrongful use of authority, right to the due processes of the law, and so on.Apart from authoritarianism, there is also the possibility that state institutionsbecome unable or unwilling to respond to the needs of the people because ofcorruption, inefficiency, or lack of resources. In short, there are many reasonswhy a state may not be all that it should be. Non-state actors and institutionsbecome important in this context, for they can keep a watch on the state, protestagainst its injustices or supplement its efforts.
Civil society is the name given to the broad arena which lies beyond theprivate domain of the family, but outside the domain of both state and market.Civil society is the non-state and non-market part of the public domain in whichindividuals get together voluntarily to create institutions and organisations. It isthe sphere of active citizenship: here, individuals take up social issues, try toinfluence the state or make demands on it, pursue their collective interests orseek support for a variety of causes. It consists of voluntary associations,organisations or institutions formed by groups of citizens. It includes politicalparties, media institutions, trade unions, non-governmental organisations (NGOs),religious organisations, and other kinds of collective entities. The main criteriafor inclusion in civil society are that the organisation should not be state-controlled,and it should not be a purely commercial profit-making entity. Thus, Doordarshanis not part of civil society though private television channels are; a carmanufacturing company is not part of civil society but the trade unions to whichits workers belong are. Of course these criteria allow for a lot of grey areas. For example, a newspaper may be run like a purely commercial enterprise, or anNGO may be supported by government funds.
The Indian people had a brief experience of authoritarian rule during the‘Emergency’ enforced between June 1975 and January 1977. Parliament wassuspended and new laws were made directly by the government. Civil libertieswere revoked and a large number of politically active people were arrested andjailed without trial. Censorship was imposed on the media and governmentofficials could be dismissed without normal procedures. The government coercedlower level officials to implement its programmes and produce instant results.The most notorious was the forced sterilisation campaign in which large numbersdied due to surgical complications. When elections were held unexpectedly inearly 1977, the people voted overwhelmingly against the ruling Congress Party.
The Emergency shocked people into active participation and helped energisethe many civil society initiatives that emerged in the 1970s. This period saw theresurgence of a wide variety of social movements including the women’s,environmental, human rights and dalit movements. Today the activities of civilsociety organisations have an even wider range, including advocacy and lobbyingactivity with national and international agencies as well as active participation
Forcing the State to Respond to the People:
The Right to Information Act
The Right to Information Act 2005 (Act No. 22/2005) is a law enacted by the Parliament of Indiagiving Indians (except those in the State ofJammu and Kashmir who have their own speciallaw) access to Government records. Under theterms of the Act, any person may requestinformation from a “public authority” (a body ofGovernment or instrumentality of State) which isexpected to reply expeditiously or within thirtydays. The Act also requires every public authorityto computerise their records for wide dissemination and to proactively publish certaincategories of information so that the citizens need minimum recourse to request forinformation formally.This law was passed by Parliament on 15 June 2005 and came into force on 13 October2005. Information disclosure in India was hitherto restricted by the Official Secrets Act1923 and various other special laws, which the new RTI Act now overrides.The Act specifies that citizens have a right to:
- request any information (as defined)
- take copies of documents
- inspect documents, works and records
- take certified samples of materials of work.
- obtain information in form of printouts, diskettes, floppies, tapes, videocassettes or in any other electronic mode or through printouts.
Find out about the civilsociety organisations orNGOs that are active inyour neighbourhood.What sorts of issues dothey take up? Whatsort of people work inthem? How and towhat extent are theseorganisations differentfroma)governmentorganisations;b)commercialorganisations?
in various movements. The issues taken up are diverse, rangingfrom tribal struggles for land rights, devolution in urbangovernance, campaigns against rape and violence againstwomen, rehabilitation of those displaced by dams and otherdevelopmental projects, fishermen’s struggles againstmechanised fishing, rehabilitation of hawkers and pavementdwellers, campaigns against slum demolitions and for housingrights, primary education reform, distribution of land to dalits,and so on. Civil liberties organisations have been particularlyimportant in keeping a watch on the state and forcing it toobey the law. The media, too, has taken an increasingly activerole, specially its emergent visual and electronic segments.
Among the most significant recent initiatives is the campaignfor the Right to Information. Beginning with an agitation in ruralRajasthan for the release of information on government fundsspent on village development, this effort grew into a nation-widecampaign. Despite the resistance of the bureaucracy, thegovernment was forced to respond to the campaign and pass anew law formally acknowledging the citizens’ right to information(Box 6.10). Examples of this sort illustrate the crucial importance of civil societyin ensuring that the state is accountable to the nation and its people.
1.What is meant by cultural diversity? Why is India considered to be a verydiverse country?
2.What is community identity and how is it formed?
3.Why is it difficult to define the nation? How are nation and state related inmodern society?
4.Why are states often suspicious of cultural diversity?
5.What is regionalism? What factors is it usually based on?
6.In your opinion, has the linguistic reorganisation of states helped or harmedIndia?
7.What is a ‘minority’? Why do minorities need protection from the state?
8.What is communalism?
9.What are the different senses in which ‘secularism’ has been understoodin India?
10.What is the relevance of civil society organisations today?
Bhargava, Rajeev. 1998. ‘What is Secularism for?’, in Bhargava, Rajeev. ed.Secularism and its Critic. Oxford University Press. New Delhi. Bhargava, Rajeev. 2005. Civil Society, Public Sphere and Citizenship. SagePublications. New Delhi. Bhattacharyya, Harihar. 2005. Federalism and Regionalism in India: InstitutionalStrategies and Political Accommodation of Identities. working paper No. 27, SouthAsia Institute, Dept of Political Science. University of Heidelberg.
Brass, Paul. 1974. Language, Religion and Politics in North India. Vikas PublishingHouse. Delhi. Chandra, Bipan. 1987. Communalism in Modern India. Vikas Publishing House. NewDelhi. Miller, David. 1995. On Nationality. Clarendon Press, Oxford. Sheth, D.L. 1999. ‘The Nations-State and Minority Rights’, in Sheth, D.L. andMahajan, Gurpreet. ed. Minority Identities and the Nation-State. Oxford UniversityPress. New Delhi.
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If you’ve ever had a minor injury like a sprain or muscle strain, you may have been advised by a doctor or nurse (or the Internet) to rest your injury and take NSAIDs to help with the pain and inflammation.
But what exactly are NSAIDs, and how do they help?
NSAID is an acronym for a class of drug called a Non-Steroidal Anti-Inflammatory Drug. As suggested by the name, these are drugs that have properties that reduce inflammation — the body’s white blood cell response to foreign organisms and infections — but they do not contain any steroids. The label “non-steroidal” differentiates them from anti-inflammatory drugs containing steroids. (For example, steroidal asthma inhalers or corticosteroids like Prednisone.)
NSAIDs are commonly prescribed by doctors to help with fight the effects of inflammation, but they’re also useful for lowering fevers and preventing blood clotting.
The most commonly-used over-the-counter (OTC) NSAIDs include:
- Ibuprofen (Motrin)
- Naproxen (Aleve)
However, over a dozen different NSAIDs are available in the U.S., some of them at prescription strength. Prescription-strength NSAIDs are quite commonly prescribed for arthritis pain and swelling, for example.
Why Fight Inflammation?
Although inflammation is a natural response to infection and injury, oftentimes its side effects can cause discomfort and disability. Inflammation signals the body to send more blood to an injury or infection site, and this can lead to problems like swelling, pain, and stiffness.
In particularly bad cases (such as after an acute orthopedic injury), you may be unable to use a joint or limb. Dangerous levels of swelling can also harm circulation and blood flow to surrounding tissue, which sometimes can lead to complications like necrosis (tissue death, like gangrene).
How NSAIDs Work
NSAIDs fight inflammation and pain by reducing the body’s production of chemicals called prostaglandins. Prostaglandins are derived from fatty acids, and they can be found throughout the body’s soft tissue.
Prostaglandins do the following:
- regulate inflammation by telling blood platelets where and when to aggregate (gather to create clots) — and when to stop
- lower blood pressure by dilating, or opening up, your blood vessels
- protect GI tissue (the linings of your stomach and intestines) from digestive acids
- keep the kidneys healthy by promoting normal function
Prostaglandins are also involved in smooth muscle contraction and relaxation (especially in the GI tract and the uterus, in the case of pregnancy), cell production, lung function, neurotransmission (the passing of chemical signals), pain sensitivity, and lipolysis (the breakdown of fats).
Prostaglandins do many things to protect the body from harm. When you’re sick or injured, they are responsible for promoting inflammation, fever, and pain — natural processes that fight off infection, signal the body’s need for self-repair, and let you know that you need to slow down and rest.
Sometimes, however, these symptoms become intrusive or dangerous (for example, in the cause of uncontrolled inflammation and swelling damaging surrounding tissue). In these cases, you may wish to take NSAIDs to temper these effects.
NSAIDs work on prostaglandins by blocking the enzymes involved in their production. By reducing the production of prostaglandins, your body produces less inflammation and less pain. However, the beneficial effects of prostaglandin production also go out the window. A lower level of prostaglandins means less protection from acids in the stomach and intestines. Also, when you reduce clotting, you leave your body at risk for freer bleeding.
The result: taking NSAIDs for too long may leave your body vulnerable to ulcers, excessive bleeding, and other side effects. This is why it’s important to coordinate your care with a doctor. In the case of long-term NSAID therapy, you may be able to take additional medications to counteract these risks.
When to Use NSAIDs
NSAIDs have numerous uses. Many people turn to OTC anti-inflammatory pain relievers for help with the following:
- OTC or prescription NSAIDs can be used to combat inflammation, swelling, and pain.
- Though NSAIDs cannot fix the underlying causes of a cough or cold (a virus or bacterial infection), they can help to address related inflammatory reactions like fever, cough, or irritation.
- NSAIDs may help reduce headache pain by preventing tissue swelling. They may also inhibit pain signals to the brain.
- Heart disease. If you’re in a high risk category for stroke or heart attack, your physician may suggest you take a daily low dose of aspirin for prevention of dangerous inflammation or clots.
- Menstrual cramps. NSAIDs help with pain relief. Blocking the muscle-contraction effects of prostaglandins may also help to minimize the cramping itself.
- Sports injuries. Because they’re so effective at controlling pain and inflammation, NSAIDs are commonly recommended for managing discomfort after sustaining a mild to moderate sports injury like a sprain or strain.
NSAIDs and Orthopedic Conditions/Injuries
An orthopedic physician may suggest you take NSAIDS for any of the following:
- Arthritis (osteoarthritis or rheumatoid arthritis)
- Chronic pain (like back or neck pain)
- Nerve impingement
- Sprains (of ligaments)
- Strains (of muscles)
- Tissue tears or ruptures
Safety & Risks of NSAIDS usage
If you’re considering taking NSAIDs for longer than a few days, it’s wise to consult with a doctor to be sure this is safe for you. Be sure to ask how many milligrams you can take per day and at what intervals.
Though anti-inflammatory medication is commonly prescribed and quite safe for the most part, it is still a drug. As such, it can carry some side effects and risks, all of which become more serious the longer you take the medication.
Side effects of NSAIDs can include:
- Increased bleeding, especially in the stomach, can lead to anemia (a lack of red blood cells in the body).
- Reduced platelet activity (clotting) means you may bleed more under the skin, leading to bruises.
- Kidney problems. Though the responsible use of NSAIDs should not cause problems in most people, some with low kidney function may experience kidney dysfunction or failure, especially if they take too much for too long. In most cases, the damage is reversible. If you have kidney issues or suspect you might have them, make sure you’re healthy before using anti-inflammatory drugs.
- Nausea/upset stomach. GI problems can include stomachaches, constipation, diarrhea, nausea, and gas. Taking NSAIDs with food sometimes helps to minimize these symptoms.
- Because the lining of your stomach and intestines is not as well-protected, you’re more vulnerable to the damaging effects of stomach acids, including ulceration.
Some people may also experience dizziness, ringing in the ears, or a rash. (For a complete list of possible side effects, talk to your doctor.)
Though side effects can affect anyone who takes NSAIDs over time, the risks are even more serious if you have certain health conditions. You automatically fall into a higher-risk category if any of the following apply to you:
- Over 65
- High blood pressure
- Ulcers (currently, or in the past)
- History of liver disease
- History of kidney disease
- Taking medications
This last item is important. NSAIDs need to be taken with caution if you’re using other medications, as they may amplify or diminish the effects of other drugs.
For example, if you’re already taking a prescription blood thinner (anticoagulant), taking an NSAID with it could lead to dangerously heavy bleeding if you were to cut yourself or suffer an impact blow (for example, in the case of a fall or car accident). Likewise, NSAIDs make it harder for your body to eliminate some substances like lithium. Taking NSAIDs long term can lead to a dangerous, possibly even toxic, buildup.
On the other hand, if you have high blood pressure, NSAIDs may reduce or negate the benefit of your hypertension medication. This is because taking NSAIDs can actually increase your blood pressure. Similarly, taking NSAIDs can reduce blood flow to your kidneys, which may make water pills (diuretics) less effective.
Summary of nsaid usage
Non-steroidal anti-inflammatory drugs are beneficial for managing pain and for controlling the effects of inflammation on the body.
If you’ve sustained an acute orthopedic injury or if you have chronic pain from an inflammatory condition like arthritis or nerve impingement, NSAIDs may bring you some relief.
However, as with any drug, this medication does have risks and side effects. Talk to your physician to make sure it’s the right choice for you.
To learn more about NSAID use and inflammation, give Coastal Orthopedics located in Corpus Christi, Texas a call: Telephone: 361.994.1166. Or just click the button below to request an appointment today!<|endoftext|>
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## Factoring Trinomials
Specifically, factoring trinomials where a is greater than one (assuming the form ax2+bx+c).
Everyone hates these. You can’t factor them using the method normally taught in school, so many teachers just teach “Guess and Check” for this type of trinomial, in the hopes that they’ll get good enough at guessing and checking.
This is dumb.
There IS a method to factor this sort of trinomials; it’s called The British Method. It’s the only memorable thing my Algebra II teacher taught me. Ready? Here we go.
Here it is in its generality, i.e. with variables And here it is as an example. 1. Realize that your trinomial is of the form ax2+bx+c 1. 2x2+x-1 2. Find a number s such that a*c=s. 2. 2*-1=-2. 3. Find two numbers p and q such that p*q=s and p+q=b. 3. p=2; q=-1: 2*-1=-2 and 2+-1=1. 4. Split the middle term into two binomials, thus: ax2+px and qx+c 4. 2x2+2x and –x-1 5. Factor completely the left binomial to obtain kx(ex+f). 5. 2x(x+1) 6. Factor the right binomial such that you get m(ex+f). 6. –(x+1) 7. Create two binomials (kx+m) and (ex+f); these are your factors! 7. (2x-1)(x+1)
• Darwin
• June 30th, 2011
That technique might be good to get even more intuition however we’re all taught the technique of completing the square, thus:
2x^2 + x -1
2(x^2 +(1/2)x – (1/2))
2((x+1/4)^2 – 1/16 – 1/2)
2((x+1/4)^2 – 9/16)
2(x+1/4+3/4)(x+1/4-3/4)
2(x+1)(x-1/2)
which will work every time with quadratic polynomials.
Edit: the page didn’t parse my reply correctly
1. Have you ever considered writing an e-book or guest authoring on other websites?
I have a blog based on the same topics you discuss
and would really like to have you share some stories/information.
I know my subscribers would appreciate your work.
If you’re even remotely interested, feel free to send me an e mail.<|endoftext|>
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This post is also available in: Italian
In geography, desert is defined as an area of the earth’s surface in which the precipitations are unlikely to exceed 250 millimeters per year, and the soil is mostly arid, with little or no vegetation.
Among the deserts, according to this definition, must also be considered Polar Regions, in addition to the more familiar arid areas that are located at medium and low latitudes.
You can distinguish three main types:
- Hot deserts, rocky deserts where the soil is made up of stones or pebbles called by the Arabic word Hamada; it can also be gravelly, called Reg, or sandy dunes, called Erg, present in tropical regions, characterized by enhanced aridity, vegetation reduced or absent, lack of perennial streams, tendency to drought; climate which is associated with that environment is the hot desert climate (according to the Köppen climate);
- Cold deserts (also called, somewhat inaccurately, “temperate deserts”), present in continental temperate regions, characterized by strong aridity and remarkable annual thermal excursions, with hot summers and cold winters; climate which is associated with that environment is the cold desert climate (according to the Köppen climate);
- Polar deserts (White Deserts), present in the northern and the southern margin of the boreal and southern continents (Greenland, the Arctic and Antarctica), characterized by intense cold and perennial expanses of snow and ice; climate which is associated with that environment is the glacial climate (according to the Köppen climate).
Hot Desert Erg, Tunisia.<|endoftext|>
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# Let p be a non singular matrix 1+p+p^2+p^3+cdots+p^n=O (O denotes the null matrix), then p^-1 is?
Oct 25, 2017
The answer is $= - \left(I + p + \ldots \ldots \ldots {p}^{n - 1}\right)$
#### Explanation:
We know that
${p}^{-} 1 p = I$
$I + p + {p}^{2} + {p}^{3.} \ldots . {p}^{n} = O$
Multiply both sides by ${p}^{-} 1$
${p}^{-} 1 \cdot \left(1 + p + {p}^{2} + {p}^{3.} \ldots . {p}^{n}\right) = {p}^{-} 1 \cdot O$
${p}^{-} 1 \cdot 1 + {p}^{-} 1 \cdot p + {p}^{-} 1 \cdot {p}^{2} + \ldots \ldots {p}^{-} 1 \cdot {p}^{n} = O$
${p}^{-} 1 + \left({p}^{-} 1 p\right) + \left({p}^{-} 1 \cdot p \cdot p\right) + \ldots \ldots \ldots \left({p}^{-} 1 p \cdot {p}^{n - 1}\right) = O$
${p}^{-} 1 + \left(I\right) + \left(I \cdot p\right) + \ldots \ldots \ldots \left(I \cdot {p}^{n - 1}\right) = O$
Therefore,
${p}^{-} 1 = - \left(I + p + \ldots \ldots \ldots {p}^{n - 1}\right)$
Oct 25, 2017
See below.
#### Explanation:
$p \left({p}^{-} 1 + p + {p}^{2} + \cdots + {p}^{n - 1}\right) = 0$ but $p$ by hypothesis is non singular then exists ${p}^{-} 1$ so
${p}^{-} 1 p \left({p}^{-} 1 + p + {p}^{2} + \cdots + {p}^{n - 1}\right) = {p}^{-} 1 + p + {p}^{2} + \cdots + {p}^{n - 1} = 0$
and finally
${p}^{-} 1 = - {\sum}_{k = 1}^{n - 1} {p}^{k}$
Also can be solved as
${p}^{-} 1 = - p \left({\sum}_{k = 0}^{n - 2} {p}^{k}\right) = p \left({p}^{n - 1} + {p}^{n}\right) = {p}^{n} \left(1 - p\right)$<|endoftext|>
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# 5.1 - Graphs for Two Different Measurement Variables
5.1 - Graphs for Two Different Measurement Variables
In a previous lesson, we learned about possible graphs to display measurement data. These graphs included: dotplots, stemplots, histograms, and boxplots view the distribution of one or more samples of a single measurement variable and scatterplots to study two at a time (see section 4.3).
## Example 5.1 Graph of Two Measurement Variables
The following two questions were asked on a survey of 220 STAT 100 students:
1. What is your height (inches)?
2. What is your weight (lbs)?
Notice we have two different measurement variables. It would be inappropriate to put these two variables on side-by-side boxplots because they do not have the same units of measurement. Comparing height to weight is like comparing apples to oranges. However, we do want to put both of these variables on one graph so that we can determine if there is an association (relationship) between them. The scatterplot of this data is found in Figure 5.2.
Figure 5.2. Scatterplot of Weight versus Height
In Figure 5.2, we notice that as height increases, weight also tends to increase. These two variables have a positive association because as the values of one measurement variable tend to increase, the values of the other variable also increase. You should note that this holds true regardless of which variable is placed on the horizontal axis and which variable is placed on the vertical axis.
## Example 5.2 Graph of Two Measurement Variables
The following two questions were asked on a survey of ten PSU students who live off-campus in unfurnished one-bedroom apartments.
1. How far do you live from campus (miles)?
2. How much is your monthly rent (\\$)?
The scatterplot of this data is found in Figure 5.3.
Figure 5.3. Scatterplot of Monthly Rent versus Distance from campus
In Figure 5.3, we notice that the further an unfurnished one-bedroom apartment is away from campus, the less it costs to rent. We say that two variables have a negative association when the values of one measurement variable tend to decrease as the values of the other variable increase.
## Example 5.3 Graph of Two Measurement Variables
The following two questions were asked on a survey of 220 Stat 100 students:
1. About how many hours do you typically study each week?
2. About how many hours do you typically exercise each week?
The scatterplot of this data is found in Figure 5.4.
Figure 5.4. Scatterplot of Study Hours versus Exercise Hours
In Figure 5.4, we notice that as the number of hours spent exercising each week increases there is really no pattern to the behavior of hours spent studying including visible increases or decreases in values. Consequently, we say that that there is essentially no association between the two variables.
[1] Link ↥ Has Tooltip/Popover Toggleable Visibility<|endoftext|>
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# Mock AIME 4 2006-2007 Problems/Problem 10
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
## Problem
Compute the remainder when
${2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}$
is divided by 1000.
## Solution
Let $\omega$ and $\zeta$ be the two complex third-roots of 1. Then let
$S = (1 + \omega)^{2007} + (1 + \zeta)^{2007} + (1 + 1)^{2007} = \sum_{i = 0}^{2007} {2007 \choose i}(\omega^i + \zeta^i + 1)$.
Now, if $i$ is a multiple of 3, $\omega^i + \zeta^i + 1 = 1 + 1 + 1 = 3$. If $i$ is one more than a multiple of 3, $\omega^i + \zeta^i + 1 = \omega + \zeta + 1 = 0$. If $i$ is two more than a multiple of 3, $\omega^i + \zeta^i + 1 = \omega^2 + \zeta^2 + 1= \zeta + \omega + 1 = 0$. Thus
$S = \sum_{i = 0}^{669} 3 {2007 \choose 3i}$, which is exactly three times our desired expression.
We also have an alternative method for calculating $S$: we know that $\{\omega, \zeta\} = \{-\frac{1}{2} + \frac{\sqrt 3}{2}i, -\frac{1}{2} - \frac{\sqrt 3}{2}i\}$, so $\{1 + \omega, 1 + \zeta\} = \{\frac{1}{2} + \frac{\sqrt 3}{2}i, \frac{1}{2} - \frac{\sqrt 3}{2}i\}$. Note that these two numbers are both cube roots of -1, so $S = (1 + \omega)^{2007} + (1 + \zeta)^{2007} + (1 + 1)^{2007} = (-1)^{669} + (-1)^{669} + 2^{2007} = 2^{2007} - 2$.
Thus, the problem is reduced to calculating $2^{2007} - 2 \pmod{1000}$. $2^{2007} \equiv 0 \pmod{8}$, so we need to find $2^{2007} \pmod{125}$ and then use the Chinese Remainder Theorem. Since $\phi (125) = 100$, by Euler's Totient Theorem $2^{20 \cdot 100 + 7} \equiv 2^7 \equiv 3 \pmod{125}$. Combining, we have $2^{2007} \equiv 128 \pmod{1000}$, and so $3S \equiv 128-2 \pmod{1000} \Rightarrow S\equiv \boxed{042}\pmod{1000}$.
## Solution 2
$\sum_{k=0}^{n} {n \choose k} =2^n$, and $\sum_{k=0}^{3n} {3n \choose k} =2^{3n}$
$\sum_{k=0}^{3n} {3n \choose 3k}=\frac{\sum_{k=0}^{3n} {3n \choose k}+q(n)}{3}$ where $q(n)$ is an integer $-2 \le q(n) \le 2$ that depends on the value of $n$ and will make the sum an integer. The division by 3 comes from the fact that we're skipping 2 out of every 3 terms in the binomial. So we divide the whole sum by 3 and we add or subtract $q(n)$ to correct for the integer based on the modularity of the sum with 3
$\sum_{k=0}^{3n} {3n \choose 3k}=\frac{2^{3n}+q(n)}{3}$
When $n$ is odd, $3n$ is odd, $2^{odd} \equiv (-1)^{odd}\;(mod\;3)\equiv -1\;(mod\;3)\equiv 2\;(mod\;3)$, Therefore $q(n)=-2$ because we need to subract 2.
When $n$ is even, $3n$ is even, $2^{even} \equiv (1)^{even}\;(mod\;3)\equiv 1\;(mod\;3)\equiv -2\;(mod\;3)$, Therefore $q(n)=2$ because we need to add 2.
So, the equation becomes:
$\sum_{k=0}^{3n} {3n \choose 3k}=\frac{2^{3n}+(-1)^n2}{3}$
${2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}=\sum_{k=0}^{3\times 669} { 3\times 669 \choose 3k }=\frac{2^{2007}+(-1)^{669}2}{3}=\frac{2^{2007}-2}{3}$
$2^{2007} \equiv 2^7\;(mod\;1000)\equiv 128\;(mod\;1000)$
$2^{2007}-2 \equiv 128-2\;(mod\;1000)\equiv 126\;(mod\;1000)$
$\frac{2^{2007}-2}{3} \equiv \frac{126}{3}\;(mod\;1000)\equiv 42\;(mod\;1000)$
Therefore, the remainder when ${2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}$ is divided by 1000 is $\boxed{042}$
~Tomas Diaz. [email protected]<|endoftext|>
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# College math calculators
There is College math calculators that can make the process much easier. Our website can solve math word problems.
## The Best College math calculators
Apps can be a great way to help students with their algebra. Let's try the best College math calculators. To find the answer, start with a whole number (e.g., 17) and a divisor (e.g., 5). Then, divide the divisor by the whole number (17 ÷ 5 = 4). Next, multiply the result by the dividend (4 × \$5 = \$20). Finally, add up all of your answers to find the total value of your item (20 + 4 + \$5 = \$25). The answer always works out to be one more than that original number because of rounding errors.
Solving for the "intercept" is a common thing to do when you are trying to find the best fit line to an equation. The intercept will tell you where the y=0 value is. This is going to be the value that you would expect if you were trying to solve for the y-axis of an equation by taking the x-axis and adding it to itself (y = y + x). On a graph, you might expect this value to be where the x-axis intersects with the y-axis. You can also think of it as being at the origin. If we are solving for y in our equation, then the intercept would be 0 on both axes. It might also be important as it will give us a good idea for how long our graph should be in order for our data points to fall within that range. If we have a very short range (like on a log scale), we will need to make sure that our x-axis intercept is much higher than our y-axis intercept so that our data points fall well above or below that line.
When solving linear equations by graphing, you can use a coordinate plane to graph the coordinates of each number. The coordinates will then represent points on the graph. For example, if there are two numbers and you know their coordinates, then you can draw a line between them to represent their relationship in the equation. This is called graphing a linear equation by intercepts. Another way to graph linear equations is by using an equation sheet. In this case, you need to enter all of the values in the problem before graphing it. Linear equations with more than one variable can also be graphed by using a table or matrix form. When graphing these types of equations, it’s important to include all of the variables as well as their corresponding values in the table or matrix format. This will ensure that your results match what was stated in your problem statement and that your solution gives you an accurate depiction of what’s happening in the equation.
This really helps me a lot. And when I usually say a lot, it's like. just a tiny amount, but this, this one helps me a lot. Makes my pain go away, saves me lots of time and for sure these Problems are easy to solve when you have the app. Fast-Solving App, Needs 5 Stars and 100% Fantastic.
Williamina Richardson
This is a great app. It picks up blurry images extremely well and filters out things you don't need and then tells you the steps to solve the problem. Seriously, this program has helped me in the most important and sensitive moments of my studies, a clean 10/10.
Prudence Foster<|endoftext|>
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Chain-Base Index Numbers, Business Mathematics and Statistics
# Chain-Base Index Numbers, Business Mathematics and Statistics - Business Mathematics and Statistics - B Com
CHAIN INDEX NUMBERS
In the fixed base method which is discussed so far the base remains the same & does not change whole throughout the series. But with the passage of time some items may have been included in the series & other ones might have been deleted, & hence it becomes difficult to compare the result of present conditions with those of the old remote period. Hence the fixed base method does not suit when the conditions change. In such a case the changing base period may be more suitable. Under this method the figures for each year are first expressed as a percentage of the preceding year (called link relatives) then they are chained together by successive multiplication to form a chain index.
STEPS IN CONSTRUCTION OF CHAIN INDEX
(1) The figures are to be expressed as the percentage of the preceding year to get link relatives.
Link Relatives of current year =
(2) Chain index is obtained by the formula :
Example 8: From the following data find the index numbers by taking (1) 2005 as base (ii) by chain base method.
Solution
Table :
Construction of Index Number taking 2005 as base
This means that from 2005 to 2006 there is an increase of 3.33% (103.33 - 100) from 2005 to 2007 there is an increase of 8.33% (108.33 - 100) increase from 2005 to 2008 there is an increase of 20% (120 - 100) & so on ..........
Table : Calculation of Chain Base Index
Note: The results obtained by the fixed base & chain base are almost similar. The difference is only due to approximation. Infact the chain base index numbers are always equal to fixed base index numbers if there is only one series.
Example 9 :
From the following data compute chain base index number & fixed base index number.
Solution :
Table :
Computation of Chain Base Index Number Calculation of Link Relatives
Table : Computation of Fixed base Index No.
Note : The two series of index numbers obtained by fixed base & chain base method are different except in the first two years. This would always be so in case of more than one series.
Example 10 : The price index & quantity index of a commodity were 120 & 110 respectively in 2012 with base 2011. Find its value index number in 2012 with base 2011.
Solution :
The document Chain-Base Index Numbers, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com is a part of the B Com Course Business Mathematics and Statistics.
All you need of B Com at this link: B Com
115 videos|142 docs
## FAQs on Chain-Base Index Numbers, Business Mathematics and Statistics - Business Mathematics and Statistics - B Com
1. What are Chain-Base Index Numbers?
Ans. Chain-Base Index Numbers are a type of index number that measure the changes in the value or quantity of a variable relative to a base period. They are computed by linking together the index numbers of consecutive time periods, using the value of the previous period as the new base for the next period.
2. What is the purpose of using Chain-Base Index Numbers?
Ans. The purpose of using Chain-Base Index Numbers is to provide a more accurate measure of changes in the value or quantity of a variable over time. By using a base period that changes with each time period, Chain-Base Index Numbers take into account any shifts in the underlying data and can better reflect the current state of the economy or market.
3. How are Chain-Base Index Numbers calculated?
Ans. Chain-Base Index Numbers are calculated by linking together the index numbers of consecutive time periods. To do this, the index number for the current period is divided by the index number for the previous period, and then multiplied by 100. This gives the percentage change in the value or quantity of the variable between the two periods.
4. What are some advantages of using Chain-Base Index Numbers?
Ans. Some advantages of using Chain-Base Index Numbers include: - They provide a more accurate measure of changes in the value or quantity of a variable over time. - They take into account any shifts in the underlying data and can better reflect the current state of the economy or market. - They are less affected by the choice of base period than fixed-base index numbers. - They can be updated easily and frequently, allowing for more timely and accurate analysis.
5. What are some limitations of using Chain-Base Index Numbers?
Ans. Some limitations of using Chain-Base Index Numbers include: - They can be more complex to calculate than other types of index numbers. - They may be more sensitive to short-term changes in the data, which can lead to greater volatility and less stability over time. - They may not provide as clear of a picture of long-term trends as fixed-base index numbers, which may be better suited for certain types of analysis.
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# Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 2
## Theorem
Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be a Cauchy sequence.
Then $\sequence {x_n}$ is convergent.
## Proof
Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.
By Real Cauchy Sequence is Bounded, $\sequence {a_n}$ is bounded.
By the Bolzano-Weierstrass Theorem, $\sequence {a_n}$ has a convergent subsequence $\sequence {a_{n_r} }$.
Let $a_{n_r} \to l$ as $r \to \infty$.
It is to be shown that $a_n \to l$ as $n \to \infty$.
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.
Then $\dfrac \epsilon 2 > 0$.
Hence:
$(1): \quad \exists R \in \R: \forall r > R: \size {a_{n_r} - l} < \dfrac \epsilon 2$
We have that $\sequence {a_n}$ is a Cauchy sequence.
Hence:
$(2): \quad \exists N \in \R: \forall m > N, n > N: \size {x_m - x_n} \le \dfrac \epsilon 2$
Let $n > N$.
Let $r \in \N$ be sufficiently large that:
$n_r > N$
and:
$r > R$
Then $(1)$ is satisfied, and $(2)$ is satisfied with $m = n_r$.
So:
$\ds \forall n > N: \,$ $\ds \size {a_n - l}$ $=$ $\ds \size {a_n - a_{n_r} + a_{n_r} - l}$ $\ds$ $\le$ $\ds \size {a_n - a_{n_r} } + \size{a_{n_r} - l}$ Triangle Inequality for Real Numbers $\ds$ $<$ $\ds \dfrac \epsilon 2 + \dfrac \epsilon 2$ $\ds$ $=$ $\ds \epsilon$
So, given $\epsilon > 0$, we have found $n \in \R$ such that:
$\forall n > N: \size {a_n - l} < \epsilon$
Thus:
$x_n \to l$ as $n \to \infty$.
$\blacksquare$<|endoftext|>
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### 6.2.5 Ray-box intersection algorithm
This section describes the ray-box intersection algorithm.
The following figure shows a graph with line equations:
Figure 6-22 Graph with line equations
Equation of a line
```y = mx + b
```
The vector form of this equation is:
```r = O + t*D
```
Where:
`O` is the origin point
`D` is the direction vector
`t` is the parameter
The following figure shows an axis aligned bounding box:
Figure 6-23 Axis aligned bounding box
An axis aligned bounding box `AABB` can be defined by its min and max points `A` and `B`
`AABB` defines a set of lines parallel to coordinate axis. Each component of line can be defined by the following equation:
```x = Ax; y = Ay; z = Az
x = Bx; y = By; z = Bz
```
To find where a ray intersects one of those lines, equal both equations. For example:
```Ox + tx*Dx = Ax
```
You can write the solution as:
```tAx = (Ax – Ox) / Dx
```
Obtain the solution for all components of both intersection points in the same manner:
```tAx = (Ax – Ox) / Dx
tAy = (Ay – Oy) / Dy
tAz = (Az – Oz) / Dz
tBx = (Bx – Ox) / Dx
tBy = (By – Oy) / Dy
tBz = (Bz – Oz) / Dz
```
In vector form these are:
```tA = (A – O) / D
tB = (B – O) / D
```
This finds where the line intersects the planes defined by the faces of the cube but it does not guarantee that the intersections lie on the cube.
The following figure shows a 2D representation of ray-box intersection:
Figure 6-24 Ray-box intersection 2D representation
To find what solution is really an intersection with the box, you require the greater value of the `t` parameter for the intersection at the min plane.
```tmin = (tAx> tAy) ? tAx: tAy
```
You require the smaller value of the parameter `t` for the intersection at the max plane.
```tmin = (tAx> tAy) ? tAx: tAy
```
You also must consider those cases when you get no intersections.
The following figure shows a ray-box with no intersection:
Figure 6-25 Ray-box with no intersection
If you guarantee that the reflective surface is enclosed by the BBox, that is, the origin of the reflected ray is inside the BBox, then there are always two intersections with the box, and the handling of different cases is simplified.
The following figure shows a ray-box intersection in BBox:
Figure 6-26 Ray-box intersection in BBox<|endoftext|>
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Often the first step in identifying a tree species is to observe its silhouette. There are in fact many different shapes of trees. In general, conifers have a well-known silhouette that differs from that of broadleaf trees. Observe the conifers in your area and determine what silhouette they have. Are they sphere- or cone-shaped?
Needles also come in different shapes and lengths, and are very important in identifying conifer species. Furthermore, the way they are bundled together and their location on the twigs help distinguish different kinds of trees. Trees can often be correctly identified by observing their needles.
Cones, the fruit of conifers, contain the seeds for reproduction of the species. The trunk and branches are enveloped in a protective covering called bark. It serves the same function as skin does for humans. The fruit and bark are often used to help identify a tree.
Did you know that it is possible to identify a tree simply by its silhouette? The silhouette is very important in dendrology. But it does take an expert eye to identify the exact species of a tree by its silhouette alone. Often, you will need to observe the leaves, fruit and bark to correctly identify a tree.
Although conifers are generally all cone-shaped, there are differences between the species. How does the silhouette of a black spruce differ from that of a balsam fir? Do the eastern hemlock and eastern redcedar have the same silhouette? Are branches on the trunk of a lodgepole pine distributed in the same way as those on a white spruce? By asking yourself these kinds of questions, you will begin to see the different silhouettes.
The lodgepole pine and the balsam fir are both cone-shaped. However, the lodgepole pine does not have many branches near the base of the trunk, unlike the fir. Notice how the fir is tapered and pointed at the top!
The Douglas-fir is neither quite cone- nor cylinder-shaped. Its silhouette is somewhat similar to that of the black spruce, but without branches at the base of the trunk.
The larch has a lot fewer needles than other conifers. Moreover, this cone-shaped tree is the only conifer that loses its needles in winter.
The branches of the eastern white pine branches are at right angles to the trunk, giving the tree the shape of a cylinder. The eastern redcedar has branches that shoot upward, giving it a cone-shaped silhouette.
The eastern hemlock has a distinctive silhouette. It becomes narrow toward the base, which makes it appear diamond-shaped.
When observing trees outdoors, it is recommended that you look at a single, isolated tree. This will enable you to observe the shape of its silhouette, without interference from other trees.
The Shape of Needles
Observing the needles on conifers is the most efficient means of identifying these trees. The shape and length of the needles are key to differentiating these species.
You can determine the shape of a needle (i.e., whether it is four-sided, round or flat) simply by rolling it between your thumb and index finger. Or you can cut the needle width wise, that is, with your scissors at a right angle to the needle, and examine the shape of the cut edge. The illustrations will help you understand how to determine the shape of needles.
Almost all conifers have evergreen needles that stay on the tree year round. Only the larch, which has deciduous needles, is an exception to the rule. Conifers retain their green coats year round, embellishing winter landscapes. Furthermore, this quality helps to significantly reduce wind speeds, much to the delight of those who love winter sports.
Types of Needle Bundles
The pine familly has needles that are grouped together in bundles, consisting of 2, 3 or 5 needles joined together at the base.
In other species, such as the larch, the bundles may consist of more than 5 needles.
Because the base of the needles or needle bundle in some species is covered in a very thin sheath, you should use a twig for identification, not just a single needle. You will therefore be able to properly observe how the needles or needle bundles are arranged.
Conifers such as firs and spruces have single needles.
The arrangement of needles
The way needles are arranged on a twig is a key feature. Many conifers are characterized by this observation criterion.
As you can see, needles are either arranged in bundles or singly placed on the twig. Identifying conifers with needles in bundles is easy to do. However, to identify conifers with single needles, you need to look at other features such as the way the needles are arranged on the twig.
Needles may be arranged in pairs opposite each other at the same position on the twig. These needles are said to be opposite needles. They can be observed on the eastern white cedar.
When needles are placed around the twig at the same position, they are said to be alternate. Take a look at a black spruce branch and check to see whether the needles alternate. Then try to find other species with alternate needles.
If the needles are placed star-like around the twig at the same position, they are said to be whorled. Although you may occasionally see whorled needles on some junipers, in most cases they are in opposite pairs.
Needles are often arranged on one plane, not all round the twig. This occurs mainly when the needles are flat. You will see this feature if you look at a balsam fir branch.
The Fruits of Conifers
Conifers have two types of cones: pollen cones and seed cones. Depending on the tree species, you may find both types of cones on the same tree or on different trees. When seed and pollen cones are found on different trees, the species is called dioecious.
Pollen cones are often smaller than seed cones and have a catkin-like structure. Pollen cones produce the pollen that is released to fertilize seed cones. Once the pollen has been shed, the pollen cone whithers away and is no longer useful for identification purposes
Female cones have scales under which the ovules are found. Once fertilized by the pollen, the ovules become seeds. As of that time, the seed cone is considered a fruit because it has everything needed to reproduce a tree.
Seed cones are useful means of identification because they persist on the tree or on the ground around it. Seed cones are often called simply cones. Cones have different forms. They may curve inward like those of the jack pine. They may be very small like eastern white cedar and tamarack cones. Or they may be of varying lengths depending on the species.
Cones are valued for their decorative qualities. When you have a chance to gather some, take one apart to observe the seeds. You will find cones come apart easily when dry. Also, many animals feed on conifer seeds.
Many conifers have arils instead of cones. Arils are fruits composed of a fleshy membrane that partially encloses the seed. You will find arils on the Canada yew. But beware. The fruit of the yew is toxic. So make sure you do not eat any.
Protective Layers of Bark
The bark on trees has two layers: an outer layer of dead wood and an inner layer of living tissues. The inner layer is made up of living cells that are continually dividing. The inner cells need water to live. They take in water through pores and lenticels. When the cells are deprived of water, they die and become part of the outer dead layer, which serves as an effective barrier against injuries and environmental stresses. Since this layer consists of dead cells, it can no longer grow. It cracks or breaks away as the inner cells continue to grow and push the older cells outward.
You have certainly noticed that the bark on different species varies considerably. It can be sticky because resin or sap is secreted by the tree. The bark on fir trees, for instance, is covered with balls of resin that can be easily pierced with a fingernail. This represents a very useful criterion for identifying fir trees.
To identify a tree by its bark, you have to observe the texture, colour and pattern of the bark.
There are four major characteristics to consider in observing bark. Some bark sheds in strips or flakes. Look at the white birch; it sheds in long horizontal strips. Bark can also be scaly as on the white spruce . Bark may also have shallow or deep grooves depending on the species. The white ash has straight grooves. Bark can also be covered in cracks as in the white elm.
To practise identifying bark, try feeling different types of bark with your eyes closed. Touching is an excellent way of developing your ability to identify bark.<|endoftext|>
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The cotangent function, in modern notation written as cot(x), is a trigonometric function. Trigonometric functions are commonly established as functions of angle, in the context of right triangle geometry. This way, the cotangent of an angle φ is defined as the ratio of the adjacent side of a right triangle containing φ, divided by the opposite side (see figure):
Unfortunately the above definition is limited to a range of angles between 0 and 90°. Extending that range, a convenient definition employs a unit circle (radius equal to 1). Any point of the circle, corresponds to a pair of sine and cosine values, of the angle that is contained by the horizontal positive axis and the radial segment towards that point, as shown in the figure below. The absolute value of cotangent is represented by the length of the tangent segment from the circle point towards the vertical axis. The angle φ is assumed positive in counter-clockwise direction. Angles larger than 90°, as well as negative ones, are possible, if the appropriate sign of the coordinates is respected.
The above definitions of cotangent function are based on a geometrical construct (right triangle or unit circle) and assume that its argument is an angle. This association to an angle is restrictive, considering the broad use of trigonometric functions in mathematics, physics, engineering etc. Applications may accept any real value as argument, with any imaginable meaning given to it, or no meaning at all. In that case, x should be better measured in radians (). Thus, derivation and integration rules are more conveniently applied.
All trigonometric functions can be defined in an infinite series form. Cotangent function can be written as:
The above series converges for . Bndenotes the n-th Bernulli number.
The cotangent function is periodic, with a period equal to π. Therefore:
Cotangent is associated with sine and cosine and tan functions with the formulas:
The derivative of the cotangent function (x in radians) is given by the following expressions:
The integral of the cotangent function (x in radians) is given by:<|endoftext|>
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There is a boundary to the Solar System. It exists in the spatial area where the sun's "bubble" of charged particles becomes less than the neutral atoms that stream into our star's system from outside said system. And the Voyager 1 spacecraft is about to cross that boundary -- sooner than scientists originally expected.
In an article in Nature revealing the findings of scientists monitoring the progress of the Voyager 1 and the Cassini spacecraft, data from Voyager's low-energy charged particle instrument, which has been collected since December 2010, indicates that the outward speed of charged particles that emanate from the sun has slowed to zero.
Tom Krimigis, prinicipal investigator for Voyager's low-energy charged particle instrument and Cassini's magnetospheric imaging instrument, and his colleagues at the Johns Hopkins University Applied Physics Laboratory in Laurel, Md., combined the data from the two spacecraft. Their analysis indicated that the boundary should exist somewhere from 11 billion to 14 billion miles out from the sun. Voyager 1 has traveled nearly 11 billion miles (which is the scientists' best estimate) thus far, so Krimigis believes that the boundary will soon be crossed.
"There is one time we are going to cross that frontier, and this is the first sign it is upon us," Krimigis said, according to PhysOrg.com.
Ed Stone, who is a Voyager project scientist, also looks forward to the "crossing" of the boundary. "These calculations show we're getting close, but how close?" he asked rhetorically. "That's what we don't know, but Voyager 1 speeds outward a billion miles every three years, so we may not have long to wait."
The scientists will study the incoming data from the two space probes until they get confirmation that the frontier has actually been crossed. Then they will know just how large an area the Solar System covers. And Voyager 1 will have entered into an interstellar trip, the first of Earth's spacecraft to leave the sun's territory.
Voyager 1 launched from Cape Canaveral, Florida, on September 5, 1977. The spacecraft was originally designed to study the gas giants Jupiter and Saturn, with its extended mission eventually taking it into interstellar space. It sent back data on Jupiter and its moons in 1979. A flyby of Saturn occurred in 1980.
As for the extended mission, Voyager 1 has been charged to ascertain and help map the edge of the Solar System, locate the edges of the Kuiper Belt, a massive asteroid-belt-like formation that surrounds the Solar System, and the heliosphere, which is the aforementioned charged particles "bubble" created in space by the sun's solar wind.
Voyager 2 is headed for the edge of the Solar System as well. However, it has only traveled 9 billion miles so far. It will take a few more years before it, too, crosses the "border" into interstellar space.
(photo credit: Harman Smith and Laura Generosa (nee Berwin), NASA.gov, Wikimedia Commons)<|endoftext|>
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If you have a budding scientist on your hands, you no doubt appreciate watching their wide-eyed discovery of the natural world and their efforts to unlock the mysteries of the universe. You also probably hear the words, “But, why?” a lot! Below we’ve gathered 10 tried-and-true science experiments, with something to captivate any young Einstein aged from three to tween. All the experiments use simple, readily available household materials or foodstuffs, and will be made even more engaging by applying the scientific method to the process to really get kids thinking about what’s happening.
1. Technicolor crystal landscape
Let’s start out with something spectacular, shall we? This gorgeous crystal landscape project from Babble Dabble Do uses simple household products such as sponges, salt and laundry blue, and will keep kids enthralled for around five days as the crystals grow into their final form. Fascinating!
2. Eggs and osmosis
Playdough to Plato‘s astounding three-stage science experiment uses an egg to teach kids about the process of osmosis. Along the way they get to be surprised when the fragile-looking egg confounds expectations about when it will or will not break.
3. Magnetic maze
Especially good for four- to six-year-olds, this charming project from My Cakies teaches children about the magic of magnetism. The lovely illustrations are downloadable printables, and if they survive their trip through the maze, they can be recycled as fridge magnets.
4. How sight influences smell
Do you connect the scent of lemon with the color yellow? This oh-so-simple project from Left Brain Craft Brain will give your child a valuable life lesson on the vagaries of perception. Left Brain Craft Brain also gives helpful, step-by-step guidelines to following the scientific method that will set your child in good stead for applying it to other experiments.
5. Capillary action with “walking water”
One of the simplest (and fastest) experiments in this list, this demonstration of both capillary action and color mixing transforms ordinary materials into something wonderful right before your eyes! Coffee Cups and Crayons has easy-to-follow step-by-step instructions, and this video shows just how elaborate you can make the experiment.
6. DIY lava lamp
I wanted a lava lamp so badly when I was a kid—if only I had known how easy it was to make one myself! The great thing about this experiment is that kids can repeat the process over and over, and when they tire of it, you can pop a lid on the bottle and put the lamp aside to play with again another day. Check out Inhabitots’ very own lava lamp DIY here, and this video shows the awesome experiment in action.
7. Dyed flowers
There’s more to a flower than meets the eye, and this easy experiment to demonstrate plants’ vascular systems reveals that clearly in living color. The Imagination Tree explains how to perform this experiment using either white flowers or celery sticks. Red Tricycle ups the ante with a more complex process that results in a two-tone flower (warning: sharp knife and a steady hand involved for the latter).
8. Egg head seed germinators
Part craft project, part science project, these cute little egg head seed germinators from Playdough to Plato are a great way to introduce kids to the concepts of germination, life cycles, nutrition and where their food comes from. Choose a variety of large and small seeds to illustrate different growth habits: peas, sunflowers, wheat, kale, and chives or onions will give a good variety of “hair,” and arugula and radishes will often sprout within 24–36 hours to give quick results.
9. Glow-in-the-dark Oobleck
Oooh, Oobleck! So simple to make, yet so bizarre! This DIY from The King of Random turns it up to 11 by not only showing you how to make Oobleck from scratch using potatoes, but also how to make it glow in the dark under black light, simply by using tonic water instead of regular water. My suggestion? Put the black light globe in the bathroom where it will be easier to clean up the inevitable mess afterwards!
10. Dancing Oobleck
If you want to take the shortcut and make your Oobleck from a more conventional cornstarch base, you can still keep it interesting with this fun experiment with sound waves from Housing a Forest. By placing your Oobleck on a tray on top of a subwoofer, you can make it “dance” with the vibration of the sound waves. Then add drops of food coloring to reveal just how dynamic it is!
If your child is particularly enthralled by science (or you would like to give them an encouraging nudge in that direction), check out Camp Google for a free, four-week program of online science activities and experiments covering the themes of oceans, space, nature and music for children up to 10 years of age. Program partners include National Geographic, Khan Academy and the National Park Service, with input from NASA too.
Lead image via Babble Dabble Do. Photos by 1. Babble Dabble Do, 2. Playdough to Plato, 3. My Cakies, 4. Left Brain Craft Brain, 5. Coffee Cups and Crayons, 6. Jennie Lyon for Inhabitots, 7. The Imagination Tree, 8. Playdough to Plato, 9. YouTube screenshot from The King of Random, 10. Housing a Forest.<|endoftext|>
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# Hasil perhitungan rumus
Rumus
Hitunglah integralnya
Jawaban
$$\displaystyle\int { \tan\left( x \right) } d { x }$$
$\dfrac { 1 } { 2 } \ln { \left( | \tan ^ { 2 } \left ( x \right) + 1 | \right) }$
Hitunglah integralnya
$\displaystyle\int { \color{#FF6800}{ \tan\left( \color{#FF6800}{ x } \right) } } d { \color{#FF6800}{ x } }$
Substitusikanlah menjadi $u = \tan\left( x \right)$ dan hitunglah integralnya.
$\left [ \displaystyle\int { \color{#FF6800}{ \frac { \color{#FF6800}{ u } } { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ \tan\left( \color{#FF6800}{ x } \right) } }$
$\left [ \displaystyle\int { \color{#FF6800}{ \frac { \color{#FF6800}{ u } } { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } } d { \color{#FF6800}{ u } } \right ] _ { u = \tan\left( x \right) }$
Hitunglah integral saat pembilang membagi turunan penyebutnya.
$\left [ \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \ln { \left( | \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } | \right) } \right ] _ { u = \tan\left( x \right) }$
$\left [ \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \ln { \left( | \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } | \right) } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ \tan\left( \color{#FF6800}{ x } \right) } }$
Kembalikanlah nilai yang telah disubstitusi.
$\color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \ln { \left( | \color{#FF6800}{ \tan ^ { \color{#FF6800}{ 2 } } \left ( \color{#FF6800}{ x } \right) } \color{#FF6800}{ + } \color{#FF6800}{ 1 } | \right) }$
Coba lebih banyak fitur lain dengan app Qanda!
Cari dengan memfoto soalnya
Bertanya 1:1 ke guru TOP
Rekomendasi soal & konsep pembelajaran oleh AI<|endoftext|>
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# If 2 cos2 θ = 3 sin θ, 0° < θ < 90°, then the value of $$\left(\dfrac{1}{2} \rm cosec^2 \ \theta - \cot^2 \theta \right)$$ is:
This question was previously asked in
SSC HSC Level Previous Paper (Held on: 9 Nov 2020 Shift 1)
View all SSC Selection Post Papers >
1. $$\dfrac{1}{4}$$
2. $$\dfrac{1}{2}$$
3. 0
4. -1
Option 4 : -1
Free
GK Chapter Test 1 - Ancient History
15493
15 Questions 30 Marks 10 Mins
## Detailed Solution
Given:
2 cos2θ = 3 sinθ
Formula Used:
sin2θ + cos2θ = 1
cosecθ = 1/sinθ
cotθ = cosθ/sinθ
Calculation:
We know that,
sin2θ + cos2θ = 1
⇒ cos2θ = 1 – sin2θ
Using the above result in the given condiion,
⇒ 2 (1 – sin2θ) = 3 sinθ
⇒ 2 – 2 sin2θ = 3 sinθ
⇒ 2 sin2θ + 3 sinθ – 2 = 0
⇒ 2 sin2θ + 4 sinθ – sinθ – 2 = 0
⇒ 2 sinθ (sinθ + 2) – 1 (sinθ + 2) = 0
⇒ 2 sinθ – 1 = 0 or sinθ + 2 = 0
⇒ sinθ = 1/2 or sinθ = –2
As 0° < θ < 90°,
sinθ can not be negative.
⇒ sinθ = 1/2
⇒ cos2θ = 1 – sin2θ
⇒ cos2θ = 1 – (1/2)2
⇒ cos2θ = 1 – 1/4
⇒ cos2θ = 3/4
⇒ cosθ = (√3)/2
Now,
cosecθ = 1/sinθ
⇒ cosecθ = 1/(1/2)
⇒ cosecθ = 2
⇒ cosec2θ = 4
cotθ = cosθ/sinθ
⇒ cotθ = (√3)/2/(1/2)
⇒ cotθ = √3
$$\left(\dfrac{1}{2} \rm cosec^2 \ θ - \cot^2 θ \right)$$
⇒ 1/2(4) – (√3)2
⇒ 2 – 3
⇒ –1
∴ The value given trigonometric expression is –1.
We have sinθ = 1/2
Ratio 0° 30° 45° 60° 90° Sinθ 0 1/2 1/√2 (√3)/2 1 Cosθ 1 (√3)/2 1/√2 1/2 0 Tanθ 0 1/(√3) 1 √3 N.D.
By comparing the above value we get,
⇒ θ = 30°
cosecθ = 1/sinθ
⇒ cosec 30° = 1/(1/2)
⇒ cosec 30° = 2
cotθ = 1/tanθ
⇒ tanθ = 1/(√3)
⇒ cotθ = 1/{1/(√3)}
⇒ cotθ = √3
Now by using these in the asked expression,
⇒ 1/2(2)2 – (√3)2
⇒ 1/2 × 4 – 3
⇒ 2 – 3
⇒ –1
∴ The value given trigonometric expression is –1.<|endoftext|>
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The purpose of clippers is to protect a circuit against too high or too low voltages. The most simple circuit contains one diode (Figure 2.31). As long as the input voltage is positive, the diode will not conduct. What remains, is a voltage divider. This is also the case for small negative input voltages. From the moment that the output voltage might exceed the diode’s knee voltage, the diode will conduct. The output voltage then equals the knee voltage (with a minus sign in the circuit below). In other words: from the moment the output voltage is tending to drop below the knee voltage, it will be limited to this value. This results in the input-output characteristic shown in Figure 2.32. An example wave form is shown in Figure 2.33. The function of the resistor R in the circuit is to protect the diode against too high currents.
Figure 2.31 A clipper circuit contains one or more diodes, and protects a certain load (RL) against too large (positive or negative) voltages.
Figure 2.32 The input-output characteristic of the clipper circuit above. Voltages below the knee voltage are not possible across the load.
Figure 2.33 Example waveforms for the clipper circuit.
If the diode is replaced by a Zener diode, the output voltage can also be limited at its positive peak (Figure 2.34). If the output voltage is tending to exceed the breakdown voltage of the Zener, the latter will break down, and the output voltage will be bounded to the breakdown voltage (Figure 2.35 and Figure 2.36).
Figure 2.34 A clipper circuit with a Zener diode.
Figure 2.35 Input-output characteristic of a clipper circuit with a Zener diode.
Figure 2.36 Example waveforms of a clipper circuit with a Zener diode.
More flexibility is obtained by placing two Zener diodes in series (Figure 2.37). The range of the output voltage is between VBD1 + VK2 (breakdown at positive input voltage) and –VBD2 – VK1 (at negative input voltage) (Figure 2.38).
Figure 2.37 A clipper circuit with two Zener diodes.
Figure 2.38 I/O characteristic of a clipper circuit with two Zener diodes.<|endoftext|>
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We learned in the Valley of the Temples in Agrigento that the Temple of Olympian Zeus there was extremely unique in Sicily and the larger Greek world. Of the Doric order, the temple itself would have been enormous if completed, measuring fourteen columns long by seven columns wide. This would have made it the largest Doric temple ever built. The monstrous monument likely commemorated the victory of Akragas (Agrigento) over Carthage in the battle of Himera (~480 BC), which definitely would have been a great accomplishment for the comparatively small city and its allies. Returning to the details of the temple itself, the odd width in columns would have been extremely uncommon since it would have caused there to be two separate entrances. Additionally, unlike any other temple we saw on the trip, the columns were (supposed to be) engaged with a wall, presumably what would have formed the entrance or entrances in a more natural way.
We also learned about the telamones, or giants, which would have been extremely unusual features on a Greek temple. Other than in the case of caryatid columns, figures were not known to be incorporated into the temple façade in such a way. And for those of us who had read any of the Iliad, we immediately recognized the root τελαμων, -οντος in the name of Ajax’s father, the hero Telamon (Τελαμων). Professor Boyd explained that τελαμων means “bearing” or “enduring,” which helps to characterize Telamon and his lineage, especially Ajax, who is extremely strong (enduring) and nearly a giant himself. Ajax’s patronymic actually helps to define his character.<|endoftext|>
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07 SES 11 C, Citizenship and Democratic Education
Over the past few decades, a broad consensus has evolved that to actively and responsibly participate in democratic societies, individuals do not only need to be endowed with civic and citizenship knowledge (to understand how society and democracy works) but they also need to acquire non-cognitive skills, including attitudes, norms and behavioural intentions. Although identifying such skills is not free of normative elements, efforts have been taken to select consensus values and attitudes that education should foster in order to prepare young people for civic engagement (see e.g. Schulz, 2007; Torney-Purta and Vermeer, 2004; Eidhof et. al. 2016).
From the literature, two main strands and one weaker argument emerge providing an explanatory framework to describe the main educational approaches that can promote democratic attitudes among students. First, it is argued that formal learning of civic issues is itself a crucial way in shaping values and attitudes as an increased cognitive understanding is also related to more positive civic attitudes. Second, a democratic, open school environment that allows students to experience the right to have their say, where they can openly discuss sensitive issues and can experience the connections between an activity and its consequences, is expected to promote the development of democratic attitudes (De Groof et.al. 2008) (Alivernini & Manganelli, 2011; Claes et al., 2009; De Groof et al., 2008; Knowles & McCafferty-Wright, 2015; Manganelli et.al. 2012). Finally, students’ active community involvement that is, doing unpaid service activities for the wider community either voluntarily, or as an activity arranged by the school can also promote civic attitudes. Research evidence – predominantly from outside Europe – suggests that community work can indeed help to improve students’ civic outcomes (Schmidt et al. 2007; Galston, 2001; Claes et al., 2009).
Civic attitudes develop in close associations with civic and citizenship knowledge and also with civic self-efficacy and we can expect that both civic knowledge and civic self-efficacy fulfill a mediating role in the development of civic attitudes. Civic knowledge might promote a broader understanding of social processes and thus, it can lead to openness and tolerance, increased support for democratic values and also more willingness for political participation (Milner 2002, 2007 – cited by De Groof, et al., 2008; Galston, 2001). Civic self-efficacy refers to students’ self-confidence in their ability to handle different situations and take actions related to civic issues and civic participation (Bandura, 1997). It has been suggested that civic knowledge and self-efficacy mutually reinforce each other and they both contribute to the improvement of the various civic and citizenship attitudes albeit in different ways. (Knowles & McCafferty-Wright, 2015; Manganelli et al., 2014)
It is less clear however, how much schools can do to improve knowledge and efficacy and how in turn the different educational approaches will also shape students’ civic attitudes. Our paper will therefore explore this complex process of attitude-development of young people with a special emphasis on the role of school. In particular we will investigate the extent to- and the channels through which schools might promote the development of three selected civic attitudes: (1) students’ perception of personal responsibility for citizenship, (2) intended participation in electoral voting and (3) openness towards equal rights for immigrants. We will investigate how formal learning, school democracy and community work are associated with these attitudes either directly or indirectly, throughout their associations with students’ civic and citizenship knowledge and civic self-efficacy. Limitations of establishing causal relationships will be discussed together with the possible interpretations of the associations explored.
For the analysis, The International Civic and Citizenship Education Study (ICCS) 2016 data from European Union Member States (EU MSs) is used. The ICCS 2016 collected data from 8th grade students, their teachers, and head-masters, including information on student’s socioeconomic background, civic and citizenship knowledge, attitudes and civic participation, teaching practices, and school resources. We will focus on the 12 EU MSs with reliable sample and without a large percentage of missing values. To identify the direct and the indirect associations between the educational approaches and civic dispositions, mediational analysis will be carried out with civic knowledge and civic self-efficacy as mediator variables. A series of ordinary least square (OLS) regression models, taking into account both the complex sample design of the survey, will be run by country with the three civic attitudes as dependent variables. All the models will control for a range of social background characteristics of the students as well as for some school-level contextual variables. The dependent variables measuring civic attitudes– students’ perception of personal responsibility based citizenship, expected electoral participation and attitudes toward equal rights for immigrants – will be measured by the relevant scales constructed by International Association for the Evaluation of Educational Achievement (IEA) applying Item Response Theory (IRT) models, based on a series of Likert-type items. The three educational approaches are proxied in the following way. For Formal learning, the scale on Students’ perception of civic learning in school will be used (aggregated on the class level). Democratic school environment was captured by three scales: Openness in classroom discussions; Students' participation at school and Principals’ perceptions of engagement of the school community. Finally, Students’ community involvement in organisations, clubs or groups we selected the ones that are related to activities which involve some (unpaid) activity. Civic and citizenship knowledge and skills is a scale based on student responses to the civic knowledge cognitive test created using IRT resulting in five plausible values for each student. Citizenship self-efficacy is also a constructed scale (by IEA) that reflects students’ self-confidence in active citizenship behavior.
After controlling for the individual- and school level factors, education has a moderate but non-negligible impact on students’ civic attitudes – expected electoral participation can better be predicted than the other two attitudes. Across the three educational approaches open classroom climate - a form of democratic schooling – presents the strongest positive association with all the three civic outcomes. The role that maintaining an open classroom climate has in civic education is rather complex as it both contributes to the level of civic knowledge and to civic self-efficacy and it is further related to civic outcomes through both of these channels and also indirectly. Further, formal learning is positively related both to expected electoral participation and responsibility for citizenship albeit in a few countries only. Tolerance for the immigrants remains largely independent of the amount of civic and citizenship education. Interestingly, civic knowledge plays little role in mediating the association between formal learning and any of the attitudes. Finally, students’ community involvement exhibits sporadic, mostly direct positive associations with the different outcomes in a few countries only. The analyses confirm that civic and citizenship knowledge and civic-efficacy both have crucial but different roles in the attitude-shaping process. Efficacy is more consistently positively related to the outcomes across all the countries, for civic knowledge this is only true for some attitudes and they are driven by different educational approaches. Cross-country comparison reveals that certain approaches work better in one social and cultural setting than in the other – possibly because the educational approach itself is of different content in the different countries. For example, students seem to benefit more from community involvement in Denmark than in other countries, whereas an open classroom climate is associated with less positive outcomes in Latvia and Bulgaria than elsewhere.
Alivernini, F., & Manganelli, S. (2011). Is there a relationship between openness in classroom discussion and students’ knowledge in civic and citizenship education? Procedia - Social and Behavioral Sciences, 15, 3441–3445. https://doi.org/10.1016/j.sbspro.2011.04.315 Bandura, A. (1997). Self-efficacy: the exercise of control. New York: W H Freeman/Times Books/ Henry Holt & Co.Claes, E., Hooghe, M., & Stolle, D. (2009). The Political Socialization of Adolescents in Canada: Differential Effects of Civic Education on Visible Minorities. Canadian Journal of Political Science, 42(03), 613. https://doi.org/10.1017/S0008423909990400 De Groof, S., Elchardus, M., Franck, E., & Kavadias, D. (2008). The Influence of Civic Knowledge versus Democratic School Experiences on Ethnic Tolerance of Adolescents A multilevel analysis. Retrieved from http://www.academia.edu/download/3457867/Paper_Brugge_DK_EF_SDG_ME.pdf Eidhof, B. B., ten Dam, G. T., Dijkstra, A. B., & van de Werfhorst, H. G. (2016). Consensus and contested citizenship education goals in Western Europe. Education, Citizenship and Social Justice, 11(2), 114–129. Galston, W. A. (2001). Political knowledge, political engagement, and civic education. Annual Review of Political Science, 4(1), 217–234. Isac, M. M., Maslowski, R., Creemers, B., & van der Werf, G. (2014). The contribution of schooling to secondary-school students’ citizenship outcomes across countries. School Effectiveness and School Improvement, 25(1), 29–63. https://doi.org/10.1080/09243453.2012.751035 Isac, M. M., Maslowski, R., & van der Werf, G. (2011). Effective civic education: an educational effectiveness model for explaining students’ civic knowledge. School Effectiveness and School Improvement, 22(3), 313–333. https://doi.org/10.1080/09243453.2011.571542 Knowles, R. T., & McCafferty-Wright, J. (2015). Connecting an open classroom climate to social movement citizenship: A study of 8th graders in Europe using IEA ICCS data. The Journal of Social Studies Research, 39(4), 255–269. https://doi.org/10.1016/j.jssr.2015.03.002 Manganelli, S., Alivernini, F., Lucidi, F., & Di Leo, I. (2012). Expected Political Participation in Italy: a Study based on Italian ICCS Data. Procedia - Social and Behavioral Sciences, 46, 1476–1481. https://doi.org/10.1016/j.sbspro.2012.05.324 Schulz, W., Ainley, J., Fraillon, J., Losito, B., Agrusti, G., & Friedman, T. (2017). Becoming Citizens in a Changing World IEA International Civic and Citizenship Education Study 2016 International Report. IEA. Retrieved from http://iccs.iea.nl/fileadmin/user_upload/Editor_Group/Downloads/ICCS_2016_International_report.pdf Schmidt, J. A., Shumow, L., & Kackar, H. (2007). Adolescents’ Participation in Service Activities and Its Impact on Academic, Behavioral, and Civic Outcomes. Journal of Youth and Adolescence, 36(2), 127–140. https://doi.org/10.1007/s10964-006-9119-5
00. Central Events (Keynotes, EERA-Panel, EERJ Round Table, Invited Sessions)
Network 1. Continuing Professional Development: Learning for Individuals, Leaders, and Organisations
Network 2. Vocational Education and Training (VETNET)
Network 3. Curriculum Innovation
Network 4. Inclusive Education
Network 5. Children and Youth at Risk and Urban Education
Network 6. Open Learning: Media, Environments and Cultures
Network 7. Social Justice and Intercultural Education
Network 8. Research on Health Education
Network 9. Assessment, Evaluation, Testing and Measurement
Network 10. Teacher Education Research
Network 11. Educational Effectiveness and Quality Assurance
Network 12. LISnet - Library and Information Science Network
Network 13. Philosophy of Education
Network 14. Communities, Families and Schooling in Educational Research
Network 15. Research Partnerships in Education
Network 16. ICT in Education and Training
Network 17. Histories of Education
Network 18. Research in Sport Pedagogy
Network 19. Ethnography
Network 20. Research in Innovative Intercultural Learning Environments
Network 22. Research in Higher Education
Network 23. Policy Studies and Politics of Education
Network 24. Mathematics Education Research
Network 25. Research on Children's Rights in Education
Network 26. Educational Leadership
Network 27. Didactics – Learning and Teaching
The programme is updated regularly (each day in the morning)
- Search for keywords and phrases in "Text Search"
- Restrict in which part of the abstracts to search in "Where to search"
- Search for authors and in the respective field.
- For planning your conference attendance you may want to use the conference app, which will be issued some weeks before the conference
- If you are a session chair, best look up your chairing duties in the conference system (Conftool) or the app.<|endoftext|>
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The devastating tsunami that struck the Tohoku Coast of Japan in March of 2011 caused destruction on a scale that is unfathomable to most people. A report published shortly after the tsunami found that nearly 20,000 people had died or gone missing, nearly 369,000 homes had been either totally destroyed or severely damaged, and the total cost of the disaster (not including clean-up costs) was estimated to be as high as 225 billion USD (Kazama and Noda, 2012). Eight years later, cleanup efforts at the Fukushima Daiichi nuclear plant are still underway. In recent years, however, debris from the tsunami has been finding its way far, far away from Japan, and along with it some oceanic travelers.
Floating debris, or flotsam, has long been recognized as important habitat for some species of high-seas fish. Natural floating rafts of debris, such as felled trees, typically provide refuge for some fish from predators on the open ocean. Rarely, however, is marine debris flung so far into the open ocean that it makes its way to another continent. Nevertheless, that is exactly what has happened in the aftermath of the 2011 tsunami.
Derelict fishing vessels, floating docks, and other collections of human-made structures started washing up on the U.S. West Coast as early as 2013, carrying with them a variety of marine invertebrates and fish never previously seen in the U.S. Let’s take a look at a few examples of the species researchers have been finding washed ashore from the other side of the Pacific Ocean:
- The barred knifejaw (Oplegnathus fasciatus; Figure 1) is a tropical species native to the coasts of Japan, Korea, and China. It has been reported in Washington, Oregon, and California both swimming in the wild and onboard derelict fishing vessels that were washed out to sea by the tsunami. Fortunately, because temperatures on the U.S. west coast are thought to be too cold for it to reproduce, it is at low risk of becoming an invasive species.
- Several Japanese yellowtail jacks (Seriola aureovittata; Figure 2) were discovered in the hold of a fishing vessel washed ashore near Seal Rock, Oregon in April of 2015. A series of genetic tests were performed on these fish to confirm that they were indeed of Japanese origin and not specimens of the closely related North American yellowtail jack (S. dorsalis).
- Finally, a new species of bryozoan (Bugula tsunamiensis; Figure 3) was described after being collected from numerous pieces of marine debris washed ashore from the Japanese tsunami.
These are just a few examples of the 289 or so species that have been found transplanted by the tsunami. The probability of any individual species arriving from the tsunami-swept debris becoming a problematic invader is low. However, as many organisms from marine debris continue to wash ashore, the probability of any one of them becoming an invasive species increases. As a result, it is in our best interest to continue monitoring these new arrivals and to study their potential impacts and pathways of spread. Research into these new arrivals could help us to predict their impacts and safeguard native marine ecosystems.
Craig, M.T., Burke, J., Clifford, K., Mochon-Collura, E., Chapman, J.W., and Hyde, J.R. 2018. Trans-Pacific rafting in tsunami associated debris by the Japanese yellowtail jack, Seriola aureovittata Temminck & Schlegel, 1845 (Pisces, Carangidae). Aquatic Invasions. 13(1): 173 – 177.
Kazama, M., and Noda, T. Damage statistics (Summary of the 2011 off the Pacific Coast of Tohoku Earthquake damage). Soils and Foundations. 52(5): 780 – 792.
McCuller, M.I., Carlton, J.T., and Geller, J.B. 2018. Bugula tsunamiensis n. sp. (Bryozoa, Cheilostomata, Bugulidae) from Japanese tsunami marine debris landed in the Hawaiian Archipelago and the Pacific Coast of the USA. Aquatic Invasions. 13(1): 163 – 171.
Ta, N., Miller, J.A., Chapman, J.W., Pleus, A.E., Calvanese, T., Miller-Morgan, T., Burke, J., and Carlton, J.T. 2018. The Western Pacific barred knifejaw, Oplegnathus fasciatus (Temminck & Schlegel, 1844) (Pisces: Oplegnathidae), arriving with tsunami debris on the Pacific coast of North America. Aquatic Invasions. 13(1): 179 – 186.<|endoftext|>
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7.4.1 Computations
Quiz
Circular Slide Rule
The Circular Slide Rule The circular slide rule found on the reverse of the CRP5, if used effectively, can give reasonably accurate answers to calculations needed for both Flight Planning and General Navigation. The General Navigation examination will have numerous calculations which involve the CRP5. This document is an aide-memoire to help you in solving these questions.
The Slide Rule consists of two scales, an outer fixed scale and an inner moveable scale. Numbers are printed on both scales from 10 to 99.9. When doing any calculation you have to mentally place the decimal point before reading your answer off the slide rule. So 25 can represent any number you wish, for example .0025, .025, .25, 2.5, 25 etc. Note that the scale around the slide rule is not constant but logarithmic.
Multiplication, Division and Ratios
Multiplication
Consider the simple multiplication 8 X 1.5. By mental arithmetic we can easily see that the answer is 12. But we will use simple questions like this to illustrate how the CRP5 is used.
1. Rotate the inner scale so that the number 10 is under the number 80 (We are using 80 to represent 8 and 10 to represent 1).
2. On the inner scale go to the number 15 (1.5).
Division
Division is the exact opposite of multiplication. So using the same numbers for the multiplication let us divide 12 by 1.5. The answer is obviously 8.
1. Place 15 on the inner scale under 12 on the outer scale.
2. On the inner scale follow the numbers to 10.
Ratios
Any ratio can be read off the slide rule direct, so for A/B = C/D let us assume that A = 30, B = 15, D = 25 what is C?
1. Place 15 on the inner scale under 30 on the outer scale.
2. Follow the inner scale to 25.
Conversions
Conversions use the same principal as the multiplication, division and ratio calculations.
To ensure accuracy the following rough conversions should be used from the ERSA.
The above datums are printed in red on the outer scale of the slide rule.
Feet – Metres – Yards Convert 3 feet into yards and metres.
1. Under the feet arrow on the outer scale, place 3 on the inner scale.
2. On the inner scale opposite the yards and metres datum arrows read off the answers 1 yard .915 metres.
Take-Off and Landing Wind Component
Aircraft are subject to crosswind and tailwind maxima. Both can be calculated using the square scale on the CRP 5.
Runway 31 is in use and the wind velocity reported by ATC is 270/40. Remember that the runway direction is in magnetic and the wind velocity reported by ATC is in magnetic. Find the crosswind and headwind component.
1. Set the grommet on the zero point of the squared section as shown.
2. Mark in the wind velocity as normal.
3. Set the runway direction of 310M against the heading index.
• The headwind is read from the horizontal zero line 30 knots
• The crosswind from the vertical centre line 26 knots
Tailwind Component
Suppose that the wind velocity is 210/40 with runway 31 in use. Using the procedure above the answer shows that the wind point is above the zero line. This indicates a tailwind.
Bring the wind point to the zero horizontal line.
• The grommet will give a tailwind of 7 knots.
Runway 21 in use. The wind direction is 180°M. A minimum headwind of 10 knots and maximum crosswind is 16 knots for this runway. What is the minimum and maximum windspeed.
1. Set the runway direction against the True Heading index and place the grommet on the zero point.
2. Mark in the maximum crosswind and minimum headwind for the runway as shown. The crosswind is blowing from the left. Wind always blows away from the grommet so the crosswind is drawn on the right.
3. Set the wind direction against the true heading index.
Read off the maximum and minimum windspeed as shown:
• Min wind speed 12kt.
• Max wind speed 32kt.
Speed, Distance and Time
To calculate any of the variables remember that minutes is always on the inner scale. To remind you, the inner scale has minutes written in red between 30 and 35. The calculations work on the factor 60.
All speeds are a distance traveled in 60 minutes so all calculations revolve around this number. To help you with these calculations the number 60 is in white surrounded by a black triangle.
Groundspeed
An aircraft flies 210 nm in 25 minutes, what is the groundspeed
1. Align the 25 on the inner scale against 210 on the outer scale
2. Read off the groundspeed against the 60 triangle.
• 503 knots.
Time
Using the same example. If the groundspeed is 503 knots, how long will it take the aircraft to travel 210 nautical miles.
1. Align the 60 triangle on the inner scale against 503 on the outer scale
2. On the outer distance scale go to 210. Read off the time on the inner scale.
• 25 minutes
Distance Travelled
For a groundspeed of 503 knots, how far will the aircraft travel in 35 minutes.
1. Align the 60 triangle on the inner scale against 503 on the outer scale
2. On the inner minutes scale go to 35. Read off the distance traveled on the outer scale.
• 294 nautical miles
Fuel consumption, fuel and time calculations are done in the same manner.
Calculation of TAS up to 300 Knots
Assume that the Pressure Altitude is 35 000 ft and the Corrected Outside Air temperature (COAT) is – 65°C. What is the TAS if the RAS (CAS) is 160 knots.
1. Against the COAT of –65° C place the altitude of 35 000 ft as shown in the diagram.
2. RAS (CAS) is found on the inner scale (To remind you this is written in red between 35 and 40.
3. Read off the TAS against the RAS (CAS) of 160 knots
• 275 knots
Triangle of Velocities
Computer Terminology
1. Grid Ring: The scale around the rotatable protractor.
2. Computer Face: The transparent plastic of the rotatable protractor.
3. True or True Course Pointer: The reference mark at the top of the stock, reading against the grid ring.
4. Drift Scale Scale on the top of the stock to the left and right of the true index. Note that the graduations are equal to those on the grid ring.
5. The Grommet; The point or circle at the exact centre of the rotatable protractor.
6. Drift Line: All the drift lines originate from one origin. The numbers on the drift lines indicate the degree of inclination to the centre line.
7. Heading Line: The central line or zero drift line.
8. Speed Circles: The arcs of concentric circles around the drift lines are equally spaced and graduated from zero knots up to any required speed. The scale is quite arbitrary. Each side of the sliding scale has a different speed scale, for the CRP5 this is:
• Side shown: 40 to 350 knots
• Other side not shown: 300 to 1050 knots
Before you start a flight along a track line you will need to prepare a heading to hold the aircraft on it that will prevent been blown off track by wind and compensate for any headwind or tailwind allowances in time and fuel by using groundspeed.
To help us to prepare a planned flight we calculate the Triangle of Velocities on the back of a Graphical Slide Fight Computer.
The Group of in the Triangle of Velocities are:
• W / V (Wind Velocity)
• TR / GS (Track and Groundspeed)
• HDG / TAS (Heading and True Air Speed)
Use the Triangle of Velocities to find the expected Heading, Drift and Groundspeed with the following data:
Example:
A Track line measured with a protractor from the WAC is 290 degrees relative from true north. A forecast wind is 330T/30kt is relative from true north, and the area has a 10 degrees east magnetic variation. The TAS is 125kt taken from the aircrafts flight manual.
• W/V 330T 30kt
• TR 290T
• TAS 125kt
When using a compass to navigate it always points to the magnetic poles (ignoring errors), so we will need to convert true bearings to magnetic bearings. When given wind directions as True North bearings covert it by adding the magnetic vaiation if it is on the west side of the nil magnetic variation line or subtract it if on the East side (most of Australia is on the east side). The lines of magnetic variation are found on a WAC for your area. Tracks drawn on a WAC are also relative to true north and need to be converted to magnetic bearings.
• Wind 330T - 10E = 320M
• TR 290T - 10E = 280M
• TAS 125kt
Plotting Triangle of Velocities
• Rotate the centre disc so the Wind Direction is under the index at the top of the flight computer.
• From the grommet count UP the centre and draw a line to represent the Wind Speed (You could just put a dot to represent the wind speed).
• At the top of the line is called the Wind Dot. The wind blows from the Wind Dot to the Grommet, as shown with green arrows.
2. Rotate the centre disc so the Track is under the True Index at the top of the flight computer:
3. Sliding the disk down the plate until the TAS is under the Wind Dot (TAS is represented as the arc lines).
You are done. Now it is just a matter of interpreting the numbers.
Read the Drift Angle (Degrees) where it crosses the TAS (Knots) under the Wind Dot. The wind is pushing to the left. The aircraft will have to balance the push by pointing the heading into the wind by 9 degrees right.
• Drift 9 degrees left
Read the Groundspeed under the centre grommet. There is a headwind slowing the aircraft down relative to the ground.
• Groundspeed 100kt
To find the Heading add the Drift to the Track if the Wind Dot is on the right of the centre line or subtract the drift to the Track if the Wind Dot is on the left of the centre line.
Wind Velocity Vector Plotting
• Rotate the centre disc so the Wind Direction is under the index at the top of the flight computer.
• From the grommet count DOWN the centre line and mark the Speed of the Wind. This mark is called the Wind Dot (The wind blows from the Mark to the Grommet).
2. Rotate the centre disc so the HDG is under the index at the top of the flight computer.
3. Sliding the TAS (that is written on the plate) under the Grommet.
4. Read the Ground Speed under the Wind Dot.
5. Read the Drift Angle (Degrees) where it crosses the GS (Knots) under the Wind Dot.
6. To find the Track:
• Add the drift to the HDG if the Wind Dot is on the left of centre line or minus if on right the right of the centre line.
Aircraft take-off into wind. This gives the best climb gradient over the shortest distance. As the aircraft gets higher, the wind is stronger. This gives a much better rate of climb over the distance, than an aircraft taking off with a tailwind. In a tailwind take off the distance is increased across the ground.
The surface wind is slowed by friction. On average the surface wind slows by 2/3rd and changes direction by veering right.
Each aircraft has a best climb speed and this should always be used after take-off until a safe height is reached. Pilots of aircraft should read the aircrafts handbook for best climb speeds. Only when at a safe height should you adopt the cruise climb. For pilots the climb gradient must be that shown in the SID charts as a minimum. Noise abatement also comes in to play here. Remember the higher you are the less noise.
Let’s look at two aircraft taking off with the same airspeed and rate of climb. The first one takes off into wind, and the second one with a tailwind.
The aircraft taking off into wind will have travelled a much shorter ground distance to a height 2000ft compared to the aircraft taking off with a tailwind. The climb angle would also be bettered by at least 3 degrees. We will assume the wind to be 20 knots on the ground and at 2000ft it would be double. It is true the faster the aircraft fly’s the more lift it has, but you also have to realize the more ground you will cover before you get to the same height.
Let’s assume the wind is 20 knots and you take off at 140 knots. This means your groundspeed is 140 - headwind speed = 120 knots.
Now let’s assume you take off with a tailwind of 20 knots at a climb out speed of 140 knots, your groundspeed now will be 160 knots 140 +20. Some airfields have minimum climb gradients published for pilots in chart form. This will tell the pilot the minimum climb for the VSI and airspeed allowed for safety at different speeds. Remember a hot day and a high altitude airfield will seriously affect your climb performance.
It is essential that the aircraft manual is checked and the correct speed flown. Also watch your Vertical Speed Indicator.
Convert Climb Gradient to Climb Rate (RoC)
To convert climb gradient to climb rate, multiply the gradient by the airspeed in knots.
Climb rate (fpm) = Climb gradient (%) x Airspeed (kts)
Assumes:
• 1% climb gradient over a mile = 60 ft (1% of 6000ft=1nm)
• No wind; groundspeed = airspeed
Example:
Airspeed = 220 knots
Climb rate = 5.5 x 220 = 1210 feet per minute
To convert the climb gradient to the climb rate in hundreds of feet, divide your current ground speed by 60 and multiply by climb gradient.
Example:
If you want to gain 200 ft per nm and have a 150 kts ground speed, your rate of climb in hundreds of feet is 500. (150/60 * 200 = 500)
Another way you can do it is to take 500 feet per nm multiplied by your speed in nm per minute. 150kts is 2.5 nm per min.
500ft per nm X 2.5nm per min = 1250ft per min.
Don't forget that for larger climbs, your true airspeed will increase even though your calibrated/indicated speed remains constant.
TOPC and TOPD
An aircraft is cruising at A080 and will arrive 2500ft AGL at an airfield. While adopting the standard rate of decent, find the distance in nautical miles from the airfield which has a 500ft elevation the aircraft will begin its ROD using a 120kt ground speed.
An aircraft is at “X” with an altitude of 3000ft and needs to climb to 9000ft (Refer to Fig 29 of the work booklet). To remain OCTA using a constant ROC of 800fpm at 150kt ground speed find the distance at which the aircraft should commence its ROC from “A”<|endoftext|>
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Blacks in Mexico – A Brief Overview
To begin a discussion of the Black Experience in Mexico, it is important to establish the quantitative significance of the black slave population in the colonial era. One of the most frequent responses I get when discussing my research with Mexicans, or Americans for that matter, is “there couldn’t have been more than a handful of slaves in Mexico.” This assumption is made because in most parts of Mexico, today, you don’t see many black people at all. The assumption is made that if there aren’t The first African slave brought to Mexico is said to be one Juan Cortés, a slave who accompanied the conquistador Hernán Cortés in 1519. The Indians, reportedly astonished by his dark skin, having never seen an African before, took him for a god! Another of the early conquistadores, Pánfilo Narvaez, brought a slave who has been credited with bringing the devastating smallpox epidemic of 1520. Mexican anthropologist Gonzalo Aguirre Beltrán estimates that there were 6 blacks who took part in the conquest of Mexico.
These early slaves were more personal servants of their masters, who may be thought of as squires. These slaves were most likely taken from Africa, then transported to Seville, where early slaves were christianized, and they probably spoke Spanish by the time they reached the New World. These slaves didn’t come over on slave ships as part of an overt slave trade. The slave trade that changed the demographic face of Mexico began when King Carlos V began issuing more and more asientos, or contracts between the Crown and private slavers, in order to expedite the trans-atlantic trade. At this point, after 1519, the New World received bozales, or slaves brought directly from Africa without being christianized. The Spanish Crown would issue these asientos to foreign slavers, who would then make deals with the Portuguese, for they controlled the slave “factories” on the West African coast. Aside from these asientos, the Crown would grant licenses to merchants, government officials, conquistadores, and settlers who requested the privilege of importing slaves.
The Crown had very few problems doling out these asientos and licenses, as a direct correlation was seen between the number of slaves imported to the new colony and the colonization and economic development of the colony. For these economic reasons, the black population soared to over 20,000 by 1553, 35,000 by 1646, and numbered some 16,000 as late as 1742.
The numerical significance of these figures becomes clear when we compare them to the Spanish population of the colonial era. In the early colonial period, European immigration was extremely small–and for good reason. There were great risks and many uncertainties in the New World, and few families were willing to immigrate until some assurance of stability was demonstrated. Because of this hesitance, very few European women immigrated, thus preventing the natural growth of the Spanish population.
The point that must be made here is the fact that the black population in the early colony was by far larger than that of the Spanish. In 1570 we see that the black population is about 3 times that of the Spanish. In 1646, it is about 2.5 times as large, and in 1742, blacks still outnumber the Spanish. It is not until 1810 that Spaniards are more numerous.
There has been a fair amount of scholarly work done on blacks in Mexico, the majority of which is historical. Some work highlights the different types of labor blacks performed in Mexico, and other work focuses on general aspects of Mexican slave society. I have included a very partial list of sources that would be a good starting point for further exploration. In addition, the majority of this historical work concerns either blacks in Veracruz or are more general works, with much less having been done on the Costa Chica.<|endoftext|>
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Pre-Algebra, Algebra 1, and Algebra 2 all require you to master new math skills. Do you find solving equations and word problems difficult in Algebra class? Are the exponents, proportions, and variables of Algebra keeping you up at night? Intercepts, functions, and expressions can be confusing to most Algebra students, but a qualified tutor can clear it all up! Our Algebra tutors are experts in math and specialize in helping students like you understand Algebra. If you are worried about an upcoming Algebra test or fear not passing your Algebra class for the term, getting an Algebra tutor will make all the difference.
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Algebra II - A primary goal of Algebra II is for students to conceptualize, analyze, and identify relationships among functions. Students will: develop proficiency in analyzing and solving quadratic functions using complex numbers; investigate and make conjectures about absolute value, radical, exponential, logarithmic and sine and cosine functions algebraically, numerically, and graphically, with and without technology; extend their algebraic skills to compute with rational expressions and rational exponents; work with and build an understanding of complex numbers and systems of equations and inequalities; analyze statistical data and apply concepts of probability using permutations and combinations; and use technology such as graphing calculators.
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# What Is The Quantity Used To Measure An Object’S Resistance To Changes In Rotational Motion?
Contents
## What Is The Quantity Used To Measure An Object’s Resistance To Changes In Rotational Motion??
Moment of inertia is a measure of an object’s resistance to changes to its rotation: … Measure of such resistance is product of the mass of a segment and the square of its distance from the axis of rotation i.e. moment of inertia.
## How do we measure rotational motion?
We define angular velocity ω as the rate of change of an angle. In symbols this is ω=ΔθΔt ω = Δ θ Δ t where an angular rotation Δθ takes place in a time Δt. The greater the rotation angle in a given amount of time the greater the angular velocity. The units for angular velocity are radians per second (rad/s).
## What is an object’s resistance to change in rotation?
This resistance to change in rotational motion is called the moment of inertia denoted 1 the units are kg m². The moment of inertia is calculated according to the mass.
## Which quantity is used to measure something that is rotating?
The amount of rotation is called the angle of rotation and it is measured in degrees. You can use a protractor to measure the specified angle counterclockwise. Consider the figure below. Here ΔA’B’O is obtained by rotating ΔABO by 180° about the origin.
## What are the quantities involved in rotational motion?
As we use mass linear momentum translational kinetic energy and Newton’s 2nd law to describe linear motion we can describe a general rotational motion using corresponding scalar/vector/tensor quantities. Angular and linear velocity have the following relationship: v=ω×r v = ω × r .
## How do you calculate rotational velocity?
We can rewrite this expression to obtain the equation of angular velocity: ω = r × v / |r|² where all of these variables are vectors and |r| denotes the absolute value of the radius. Actually the angular velocity is a pseudovector the direction of which is perpendicular to the plane of the rotational movement.
## How do you calculate the number of revolutions?
To do this use the formula: revolutions per minute = speed in meters per minute / circumference in meters. Following the example the number of revolutions per minute is equal to: 1 877 / 1.89 = 993 revolutions per minute.
## What is the quantity used to measure an object’s resistance to changes in rotation write in small letters?
The moment of inertia of an object is the measure of its resistance to being rotated about an axis.
## Which quantity is the measure of the object’s mass?
The SI base unit of mass is the kilogram (kg). In physics mass is not the same as weight even though mass is often determined by measuring the object’s weight using a spring scale rather than balance scale comparing it directly with known masses.
Mass
Common symbols m
SI unit kg
Extensive? yes
Conserved? yes
## What is the measure of resistance to angular acceleration?
moments-of-inertia. Frequency: A measure of a body’s resistance to angular acceleration equal to: The product of the mass of a particle and the square of its distance from an axis of rotation. The sum of the products of each mass element of a body multiplied by the square of its distance from an axis.
## Is used to change the speed of rotation?
For example a stepper motor might turn exactly one complete revolution each second. Its angular speed is 360 degrees per second (360°/s) or 2π radians per second (2π rad/s) while the rotational speed is 60 rpm.
Rotational speed
Derivations from other quantities ω = v / r
Dimension
## Which instrument is used for speed of rotation of a rotating body?
Tachometer
A tachometer is an instrument used to measure the angular/rotational speed of a shaft or disk.
Detailed Solution.
Instruments Uses
Speedometer Used to measure the speed of vehicles
Tachometer Used to measure the rotational speed of shaft
## How do you calculate the force of a rotating object?
Follow these simple steps:
1. Find the mass of the object – for example 10 kg .
2. Determine the radius of rotation. Let’s assume it’s 2 m .
3. Determine the velocity of the object. It can be equal to 5 m/s . …
4. Use the centrifugal force equation: F = m v² / r . …
5. Or you can just input the data into our calculator instead
## What is a linear quantity?
Linear motion is motion in a straight line. This type of motion has several familiar vector quantities associated with it including linear velocity and momentum. These vector quantities each have a magnitude (a scalar or number) and direction associated with them.
See also how does the way that matter flows through an ecosystem differ from the way that energy flows
## What is the rotational quantity that is analogous to the linear quantity of force?
Solution for problem 1CQ Chapter 6
The rotational angle (rotational quantity) is the amount of rotation and is analogous to distance (linear physical quantity). We define rotational angle to be the ratio of the ar length to the radius of curvature.
## What are the rotational equivalent for the physical quantities?
The rotational equivalents for (i) mass and (ii) force are a moment of inertia and torque respectively.
## How do you calculate rotational kinetic energy?
Rotational kinetic energy can be expressed as: Erotational=12Iω2 E rotational = 1 2 I ω 2 where ω is the angular velocity and I is the moment of inertia around the axis of rotation. The mechanical work applied during rotation is the torque times the rotation angle: W=τθ W = τ θ .
## How do you calculate rotational acceleration?
In equation form angular acceleration is expressed as follows: α=ΔωΔt α = Δ ω Δ t where Δω is the change in angular velocity and Δt is the change in time. The units of angular acceleration are (rad/s)/s or rad/s2.
## How do you find the number of rotations in physics?
Here we are asked to find the number of revolutions. Because 1 rev=2π rad we can find the number of revolutions by finding θ in radians. We are given α and t and we know ω is zero so that θ can be obtained using θ=ω0t+12αt2 θ = ω 0 t + 1 2 α t 2 .
## How many revolutions are there?
Key characteristics of a revolution
As an historian of the French Revolution of 1789-99 I often ponder the similarities between the five great revolutions of the modern world – the English Revolution (1649) American Revolution (1776) French Revolution (1789) Russian Revolution (1917) and Chinese Revolution (1949).
## How do you calculate revolutions per mile?
For example if the tire has a 20 inch diameter multiply 20 by 3.1416 to get 62.83 inches. Finally divide 63 360 inches per mile by the tire circumference to find the revolutions per mile. Finishing the example you would divide 63 360 by 62.83 to get 1 008.44 revolutions per mile.
## Is a measure of the object’s resistance to any type of force?
Inertial mass is a measure of an object’s resistance to changing its state of motion when a force is applied. … dividing an object’s weight by its free-fall acceleration.
## What is the physics term used to measure an object’s resistance against an applied force?
Friction – Friction is the resistance of motion when one object rubs against another. It is a force and is measured in newtons.
## What is the product of an object’s mass and its velocity?
momentum The product of an object’s mass and velocity.
## What is an object’s mass?
The mass of an object is a measure of the object’s inertial property or the amount of matter it contains. The weight of an object is a measure of the force exerted on the object by gravity or the force needed to support it. The pull of gravity on the earth gives an object a downward acceleration of about 9.8 m/s2.
## What is the measure of an object’s inertia *?
Mass is the measure of an object’s inertia.
## Is mass a measure of an object’s inertia?
3.5 Mass-A Measure of Inertia. The more mass an object has the greater its inertia and the more force it takes to change its state of motion. The amount of inertia an object has depends on its mass—which is roughly the amount of material present in the object. Mass is a measure of the inertia of an object.
## Which quantity causes a change in angular momentum?
The torque caused by the two opposing forces Fg and −Fg causes a change in the angular momentum L in the direction of that torque (since torque is the time derivative of angular momentum).
## Is the measure of a body’s resistance to bending about a given axis?
A measure of a body’s resistance to bending about a given axis. The area moment of inertia and mass moment of inertia are both identified by the variable I. The area moment of inertia and the mass moment of inertia always have the same value.
## How do you calculate rotational velocity from RPM?
Revolutions per minute can be converted to angular velocity in degrees per second by multiplying the rpm by 6 since one revolution is 360 degrees and there are 60 seconds per minute. If the rpm is 1 rpm the angular velocity in degrees per second would be 6 degrees per second since 6 multiplied by 1 is 6.
## How do you calculate the rotational speed of a motor?
How to Calculate Motor RPM. To calculate RPM for an AC induction motor you multiply the frequency in Hertz (Hz) by 60 — for the number of seconds in a minute — by two for the negative and positive pulses in a cycle. You then divide by the number of poles the motor has: (Hz x 60 x 2) / number of poles = no-load RPM.
## How do you convert rotational speed to linear speed?
Since the arclength around a circle is given by the radius*angle (l = r*theta) you can convert an angular velocity w into linear velocity v by multiplying it by the radius r so v = rw.
## Which instrument is used to measure the speed of a machine *?
A tachometer (revolution-counter tach rev-counter RPM gauge) is an instrument measuring the rotation speed of a shaft or disk as in a motor or other machine. The device usually displays the revolutions per minute (RPM) on a calibrated analogue dial but digital displays are increasingly common.
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# Euclid's Axioms and Postulates: A Breakdown
In mathematics, an axiom or postulate is a statement that is considered to be true without the need for proof. These statements are the starting point for deriving more complex truths (theorems) in Euclidean geometry. In this blog post, we'll take a look at Euclid's five axioms and four postulates, and examine how they can be used to derive some basic geometric truths.
### Euclid's Five Axioms
Euclid's five axioms are as follows:
1. Things which are equal to the same thing are also equal to one another. (Reflexive Property)
2. If things which are equal to one another are also equal to something else, then they are equal to one another. (Transitive Property)
3. There exists a unique line segment between any two points.
4. Any line segment can be extended indefinitely in either direction.
5. Given any line segment, a circle can be drawn with any point on the line segment as its center and with the line segment as its radius.
### Euclid's Four Postulates
In addition to his five axioms, Euclid also included four postulates in his work:
1. A straight line may be drawn from any point to any other point.
2. A terminated line segment can be produced in a straight line continuously in either direction.
3. Circle may be described with any point as its center and with any distance as its radius.
4. All right angles are equal to one another.
5.'If two lines intersect each other, the vertical angles formed will be equal to one another.' (playfair's axiom) https://en.wikipedia.org/wiki/Playfair%27s_axiom#Example
These are just a few of the many geometric truths that can be derived from Euclid's axioms and postulates. As you can see, these simple statements can be used to derive some complicated truths about lines, angles, and circles in Euclidean geometry. So next time you're studying for your math test, make sure to review these important principles!
## FAQ
### What is the difference between Euclid axioms and postulates?
Euclid's axioms are statements that are assumed to be true without the need for proof. On the other hand, postulates are statements that are considered to be true based on our experiences in the world.
### Why is Euclid's 5th postulate controversial?
Euclid's 5th postulate, also known as the parallel postulate, is controversial because it is not self-evident like the other postulates in Euclid's system. Many mathematicians have tried to prove the parallel postulate, but no one has been successful so far.
### What are some of the implications of the parallel postulate?
If the parallel postulate is not true, then Euclidean geometry is not the only type of geometry that is possible. In fact, there are many non-Euclidean geometries that have been developed, where the parallel postulate is not true. These non-Euclidean geometries have many applications in physics and mathematics.
### What are axioms and postulates with examples?
Axioms are statements that are assumed to be true without the need for proof. For example, one of Euclid's axioms is the statement that "things which are equal to the same thing are also equal to one another." A postulate is a statement that is considered to be true based on our experiences in the world. For example, Euclid's first postulate is that "a straight line can be drawn between any two points."<|endoftext|>
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Plug in the value of $$x$$ and $$y$$ and use order of operations rule. $$x=2$$ and $$y=-3$$
$$5(4x-3y)-7y^2=5(4(2)-3(-3))-7(-3)^2=5(8+9)-7(9)=5(17)-63=85-63=22$$
For one hour he earns $18, then for t hours he earns$18t. If he wants to earn at least $78, therefore, the number of working hours multiplied by 18 must be equal to 78 or more than 78. $$18t≥78$$ 3- Choice B is correct $$(108-(3×9))÷9=9^3÷81=9$$ 4- Choice B is correct The ratio of boys to girls is 3 ∶ 5. Therefore, there are 3 boys out of 8 students. To find the answer, first, divide the total number of students by 8, then multiply the result by 3. $$240÷8=30 ⇒ 30×3=90$$ 5- Choice A is correct Probability$$=\frac{number \space of \space desired \space outcomes}{number \space of \space total \space outcomes}=\frac{9}{9+15+14+16}=\frac{9}{54}\frac{1}{6}=0.16$$ 6- Choice D is correct Let’s compare each fraction: $$\frac{2}{3}<\frac{3}{4}<\frac{7}{9}<\frac{4}{5}$$ Only choice D provides the right order. 7- Choice B is correct Let $$y$$ be the width of the rectangle. Then; $$14×y=84→y=\frac{84}{16}=6$$ 8- Choice B is correct $$4×\frac{5}{16}=\frac{20}{16}=1.25$$ A. $$1.25>2$$ B. $$1<1.25<2$$ This is the answer! C. $$\frac{3}{8}=1.25$$ D. $$1.25=2^2$$ 9- Choice B is correct In any rectangle, The measure of the sum of all the angles equals $$180^\circ$$. 10- Choice C is correct $$\frac{824}{17}=48.5$$ ## The Absolute Best Book to Ace the 6th Grade SBAC Math Test Original price was:$23.99.Current price is: $18.99. Satisfied 139 Students 11- The answer is $$7^2$$. $$588=2^2×3^1×7^2$$ 12- Choice B is correct The area of the trapezoid is: Area $$=\frac{base \space 1+base \space 2}{2}×height=\frac{12+10}{2}x=A→ 11x=A→x=\frac{A}{11}$$ 13- Choice B is correct $$\frac{72}{8}=9, \frac{648}{72}=9, \frac{5,832}{648}=9$$, Therefore, the factor is 9. 14- Choice C is correct Simplify each option provided. A. $$13-(3×6)+(7×(-6))=13-18+(-42)=-5-42=-47$$ B. $$(\frac{25}{400})+(\frac{7}{50})=\frac{25}{400}+\frac{56}{400}=\frac{81}{400}$$ C. $$((22×\frac{30}{6})-(7×\frac{144}{12}))×\frac{18}{2}=(110-84)×9=26×9=234$$ (this is the answer) D. $$(\frac{6}{24}+\frac{12}{33})-50=(\frac{1}{4}+\frac{1}{3})-50=(\frac{3}{12}+\frac{4}{12})-50=\frac{7}{12}-50=\frac{-593}{15}$$ 15- Choice D is correct To find the discount, multiply the number ($$100\%$$- rate of discount) Therefore; $$450(100\%-16\%)=450(1-0.16)=450-(450×0.16)$$ 16- Choice A is correct 1,400 out of 11,900 equals to $$\frac{1,400}{11,900}=\frac{200}{1,700}=\frac{2}{17}$$ 17- Choice C is correct The opposite of Nicolas’s integer is $$25$$. So, the integer is $$-25$$. The absolute value of $$25$$ is also $$25$$. 18- Choice B is correct Volume of a box = length × width × height = 7 × 4 × 12 = 336 19- Choice C is correct 1 yard = 3 feet, Therefore, $$33,759 ft×\frac{1 \space yd}{3 \space ft}=11,253$$ yd 20- Choice B is correct $$16\%$$ of the volume of the solution is alcohol. Let $$x$$ be the volume of the solution. Then: $$16\%$$ of $$x=38$$ ml ⇒ $$0.16x=38 ⇒ x=38÷0.16=237.5$$ ## Best 6th Grade SBAC Math Prep Resource for 2022 Original price was:$19.99.Current price is: $14.99. Satisfied 147 Students 21- Choice C is correct $$(-2)(9x-8)=(-2)(9x)+(-2)(-8)=-18x+16$$ 22- Choice D is correct 1 pt = 16 fluid ounces. $$576÷16=36$$ Then: 576 fluid ounces = 36 pt 23- Choice D is correct 1 kg = 1000 g and 1 g = 1000 mg, 120 kg = 120 × 1000 g = 120 × 1000 × 1000 = 120,000,000 mg 24- Choice C is correct The diameter of a circle is twice the radius. Radius of the circle is $$\frac{14}{2}=7$$. Area of a circle $$= πr^2=π(7)^2=49π=49×3.14=153.86≅153.9$$ 25- Choice B is correct Average (mean) $$=\frac{sum \space of \space terms}{number \space of \space terms}=\frac{15+17+12+16+21+23}{6}=\frac{104}{6}=17.33$$ 26- Choice C is correct Prime factorizing of $$18=2×3×3$$, Prime factorizing of $$24=2×2×2×3$$ LCM $$= 2×2×2×3×3=72$$ 27- Choice B is correct The coordinate plane has two axes. The vertical line is called the $$y$$-axis and the horizontal is called the $$x$$-axis. The points on the coordinate plane are addressed using the form $$(x,y)$$. Point A is one unit on the left side of $$x$$-axis, therefore its $$x$$ value is 4 and it is two units up, therefore its $$y$$ axis is 2. The coordinate of the point is: (4, 2) 28- Choice B is correct $$α$$ and $$β$$ are supplementary angles. The sum of supplementary angles is 180 degrees. $$α+β=180^\circ→α=180^\circ-β=180^\circ-125^\circ=55^\circ,$$ Then, $$\frac{α}{β}=\frac{55}{125}=\frac{11}{25}$$ 29- Choice C is correct Opposite number of any number $$x$$ is a number that if added to $$x$$, the result is 0. Then: $$7+(-7)=0$$ and $$4+(-4)=0$$ 30- Choice C is correct $$16=-129+x$$, First, subtract 129 from both sides of the equation. Then: $$16+129=-129+x+129 →145=x$$ ## The Best Books to Ace the 6th Grade SBAC Math Test Original price was:$15.99.Current price is: $10.99. Satisfied 81 Students Original price was:$18.99.Current price is: $13.99. Satisfied 222 Students Original price was:$16.99.Current price is: $11.99. Satisfied 79 Students Original price was:$17.99.Current price is: $12.99. Satisfied 144 Students ## Related to This Article ### More math articles ### What people say about "Full-Length 6th Grade SBAC Math Practice Test-Answers and Explanations - Effortless Math: We Help Students Learn to LOVE Mathematics"? No one replied yet. X 21% OFF Limited time only! Save Over 21% SAVE$5
It was $23.99 now it is$18.99<|endoftext|>
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Solve triangle abc
In this blog post, we will be discussing how to Solve triangle abc. Our website can solving math problem.
Solving triangle abc
Are you ready to learn how to Solve triangle abc? Great! Let's get started! Solving exponential equations can be a bit tricky. Most of the time you will need to use an inverse function to get from one number to the other. However, it is possible to solve some equations without using such techniques. Here are some examples: One way to solve an exponential equation is to use a logarithm table. For example, if you have an equation of the form y = 4x^2 + 32, then you would use the logarithm table found here. Then, you would find that log(y) = -log(4) = -2 and log(32) = 2. These values would be used in the original equation to obtain the solution: 4*y = -2*4 + 32 = -16 + 32 = 16. This value is the desired answer for y in this problem. Another way to solve an exponential equation is by using a combination of substitution and elimination. You can start by putting x into both sides of the equation and simplifying: ax + b c where a c if and only if b c/a . Then, once this is done, you can eliminate b from each side (using square roots or taking logs if necessary) to obtain a single solution that does not involve x . c if and only if , then you can substitute for y in both sides, thus eliminating x
Solving calculus problems without a calculator is a great way to practice critical thinking and build confidence in your math skills. Solving calculus problems can feel like a daunting task, but it doesn’t have to be! There are several resources available online that can help you tackle any calculus problem. The key is to practice the skill in small, manageable steps so that you don’t feel overwhelmed. You can also use an online calculator such as Wolfram Alpha or Khan Academy’s free online calc tools to simplify complex equations. Using these tools will also help you practice critical thinking skills as you work through the problem step-by-step. By learning how to solve calculus problems, you’ll be better prepared for more challenging courses and more confident when you approach new tasks.
The best math with steps is when you can practice a new skill over and over again. This helps you to remember what’s going on and how to do it correctly. It also gives you the chance to make mistakes so that you can learn from them. When you’re working with steps, it can be very easy to get lost or confused. The best thing to do is take your time and make sure that you understand what you’re doing before moving on. Once you’ve got the hang of it, then you can start making bigger leaps forward in your development. When it comes to math, there are lots of different ways to practice. You could try out some different apps, such as a Sudoku app or a word search game. You could also go through a basic book like “Fifty Simple Things You Can Do To Improve Your Math Skills.” Or you could even try something more hands-on like building a tower out of blocks or designing your own LEGO model.
One of the best ways to improve at math is by learning how to solve problems. Knowing how to set up equations, work with fractions and percentages, and use arithmetic are essential skills that underlie all math. Solving problems is also a great way to challenge yourself and practice your problem-solving skills. Solving problems can be challenging at times, but it's never impossible. With practice and patience, you'll get better at solving problems every time you sit down at the table.<|endoftext|>
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$\integral{(7 x + 8)}{x}$
Evaluate
Differentiate w.r.t. x
গ্রাফ
## শেয়ার করুন
\int 7x\mathrm{d}x+\int 8\mathrm{d}x
Integrate the sum term by term.
7\int x\mathrm{d}x+\int 8\mathrm{d}x
Factor out the constant in each of the terms.
\frac{7x^{2}}{2}+\int 8\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 7 times \frac{x^{2}}{2}.
\frac{7x^{2}}{2}+8x
Find the integral of 8 using the table of common integrals rule \int a\mathrm{d}x=ax.
\frac{7x^{2}}{2}+8x+С
If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.<|endoftext|>
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If you are exposed to an allergen or chemical irritant, your skin may develop itchy, red, raised welts on the skin called hives. Often hives spread or blend together to form larger areas of raised lesions. Harmless and non-contagious, hives generally clear up on their own.
Hives develop when mast cells release histamine and other chemicals into your bloodstream, causing small blood vessels to leak. Generally, hives are an allergic reaction to a food, animal or medication. Sun exposure, stress, excessive perspiration and other more serious conditions, like lupus, can bring on hives.
Individual hives can take from 30 minutes to 36 hours to disappear. Chronic hives, referred to as urticaria, last longer than six weeks to go away, but frequently reoccur. If swelling occurs below the surface of the skin, a condition called angioedema, you should seek medical attention because angioedema can affect the internal organs.
Signs of hives include:
- Developing in batches
- Often raised
- Small, round rings or large patches with a red flare
- Usually itch
Usually, hives develop suddenly and disappear almost as quickly with no treatment. You can use over-the-counter creams and antihistamines to control itching if it becomes a problem. For cases of chronic hives, your physician may prescribe antihistamines or oral corticosteroids. The best prevention for hives is avoidance of any known triggers, such as certain foods, stress or extreme changes in weather.<|endoftext|>
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Beacon Lesson Plan Library
A Goldfish is the Best Pet
DescriptionA goldfish is the best pet. What facts support this thesis? What facts oppose it? Use graphic organizers to help students select facts which must be considered in order to persuade an audience to agree with a given point of view.
ObjectivesThe student selects and uses appropriate pre-writing strategies, such as brainstorming, graphic organizers, and outlines.
Materials- A selection of graphic organizers (Two types are provided in the associated files for this lesson.) Each student will need at least three of these. If you prefer, they can draw their own organizers and save on copying materials.
- An overhead projector, marker board, or chalk board to be used by the class secretary or by the teacher
- Markers, chalk, or pens as needed
- A list of argumentative topics for the students who have difficulty deciding what to write about (nothing heavy or mind boggling at first)
- One copy per student of the rubric you plan to use (A sample is provided in the associated files.)
Preparations1. Duplicate graphic organizers for students to use.
2. Pre-select a student to be the secretary, preferably one with a legible handwriting.
3. Be sure you have a working overhead projector, or a chalkboard/marker board with plenty of writing surface available and different colored writing utensils.
4. Prepare a list of possible topics; there will always be a few students who canít think of anything to use.
Procedures1. Begin by asking the class if they have any pets at home. Ask several students to explain why they selected that type of pet.
2. Appoint one student to act as secretary and write the information on the board.
3. After several minutes of brainstorming, ask students if they think a goldfish, or a frog, or some other critter of your choice would make a good pet. This should be an animal that we usually donít think of as a pet. You can use one of the animals they mentioned or use the goldfish. A sample T chart is provided in the associated files.
4. Have the secretary make a list of at least three of the reasons why this animal would be a good pet and a list of at least three reasons why it would not be a good pet. (Color coding the responses helps the visual learners.) These reasons should be facts, not opinions. At this point you may need to explain the difference between facts and opinions.
5. Give each student a sample of the graphic organizers which can be used to record this information (see associated files for examples of T charts and mapping charts). Explain that you will be giving them a similar organizer on which they will record information as a final assessment.
6. Have students record the selected information on the organizer as you or your secretary record it on the board or the overhead.
7. Circulate in the room to be sure each student understands how to use the graphic organizer.
8. When all students have satisfactorily completed this activity, explain that this is an arguable topic. Some people think gold fish are good pets; some people donít even consider them to be pets. The statement ďGold fish make the best pets,Ē could be used as a thesis statement in a persuasive essay.
9. Ask students to highlight the supporting fact that they think is the most important and the opposing fact that they think is most important.
10. Give each student another graphic organizer and go through the same brainstorming procedure with another topic that does not require research.
a. Pens are better than pencils.
b. Students should be required to take a computer class in high school.
c. Elementary students should not be involved in competitive sports.
You can also use a topic from literature. (Grendel was a poor, misunderstood child.)
11. Ask the class to help you formulate an antithesis (statement of the opposite opinion) and prepare a graphic organizer from that information, highlighting the most important supporting fact and the most important opposing fact.
12. Point out that some of the supporting facts for the thesis might be listed as opposing information for the antithesis.
13. Tell students that graphic organizers such as these can be used to prepare to write a persuasive paper or a persuasive speech.
14. Ask students to choose a topic of their own and a graphic organizer on which to record information. Have them go through the same process with the new topic. This could be a homework assignment. (CAUTION: you might want to ask them to steer away from extremely controversial topics, those not suitable for classroom discussions).
15. Show students a sample of the rubric you will use to evaluate their work. (see sample rubric in associated files)
16. Explain what you will be checking for in the different areas to be evaluated.
17. After you evaluate the prewriting, return the papers and assist the students with their revisions.
18. OPTIONAL: If you plan to assign a composition based on this prewriting, give students time in class to revise the prewriting.
18. OPTIONAL: Tell students that the next step will be to use this information to prepare an outline for a persuasive essay, persuasive speech, etc., which will be covered in a separate lesson.
AssessmentsStudents may be assessed using the sample organizers provided (see associated files).
Information included in the organizer: (criteria)
A. The general topic
B. The authorís opinion on the topic (thesis statement)
C. Three facts supporting the thesis with the most important fact highlighted
D. Three facts opposing the thesis
E. The oppositionís opinion on the topic (antithesis)
F. Three facts supporting the antithesis with the most important fact highlighted
G. Three facts opposing the antithesis
Extensions1. Language arts: Move from this prewriting assignment to an outline for a persuasive paper. From there you can proceed to a rough draft and eventually a final copy.
2. This could also be turned into a group assignment. Give two groups opposing viewpoints for which they will do research, prepare a T chart or mapping chart, and debate their findings.
3. Science: This could be the starting point for a persuasive paper on several different topics including environmental issues, the space program, and research topics.
4. Social studies: This is an excellent starting point for debates in the area of politics, government policies, controversial laws, or even economic concerns.
Attached FilesT Chart for Antithesis File Extension: pdf
T Chart for Thesis File Extension: pdf
Mapping for Antithesis File Extension: pdf
Mapping for Thesis File Extension: pdf
T Chart for A Goldfish Makes the Best Pet File Extension: pdf
Sample Evaluation Rubric File Extension: pdf
Return to the Beacon Lesson Plan Library.<|endoftext|>
| 3.84375 |
658 |
Refrigerants are the fluids or gases contained in refrigerating devices, which boil or expand, removing heat from objects to be cooled, then compressed, transferring heat to cooling mediums such as water and air. Refrigerants used in commercial heating, ventilating and air conditioning (HVAC), and in home air conditioning units include hydrofluorocarbons (HCFCs), chlorofluorocarbons (CFCs) and perfluorocarbons (PFCs). National governments are concerned with refrigerant properties because of evidence linking emissions of some of the gases to the depletion of the Earth's ozone layer. Others function as greenhouse gases, trapping heat within the atmosphere, and therefore have high global warming potential. The U.S. Clean Air Act regulates emissions from systems that use refrigerant gases. Refrigerants are ranked in 13 property classes, including a flammability class that has three sub-classes.
Refrigerant Flammability Classes
Class 1 refrigerants are either non-combustible or, at 70 degrees F and 14.6 psi (room temperature and sea-level atmospheric pressure), do not support the spread of a flame in a combustible environment of the gas outward from the point of ignition. Refrigerants in this class are considered the safest. Class 2 refrigerants have a lower flammability limit of more than 0.00624 lb./cubic foot (0.10kg/cubic meter) at 70 degrees F and 14.6 psi, and a heat of combustion less than 19 kilojoules/kilogram. Class 3 refrigerants are highly flammable with a lower flammability limit of less than or equal to 0.00624 lb./cubic foot (0.10 kg/cubic meter) at 14.6 psi and 70 degrees F, or a heat of combustion greater than or equal to 19 kilojoules/kilogram.
Flammability Class I
Examples of class 1 refrigerants are helium (He), neon (Ne), nitrogen (N), water, air, carbon dioxide (CO2), sulfur dioxide (SO2), carbon tetrachloride (CCl4), trichloromonofluoromethane (CCL3F) and carbon tetrafluroide (CF4).
Sciencing Video Vault
Flammability Class 2
Examples of class 2 refrigerants are ammonia (NH3), ethane (C2H6), propane (C3H8), iso-butane (iC4H10), Methyl chloride (CH3CL), acetic acid (CH3COOH) and dichloromethane (CH2CL2).
Flammabilty Class 3
Class 3 refrigerants are hydrogen (H2), methane (CH4), butane (C4H10), trifluromethane (CHF3), pentafluroethane (C2HF5), chlorodifluromethane (CHClF2), tetrafluroethane (CF3CH2F) and difluroethane (CHF2CH3).<|endoftext|>
| 3.828125 |
397 |
# How do you find the end behavior and state the possible number of x intercepts and the value of the y intercept given y=-x^3-4x?
Apr 10, 2018
See below.
#### Explanation:
To find the end behaviour of a polynomial, we only need to look at the degree and leading coefficient of the polynomial. The degree is the highest power of $x$ in this case.
$- {x}^{3}$
We now see what happens as $x \to \pm \infty$
as $x \to \infty$ , $\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus - {x}^{3} \to - \infty$
as $x \to - \infty$ , $\setminus \setminus \setminus \setminus \setminus \setminus - {x}^{3} \to \infty$
$y$ axis intercepts occur where $x = 0$:
$y = - {\left(0\right)}^{3} - 4 \left(0\right) = 0$
Coordinates:
color(blue)( (0 ,0)
$x$ axis intercepts occur where $y = 0$
$- {x}^{3} - 4 x = 0$
${x}^{3} + 4 x = 0$
Factor:
$x \left({x}^{2} + 4\right) = 0$
$x = 0$
${x}^{2} + 4 = 0$ ( this has no real solutions ).
coordinates:
color(blue)(( 0 , 0 )
The graph confirms these findings:
graph{y=-x^3-4x [-16.01, 16.02, -20,20]}<|endoftext|>
| 4.625 |
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