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# CTK Insights
• ## Pages
24 Feb
### The Joy of Homogeneity
In a recent blog A Lovely Observation Gary Davis (@RepublicOfMath) elaborated on an observation of Ben Vitale (@BenVitale) to the effect that
In the fractions both numerators and denominators are sums of successive odd numbers: the numerators start with 1, the denominators where the numerators leave off. Thus naturally derivation of the formula for the sum of successive odd numbers is the key to the explanation of Ben Vitale's observation. There are ways and ways to obtain the required expression. The most elementary one is to observe that
is the sum of an arithmetic sequence $a, a + d, a + 2d, \ldots, a + (n-1)d$,
where $a = 1$ and $d = 2$. With a nod to the young F. Gauss,
which in the case of odd numbers gives
Another way to derive that formula is to noticee that the sum of odd numbers is the sum of all numbers less the sum of the even ones. It is convenient at this point to start using the symbol of $\sum$ to make the formulas shorter and more manageable:
and, for the sum of all integers from $1$ to $n$,
Now a little of summation magic:
with the same result. Let's note that here we made a subtle use of the homogeneity property of function $f(x) = x$, namely $f(2x)=2f(x)$. Simple as it appears, it was central to the derivation. For Ben Vitale's observation we'll have to do something very similar:
Ben Vitale's observation comes to
A generalization of this would be to take, say, twice as many terms in the denominator as in the numerator. This can be done in more than one way:
or
because $\sum_{k=1}^{3n}(2k-1)=(3n)^2-n^2=8n^2$, as before. Going further, there are at least three ways to have three times as many terms in the denominator as in the numerator:
One can easily check that the three fractions are equal to $\frac{1}{9}$, $\frac{1}{3}$, and $\frac{5}{9}$, respectively.
Obviously, our success in getting a simple expression is due to the homogeneity of function $g(x)=x^2$: $g(ax)=a^{2}g(x)$. (There is a difference between the two functions $f$ and $g$. For the obvious reasons, the former is said to be homogeneous of order (degree) 1, the later of order 2.)
As Gary Davis notes at the end of his post, this kind of algebraic manipulations is accessible to middle and high school students. An additional example of the infinite series of the reciprocals of squares requires no more effort but a little hand-waving if not presented at the beginning Calculus class.
Assume we know that
Then of course, for the series that involves only even numbers,
As a reward for the effort, we get an expression for the series of the reciprocals of the squares of odd numbers:
Who would have thought that the series of the reciprocals of odd squares sums up to three times a similar series for even squares!
#### 7 Responses to “The Joy of Homogeneity”
1. 1
Eric Hsu Says:
Nice problem! The sum of odd numbers version has a nice visual answer. First, sum of 2n odd numbers can be famously represented as the Ls of a 2n x 2n square. (compare http://merganser.math.gvsu.edu/calculus/summation/odds.html). Then the numerator is the sum of first n of these, which form a smaller n x n square, which can be seen as a quarter of the larger square. So the denominator is three times larger than the numerator.
2. 2
Thank you Eric. Have not thought of that in geometric terms.
3. 3
Allen Says:
Thanks! Nice to see this problem. I've also noticed that (1+3+5+7+...+n)/(n+(n+2)+(n+4)+...+(2n-1))=a/b where a+b = 2n or n depending on if n is of the form 4i+1 or 4i+3 (where i is a natural number)
For example: let n=4i+1 = 13 (i=3), then (1+3+5+7+9+11+13)/(13+15+17+19+21+23+25)=49/133 = 7/19, and 7+19=26=2n
If n=4i+3=15 (i=3), then
(1+3+5+7+9+11+13+15)/(15+17+19+21+23+25+27+29)=64/176=4/11, and 4+11 =15=n.
These properties can be proven using the same techniques used in your blog.
4. 4
Allen, this is a curious property. But just to check, what about
(1+3+...+11)/(11+13+..+21)
or
(1+3+...+15)/(11+13+..+25) ?
5. 5
Allen Says:
For your first example, n=11 or it fits the form 4i+3 when i=2, so (1+3+...+11)/(11+13...+21)=36/96=3/8, and 3+8 = 11 = n.
Your second example doesn't seem to fit the pattern I noticed. The largest term in the numerator (15) needs to be the same as the smallest term in the denominator(11). In other words, if the numerator sum ends with n, then the denominator sum needs to begin with n.
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You are viewing an older version of this Concept. Go to the latest version.
# Simple and Compound Interest
## Use A = P(1 + r)^t to solve for A.
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Practice Simple and Compound Interest
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Simple and Compound Interest
Suppose you are re-negotiating an allowance with your parents. Currently you are given $25 per week, but it is the first of June, and you have started mowing the lawn and taking out the trash every week, and you think your allowance should be increased. You father considers the situation and makes you the following offer: "I tell you what, son. I will give you three options for your allowance, you tell me which you would like" "Option A: You keep the$25 per week"
"Option B: You take $15 this week, then$16 next week, and so on. I'll continue adding $1 per week until New Year's." "Option C: I'll give you 1 penny this week, and then double your allowance each week until the first of October, then keep it at that rate." Which option would you choose? ### Watch This Embedded Video: ### Guidance Simple interest vs. Compound Interest Simple interest is interest which accrues based only on the principal of an investment or loan. The simple interest is calculated as a percent of the principal. Simple interest: $i = p \cdot r \cdot t$ . Variable i is interest, p represents the principal amount, r represents the interest rate, and t represents the amount of time the interest has been accruing. For example, say you borrow$2,000 from a family member, and you insist on repaying with interest. You agree to pay 5% interest, and to pay the money back in 3 years.
The interest you will owe will be 2000(0.05)(3) = $300. This means that when you repay your loan, you will pay$2300. Note that the interest you pay after 3 years is not 5% of the original loan, but 15%, as you paid 5% of $2000 each year for 3 years. Now let’s consider an example in which interest is compounded. Say that you invest$2000 in a bank account, and it earns 5% interest annually. How much is in the account after 3 years?
Compound interest: $A(t) = p \cdot (1 +r)^{t}$
Here, A ( t ) is the A mount in the account after a given t ime in years, p rincipal is the initial investment, and r ate is the interest rate. Note that we use $(1 + r)$ instead of just $r$ , so we can find the entire amount in the account, not just the interest paid.
$A(t) = 2000 \cdot (1.05)^{3}$
After three years, you will have $2315.25 in the account, which means that you will have earned$315.25 in interest.
Compounding results in more interest because the principal on which the interest is calculated increased each year. Another way to look at it is that compounding creates more interest because you are earning interest on interest, and not just on the principal.
#### Example A
Use the formula for compound interest to determine the amount of money in an investment after 20 years, if you invest $2000, and the interest rate is 5% compounded annually. Solution: The investment will be worth$5306.60
A(t) = P(1 + r) t
A(20) = 2000(1.05) 20
#### Example C
What is the value of an investment after 20 years, if you invest $2000, and the interest rate is 5% compounded continuously ? Solution The more often interest is compounded, the more it increases, but there is a limit. Each time you increase the number of compoundings, you decrease the fraction of the annual interest that is applied to each compounding. Eventually, the differences become so small as to be negligible. This is known as continuous compounding . The function A ( t ) = Pe rt is the formula we use to calculate the amount of money when interest is continuously compounded, rather than interest that is compounded at discrete intervals, such as monthly or quarterly. A ( t ) = Pe rt A (20) = 2000 e .05(20) A (20) = 2000 e 1 A (20) =$5436.56
Concept question wrap-up
"Which option would you choose?"
Assume you want to make the most money possible by the end of the year. Assume also that there are 24 weeks left.
Option A = $25 \cdot 24 = 600$ total
Option B = $15 + 16 + 17... + 39 = 609$ total
Option C (assuming 16 weeks until Oct.) = $1 \cdot (2^{16}) = 655.36$ each week after Oct 1.
It is entirely possible that dear old dad didn't take exponential growth seriously enough, he may need a second job!
### Vocabulary
Simple interest is interest earned only on the initial value or principal.
Compound interest refers to interest earned on the total account at the time it is compounded, including previously earned interest.
Principal is the initial sum, before any interest is added.
Rate is the percentage at which interest accrues.
Continuous compounding refers to a loan or investment with interest that is compounded constantly, rather than on a specific schedule.
'Accrue ' means "increase in amount or value over time". If interest accrues on a bank account, you will have more money in your account. If interest accrues on a loan, you will owe more money to your lender.
### Guided Practice
1) Compare the values of the investments shown in the table. If everything else is held constant, how does the compounding influence the value of the investment?
Principal r n t
a. $4,000 .05 1 (annual) 8 b.$4,000 .05 4 (quarterly) 8
c. $4,000 .05 12 (monthly) 8 d.$4,000 .05 365 (daily) 8
e. $4,000 .05 8760 (hourly) 8 2) Determine the value of each investment. a. You invest$5000 in an account that gives 6% interest, compounded monthly. How much money do you have after 10 years?
b. You invest $10,000 in an account that gives 2.5% interest, compounded quarterly. How much money do you have after 10 years? 3) How long will it take$2000 to grow to $25,000 at a 5% interest rate? Answers 1) Use the compound interest formula. For this example, the n is the quantity that changes: $A(8) = 4000 \left (1 + \frac{.05} {n}\right )^{8n}$ Principal r n t A a.$4,000 .05 1 (annual) 8 $5909.82 b.$4,000 .05 4 (quarterly) 8 $5952.52 c.$4,000 .05 12 (monthly) 8 $5962.34 d.$4,000 .05 365 (daily) 8 $5967.14 e.$4,000 .05 8760 (hourly) 8 $5967.29 A graph of the function $f(x) = 4000 \left (1 + \frac{.05} {x}\right )^{8x}$ is shown below: The graph seems to indicate that the function has a horizontal asymptote at$6000. However, if we zoom in, we can see that the horizontal asymptote is closer to 5967.
What does this mean? This means that for the investment of $4000, at 5% interest, for 8 years, compounding more and more frequently will never result in more than about$5968.00.
2)
a. $5000, invested for 10 years at 6% interest, compounded monthly. $A(t) = P \left (1 + \frac{r} {n}\right )^{nt}$ $A(10) = 5000 \left (1 + \frac{.06} {12}\right )^{12\cdot 10}$ $A(10) = 5000 \left (1.005\right )^{120}$ $A(10) = \9096.98$ b.$10000, invested for 10 years at 2.5% interest, compounded quarterly.
Quarterly compounding means that interest is compounded four times per year. So in the equation, n = 4.
$A(t) = P \left (1 + \frac{r} {n}\right )^{nt}$
$A(10) = 6000 \left( 1 + \frac{.025} {4}\right )^{4 \cdot 10}$
$A(10) = 6000 (1.00625)^{40}$
$A(10) = \12,830.30$
In each example, the value of the investment after 10 years depends on three quantities: the principal of the investment, the number of compoundings per year, and the interest rate.
3) It will take about 50 years:
A ( t ) = Pe rt
25,000 = 2000 e .05(t)
12.5 = e .05(t) Divide both sides by 2000
ln 12.5 = ln e .05( t ) Take the ln of both sides
ln 12.5 = .05 t ln e Use the power property of logs
ln 12.5 = .05 t × 1 ln e = 1
ln 12.5 = 0.5 t Isolate t
$t = \frac{ln 12.5} {.05} \approx 50.5$
### Practice
1. What is the formula for figuring simple interest?
2. What is the formula for figuring compound interest?
3. If someone invested $4500.00, how much would they have earned after 4 years, at a simple interest rate of 2%? 4. Kyle opened up a savings account in July. He deposited$900.00. The bank pays a simple interest rate of 5% annually. What is Kyle's balance at the end of 4 years?
5. After having an account for 6 years, how much money does Roberta have in the account, if her original deposit was $11,000, and her bank's yearly simple interest rate is 8.4%? 6. Tom called his bank today to check on his savings account balance. he was surprised to find a balance of$6600, when he started the account with just $5000.00 8 years ago. Based on this data, what percentage rate has the bank been paying on the account? 7. Julie opened a 4% interest account with a bank that compounds the interest quarterly. If Julie were to deposit$3000.00 into the account at the beginning of the year, how much could she expect to have at the end of the year?
8. Susan has had a saving account for a few years now. The bank has been paying her simple interest at a rate of 5%. She has earned $45.00 on her initial deposit of$300.00. How long has she had the account?
9. What is the balance on a deposit of $818.00 earning 5% interest compounded semiannually for 5 years? 10. Karen made a decent investment. After 4 years she had$3250.00 in her account and expects to have $16,250, after another 4 years. Her savings account is a compounding interest account. How much was her original deposit? 11. What is the yearly simple interest rate that Ken earns, if after only three months he earned$16.00 on an initial $800.00 deposit? 12. Write an expression that correctly represents the balance on an account after 7 years, if the account was compounded yearly at a rate of 5%, with an initial balance of$1000.00
13. Caryl gives each of his three kids $3000.00 each, and they each use it to open up saving accounts at three different banks. Georgia, his oldest, is earning 3% annually at her bank. Kirk earns 7% annually at his bank. Lottie's bank is paying her an annual rate of 4%. At the end of 6 years show much will each of them have in their respective accounts. 14. Kathy receives an inheritance check for$3000.00 and decides to put it in a saving account so she can send her daughter to college when she gets older. After looking she finds an account that pays compounding interest annually at a rate of $14%. The balance on the account can be represented by a function, where x is the time in years. Write a function, and then use it to determine how much will be in the account at the end of 7 years. 15. Stan is late on his car payment. The finance company charges 3% interest per month it is late. His monthly payment is$300.00. What is the total amount he will owe if he pays the August first bill October first? (assuming he was able to make his September bill on - time)
Today, you get your first credit card. It charges 12.49% interest on all purchases and compounds that interest monthly. Within one day you max out the credit limit of $1,200.00. 1. If you pay the monthly accrued interest plus$50.00 towards the initial $1,200 amount every month, how much will you still owe at the end of the first 12 months? 2. How much will you have paid in total at the end of the year? You are preparing for retirement. You invest$10,000 for 5 years, in an account that compounds monthly at 12% per year. However, unless this money is in an IRA or other tax-free vehicle, with zero inflation, you also have an annual tax payment of 30% on the earned interest.
1. How much will you have in 5 years?
2. Now take into account that the money loses 3% spending value per year due to inflation, how much is what you have saved really worth at the end of the 5 years?
### Vocabulary Language: English
Accrue
Accrue
Accrue means "increase in amount or value over time." If interest accrues on a bank account, you will have more money in your account. If interest accrues on a loan, you will owe more money to your lender.
Compound interest
Compound interest
Compound interest refers to interest earned on the total amount at the time it is compounded, including previously earned interest.
Continuous compounding
Continuous compounding
Continuous compounding refers to a loan or investment with interest that is compounded constantly, rather than on a specific schedule. It is equivalent to infinitely many but infinitely small compounding periods.
Continuously compounding
Continuously compounding
Continuous compounding refers to a loan or investment with interest that is compounded constantly, rather than on a specific schedule. It is equivalent to infinitely many but infinitely small compounding periods.
Principal
Principal
The principal is the amount of the original loan or original deposit.
Rate
Rate
The rate is the percentage at which interest accrues.<|endoftext|>
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# Binomial Distribution
In statistics the so-called binomial distribution describes the possible number of times that a particular event will occur in a sequence of observations. The event is coded binary, it may or may not occur. The binomial distribution is used when a researcher is interested in the occurrence of an event, not in its magnitude. For instance, in a clinical trial, a patient may survive or die. The researcher studies the number of survivors, and not how long the patient survives after treatment. Another example is whether a person is ambitious or not. Here, the binomial distribution describes the number of ambitious persons, and not how ambitious they are.
The binomial distribution is specified by the number of observations, n, and the probability of occurence, which is denoted by p.
A classic example that is used often to illustrate concepts of probability theory, is the tossing of a coin. If a coin is tossed 4 times, then we may obtain 0, 1, 2, 3, or 4 heads. We may also obtain 4, 3, 2, 1, or 0 tails, but these outcomes are equivalent to 0, 1, 2, 3, or 4 heads. The likelihood of obtaining 0, 1, 2, 3, or 4 heads is, respectively, 1/16, 4/16, 6/16, 4/16, and 1/16. In the figure on this page the distribution is shown with p = 1/2 Thus, in the example discussed here, one is likely to obtain 2 heads in 4 tosses, since this outcome has the highest probability.
Other situations in which binomial distributions arise are quality control, public opinion surveys, medical research, and insurance problems.
## Poisson Limit
If the probability p is small and the number of observations is large the binomial probabilities are hard to calculate. In this instance it is much easier to approximate the binomial probabilities by poisson probabilities. The binomial distribution approaches the poisson distribution for large n and small p. In the movie we increase the number of observations from 6 to 50, where the parameter p in the binomial distribution remains 1/10. The movie shows that the degree of approximations improves as the number of observations increases.
For moderate values of p, the binomial distribution approaches the normal distribution if the number of observations is large. This is an example of the central limit theorem.<|endoftext|>
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Multiplication is a mathematical concept generally taught in elementary school. Although it takes time and effort to learn, memorizing the basic multiplication tables can facilitate continued academic success and allow you to learn future concepts with greater ease. When solving multiplication problems, teachers will generally ask you to show your work. There are benefits to showing your work, such as a better-organized thought process, less chance for error, and the chance to receive partial credit on a test, even if the answer if incorrect. When learning how to multiply and show your work, follow a strategic plan to ensure success.
How to Learn Multiplication
Gain an understanding of the meaning of multiplication. Recognize that multiplication is an alternative for repeated addition. For example, 4 x 3 = 12 is the same as 4 + 4 + 4 = 12. Additionally, familiarize yourself with an array, which is a diagram of rows and columns that can be used to calculate a multiplication equation.
Learn strategies that will enable you to figure out multiplication equations. For the two times-table, double the original number. For the fours, double the double. For example, for 8 x 4, think 8 x 2 = 16 and 16 x 2 = 32. For the fives, skip-count by fives — 5, 10, 15, 20. For the tens, add a zero to the original number. For example, 9 x 10 = 90.
Sciencing Video Vault
Memorize the multiplication facts. Start with the zero and one times-tables, as they are the easiest and will not take long to learn. Then move on to the twos, fours, fives, tens and nines, respectively. Last, memorize the threes, sixes, sevens and eights. Memorizing facts in this order ensures that you will first memorize the facts that have the most efficient strategies.
Practice, practice, practice. There is no substitute for a lot of practice when mastering basic multiplication facts. Aim for ten minutes each night. Use a variety of methods, such as flashcards, interactive Web sites such as Multiplication.com or having another person ask you questions.
Once you have memorized the basic multiplication facts, develop mental math strategies for calculating large numbers. For example, for 15 x 8, break up the number 15 into a ten and a five: 10 x 8 = 80 and 5 x 8 = 40. 80 + 40 = 120, so 15 x 8 = 120.
How to Show Your Work
Draw a diagram if it helps you figure out the problem or equation. For example, for the equation 4 x 5, draw an array with four columns and five rows, and subsequently use skip-counting to count the rows.
Write each step as you complete it. According to Stan Brown of Tompkins Cortland Community College, showing your work means writing down enough information so that someone can see exactly how you came up with your answer. For example, if a multiplication problem involves three separate steps in order to calculate the answer, record all three steps. Writing down this amount of information is sometimes frustrating for a student who can easily calculate an answer in his head, but it increases the chances that he will earn partial marks for completing the correct steps, even if the answer is incorrect.
Reread the question to ensure that the answer seems logical and recheck your work. Often, students get caught up in solving a problem and end up calculating a piece of information that was not asked for. Rechecking the question and your work will decrease the likelihood of this happening.<|endoftext|>
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What is Demand-Pull Inflation
Demand-pull inflation is used by Keynesian economics to describe what happens when price levels rise because of an imbalance in the aggregate supply and demand. When the aggregate demand in an economy strongly outweighs the aggregate supply, prices go up. Economists describe demand-pull inflation as a result of too many dollars chasing too few goods.
Demand-pull inflation results from strong consumer demand. Many individuals purchasing the same good will cause the price to increase, and when such an event happens to a whole economy for all types of goods, it is called demand-pull inflation.
BREAKING DOWN Demand-Pull Inflation
In Keynesian theory, an increase in employment leads to an increase in aggregate demand. Due to the increase of demand, firms hire more people to increase their output. The more people firms hire, the more employment increases. Eventually, output by firms becomes so small that the prices of their goods rise.
Demand-Pull Inflation in Contrast With Cost-Push Inflation
Cost-push inflation is when price and wage go up and are transfered from one sector of the economy to another. Though they move in practically the same manner, they work on a different aspect of the whole inflationary system. Demand-pull inflation shows how price rise starts, while cost-push inflation portrays why inflation is difficult to stop once it begins.
The main idea behind demand-pull inflation is that consumer demand that outweighs aggregate supply greatly drives inflation. In a market where there are a particular number of goods and a huge demand for those goods, the prices of those goods have to rise.
Causes of Demand-Pull Inflation
There are five causes for demand-pull inflation:
- A growing economy: When consumers feel confident, they will spend more, take on more debt by borrowing more. This leads to a steady increase in demand, which means higher prices.
- Asset inflation: a sudden rise in exports, which translates to an undervaluation of the involved currencies
- Government spending: When the government opens up its pocketbooks, it drives up prices. Military spending prices may go up when the government starts to buy more military equipment.
- Inflation expectations: forecasts and expectations of inflation, where companies increase their prices to go with the flow of the expected rise
- More money in the system: demand-pull inflation is produced by an excess in monetary growth or an expansion of the money supply. Too much money in an economic system with too few goods makes prices increase.
Example of Demand-Pull Inflation
When oil refineries work at full capacity, they cause demand-pull inflation. Environmental concerns cause regulatory problems for refineries. Because of prohibitive factors by the government, supply created by oil refineries is also limited. Rather than a lack of oil or the lack of companies to produce oil, restrictive legislation prevents the market from providing optimum efficiency in producing goods with high demand. The oil industry, then, is one of the biggest contributors of demand-supply inflation.
During the U.S. economic downturn and the eurozone debt crisis, concerned investors turned to gold, buying the precious metal as a hedge against a collapse in the U.S. dollar and the euro, increasing demand for the commodity.<|endoftext|>
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Lesson 9
Solving Rate Problems
9.1: Grid of 100 (5 minutes)
Warm-up
In this warm-up, students are asked to name the shaded portion of a 10-by-10 grid, which is equal to 1. To discourage students from counting every square, flash the image for a few seconds and then hide it. Flash it once more for students to check their thinking. Ask, “How did you see the shaded portion?” instead of “How did you solve for the shaded portion?” so students can focus on the structure of the fractional pieces and tenths in the image. Encourage students to name the shaded portion in fractions or decimals. Some students may also bring up percentages.
Launch
Tell students you will show them a 10 x 10 grid for 3 seconds and that the entire grid represents 1. Their job is to find how much is shaded in the image and explain how they saw it.
Display the image for 3 seconds and then hide it. Do this twice. Give students 15 seconds of quiet think time between each flash of the image. Encourage students who have one way of seeing the grid to consider another way to determine the size of the shaded portion while they wait.
Student Facing
How much is shaded in each one?
Student Response
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Activity Synthesis
Invite students to share how they visualized the shaded portion of each image. Record and display their explanations for all to see. Solicit from the class alternative ways of quantifying the shaded portion and alternative ways of naming the size of the shaded portion (to elicit names in fractions, decimals, and percentages). To involve more students in the conversation, consider asking:
• “Who can restate ___’s reasoning in a different way?”
• “Did anyone solve the shaded portion the same way but would explain it differently?”
• “Did anyone solve the shaded portion in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”
9.2: Card Sort: Is it a Deal? (20 minutes)
Activity
Students are given cards, each of which contains an original price and a new price, as shown.
B. Juice Boxes Original: 10 for \$3.50 New Deal: 6 for \$2.40
Their job is to sort the cards into two piles: one pile for deals they would take, and another for those they would reject. There are many paths students could use to reason about whether or not to accept a deal. For example, if the original deal was \$3.50 for 10 juice boxes and the new deal is \$2.40 for 6 juice boxes, they could:
• Find and compare the unit rates for both the original pack and the new pack. If the unit rate is the same, the deal is fair. If the unit rate is lower, the clerk is offering a discount. If the unit rate is higher, the clerk is not being fair.
number of juice boxes cost in dollars dollars per box
10 3.50 0.35
6 2.40 0.40
• Find the unit rate in the original pack, apply it to the number of items in the new pack, and compare the costs for the same number of items in the original and new pricing schemes. This can be done in two ways, one focused more on column reasoning and the other on row reasoning, as shown.
• Use an abbreviated table and bypass calculating the unit rate. Find the multiplier to get from the original to the new number of items, and use the same multiplier to find what the price would be if the deal has not changed. Compare the actual new price to this projected price.
As students work, attend to how they reason about the deals and make their decisions (deal or no deal).
Launch
Show the picture on Card A (or use an actual 4-pack of a beverage with a missing bottle.) Present the following situation:
“You’ve entered a local shop to buy a 4-pack of drinks. You find one last pack of the drink you want on the shelf and, unfortunately, only 3 bottles remain in that pack. You decide to buy it anyway. You take the 3-pack to the check-out counter and ask the clerk to consider a fair price for the incomplete pack. If the cost of a 4-pack was $3.16 and the clerk offers to sell the 3 pack for$2.25, will you take the deal?”
Poll the class for their response and display how many students would and would not take the deal. Then, ask “How could you figure out if the deal is good or not?” Give students a moment of quiet think time to come up with strategies for solving such a problem and then invite a few students to share.
Arrange students in groups of 2. Give each group a set of five cards A–E (or six cards A–F if including the extension problem). Tell students their job is to sort the cards into a ‘Deal’ pile and a ‘No Deal’ pile. Instruct partners to collaborate in finding the answer for card A and divide up the remaining cards between them. Ask students to first work on their cards individually, then share their reasoning with their partner, and lastly, sort the cards together.
Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have identified which initial cards were good deals.
Supports accessibility for: Conceptual processing; Organization
Student Facing
Your teacher will give you a set of cards showing different offers.
1. Find card A and work with your partner to decide whether the offer on card A is a good deal. Explain or show your reasoning.
2. Next, split cards B–E so you and your partner each have two.
1. Decide individually if your two cards are good deals. Explain your reasoning.
2. For each of your cards, explain to your partner if you think it is a good deal and why. Listen to your partner’s explanations for their cards. If you disagree, explain your thinking.
3. When you and your partner are in agreement about cards B–E, place all the cards you think are a good deal in one stack and all the cards you think are a bad deal in another stack. Be prepared to explain your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Time to make your own deal! Read the information on card F and then decide what you would charge if you were the clerk. When your teacher signals, trade cards with another group and decide whether or not you would take the other group’s offer.
Keep in mind that you may offer a fair deal or an unfair deal, but the goal is to set a price close enough to the value it should be that the other group cannot immediately tell if the deal you offer is a good one.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Students who are not fluent in multiplication and division computation work from grade 5 may need some review in order to be successful in this activity.
Activity Synthesis
Select 2–3 students who used different but effective strategies to share their thinking with the class. Encourage students to listen to others’ reasoning. Record the different strategies in one place and display them for all to see. Highlight any similarities and differences (e.g., whether a unit rate was used, whether students compare the original unit rate to the new quantity or the other way around, etc.)
Writing, Listening, Conversing: MLR2 Collect and Display. Listen and observe how students reason about the deals and make their decisions (deal or no deal), and make note of the different strategies students use to compare unit rates. Listen for language such as “the same,” “equal,” “unit rate,” “cost for the same number of items,” etc., and display these visually for the whole class to use as a reference. Continue to add to and refer to the display during the whole class discussion, making explicit connections between the language and the strategies used (i.e., whether students compare the original unit rate to the new quantity or the other way around). This will help students make sense of calculating and comparing unit rates while increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness
9.3: The Fastest of All (15 minutes)
Activity
In this activity, students convert between customary and metric units in order to compare lengths. To make some measurements comparable to others, students need to perform multistep conversions and activate arithmetic skills from previous grades. Support students with computations as needed and provide access to calculators as appropriate. Share the following information with students when requested.
• 1 mile = 1,760 yards = 5,280 feet
• 1 yard = 3 feet
• 1 foot = 12 inches
• 1 kilometer = 1,000 meters
• 1 meter = 100 centimeters
Expect students to choose different units of measurements to make comparisons. As students work, identify those who opt for the same unit so that they can partnered or grouped together for discussion.
Launch
Give students 1–2 minutes to read the task, and then ask how they think they could compare these lengths. Students are likely to suggest converting all the lengths into the same unit of measurement. Ask students which units might be appropriate in this case and why. (Feet, yards, and meters are better choices than inches or miles.) After discussing some appropriate options, give students quiet think time to complete the activity, and then time to share their explanation with one or more students who have chosen to use the same unit of measurement.
Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2-3 minutes of work time. Look for students who are converting all the distances to the same length.
Supports accessibility for: Memory; Organization
Student Facing
Wild animals from around the world wanted to hold an athletic competition, but no one would let them on an airplane. They decided to just measure how far each animal could sprint in one minute and send the results to you to decide the winner.
You look up the following information about converting units of length:
1 inch = 2.54 centimeters
animal sprint distance
cougar 1,408 yards
antelope 1 mile
hare 49,632 inches
kangaroo 1,073 meters
ostrich 1.15 kilometers
coyote 3,773 feet
1. Which animal sprinted the farthest?
2. What are the place rankings for all of the animals?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Some students may need to be prompted about the intermediate steps needed to compare units that require several conversions before they can be compared.
Activity Synthesis
Poll the class to see if they agree on who took first, second, third, and last place. If there is widespread agreement, invite two students to share: one student who converted all measurements to feet or yards, and another who converted everything to meters. If there are discrepancies, list the distances run by each animal in each unit of measurement and display them for all to analyze and double check. While the numerical values of the measurements in feet will all be greater than those in meters, the rank order will come out the same.
Speaking, Representing: MLR7 Compare and Connect. Use this routine when students present their strategy and representation for determining the place rankings for all of the animals. Ask students to consider what is the same and what is different about each approach. Draw students’ attention to the different units of measurements used to make comparisons, while making connections to the strategies used to make conversions. These exchanges can strengthen students’ mathematical language use and reasoning to make sense of strategies used to convert units to be able to make comparisons.
Design Principle(s): Maximize meta-awareness
Lesson Synthesis
Lesson Synthesis
Emphasize that when we want to compare rates, a straightforward way is to compare unit rates. For example, when we were comparing the best deal, an example was 10 juice boxes for \$3.50 or 6 juice boxes for \$2.40. It may be helpful to draw two tables or two double number lines to facilitate discussion. Questions to discuss:
• “What are two associated unit rates that we could compare?” (0.35 and 0.4)
• “How were they computed?” (Divide 3.5 by 10 and divide 2.4 by 6)
• “What do these numbers mean in this context?” (They are each the price per bottle for the different offers. For example, \$0.35 for 1 bottle.) 9.4: Cool-down - Tacos by the Pack (5 minutes) Cool-Down For access, consult one of our IM Certified Partners. Student Lesson Summary Student Facing Sometimes we can find and use more than one unit rate to solve a problem. Suppose a grocery store is having a sale on shredded cheese. A small bag that holds 8 ounces is sold for \$2. A large bag that holds 2 kilograms is sold for \$16. How do you know which is a better deal? Here are two different ways to solve this problem: Compare dollars per kilogram. • The large bag costs \$8 per kilogram, because $$16 \div 2 = 8$$.
• The small bag holds $$\frac12$$ pound of cheese, because there are 16 ounces in 1 pound, and $$8 \div 16 = \frac12$$.
The small bag costs \$4 per pound, because $$2 \div \frac12 = 4$$. This is about \$8.80 per kilogram, because there are about 2.2 pounds in 1 kilogram, and $$4.00 \boldcdot 2.2 = 8.80$$.
The large bag is a better deal, because it costs less money for the same amount of cheese.
Compare ounces per dollar.
• With the small bag, we get 4 ounces per dollar, because $$8 \div 2 = 4$$.
• The large bag holds 2,000 grams of cheese. There are 1,000 grams in 1 kilogram, and $$2 \boldcdot 1,\!000 = 2,\!000$$. This means 125 grams per dollar, because $$2,\!000 \div 16 = 125$$.
There are about 28.35 grams in 1 ounce, and $$125 \div 28.35 \approx 4.4$$, so this is about 4.4 ounces per dollar.
The large bag is a better deal, because you get more cheese for the same amount of money.
Another way to solve the problem would be to compare the unit prices of each bag in dollars per ounce. Try it!<|endoftext|>
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# Question 5aed5
Oct 7, 2016
$y \in \left\{\frac{2}{5} , 2 , 3 , \frac{28}{5}\right\}$
#### Explanation:
The Pythagorean theorem's converse is also true, and states that if the sum of the squares of two sides of a triangle equals the square of the third side, then the triangle is a right triangle.
Using the distance formula
"dist"((x_1, y_1)"",(x_2, y_2)) = sqrt((x_2-x_1)^2+(y_2-y_1)^2)#
we can calculate the squares of the lengths of each side:
$A {B}^{2} = {\left(3 - 2\right)}^{2} + {\left(5 - 0\right)}^{2} = 26$
$A {C}^{2} = {\left(0 - 2\right)}^{2} + {\left(y - 0\right)}^{2} = {y}^{2} + 4$
$B {C}^{2} = {\left(0 - 3\right)}^{2} + {\left(y - 5\right)}^{2} = {y}^{2} - 10 y + 34$
Now we can consider all cases in which two of those sum to the third, and what values of $y$ make that true.
Case 1: $A {B}^{2} + A {C}^{2} = B {C}^{2}$
$\implies {y}^{2} + 30 = {y}^{2} - 10 y + 34$
$\implies 10 y = 4$
$\implies y = \frac{2}{5}$
Case 2: $A {B}^{2} + B {C}^{2} = A {C}^{2}$
$\implies {y}^{2} - 10 y + 60 = {y}^{2} + 4$
$\implies 10 y = 56$
$\implies y = \frac{28}{5}$
Case 3: $A {C}^{2} + B {C}^{2} = A {B}^{2}$
$\implies 2 {y}^{2} - 10 y + 38 = 26$
$\implies 2 {y}^{2} - 10 y + 12 = 0$
$\implies {y}^{2} - 5 y + 6 = 0$
$\implies \left(y - 2\right) \left(y - 3\right) = 0$
$\implies y - 2 = 0 \mathmr{and} y - 3 = 0$
$\implies y = 2 \mathmr{and} y = 3$
By the Pythagorean theorem's converse, we know that all of the above $y$ values result in right triangles, and as those are the only values which result in the Pythagorean theorem holding true, we know that there are no other values for $y$ which work. Thus we have the solution set
$y \in \left\{\frac{2}{5} , 2 , 3 , \frac{28}{5}\right\}$<|endoftext|>
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Which rights are counted in under "women's rights" has diverse through time and cultures. Even nowadays, there is some dissimilarity about what establish women's rights. Does a woman have a right to control family size? To equality of treatment in the workplace? To equality of access to military assignments?
INTERNATIONAL AND REGIONAL WOMEN RIGHTS INSTRUMENTS
Since the founding of the United Nations, equality between men and women has been among the most fundamental guarantees of human rights. Furthermore, Article 1 of the Charter stipulates that one of the purposes of the United Nations is to promote respect for human rights and fundamental freedoms “without distinction as to race, sex, language or religion”. This prohibition of discrimination based on sex is repeated in its Articles 13 (mandate of the General Assembly) and 55 (promotion of universal human rights). Later, many international and regional instruments have been adopted which declared women rights.......
- Universal Declaration of Human Rights (1948)
- Convention on the Political Rights of Women (1952)
- International Covenant on Civil and Political Rights (1966)
- International Covenant on Economic, Social and Cultural Rights (1966)
- Declaration on the Elimination of All Forms of Discrimination against Women (1967)
- Declaration on the Protection of Women and Children in Emergency and Armed Conflict (1974)
- Convention on the Elimination of All Forms of Discrimination against Women (1979)
- Declaration on the Elimination of Violence against Women (1993)
- Inter-American Convention for the Prevention, Punishment and Elimination of Violence against Women (Belém do Pará Convention) (1995)
- Universal Declaration on Democracy (1997)
- Optional Protocol to the Convention on the Elimination of All Forms of Discrimination against Women
1. Universal Declaration of Human Rights (1948):
In 1948, the Universal Declaration of Human Rights was adopted. It, too, proclaimed the equal entitlements of women and men to the rights contained in it, “without distinction of any kind, such as ... sex, ….” In drafting the Declaration, there was considerable discussion about the use of the term “all men” rather than a gender-neutral term. The Declaration was eventually adopted using the terms “all human beings” and “everyone” in order to leave no doubt that the Universal Declaration was intended for everyone, men and women alike.
2. Convention on the Political Rights of Women (1952):
The Convention on the Political Rights of Women was approved by the United Nations General Assembly during the 409th plenary meeting, on 20 December 1952, and adopted on 31 March 1953. The Convention was the first international legislation protecting the equal status of women to exercise political rights. Moreover, it was the first international treaty to obligate its states to protect citizens' political rights. The Convention was one of the United Nations' several efforts in the postwar period to set standards of nondiscrimination against women; others were the Convention on the Nationality of Married Women and the Convention on Consent to Marriage, Minimum Age for Marriage and Registration of Marriages, brought into force in 1958 and 1964, respectively.
Continue............. Follow for next part.
Declaration: All data which are provide in this writing, All are secondary data. Which I have collected from different writing and documents......<|endoftext|>
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# 3 digit by 3 digit Multiplication
### 3 digit by 3 digit Multiplication Examples
3 digit by 3 digit Multiplication – Example 1
Find the product.
235x 624
Explanation
Whenever we multiply a three digit Multiplicand by three-digit Multiplier, we first multiply the Multiplicand with one’s place of Multiplier, then with Tens place of the Multiplier and lastly by Hundreds place.
Step I: Multiply Multiplicand with one’s place of Multiplier.
i.e, 235 x 4 = 940
Step II: Multiply Multiplicand with tens place of Multiplier.
i.e, 235 x 2 = 470
Since the Number obtained at Step II, has a Tens value, we add a “0” or “X” at the end of the answer obtained by multiplying Multiplicand with Tens place of the Multiplier.
Step III: Multiply Multiplicand with hundreds place of Multiplier.
i.e, 235 x 6 = 1410
Since the Number obtained at Step III, has a Hundreds place value, we add two “0” or “x” at the end of the answer obtained by multiplying Multiplicand with Tens place of the Multiplier.
Step III : Add all the numbers
Hence, 235 x 624 = 146640
3 digit by 3 digit Multiplication – Example 2
Find the product.
343 x 732
Explanation
Whenever we multiply a three digit Multiplicand by three digit Multiplier, we first multiply the Multiplicand with ones place of Multiplier, then with Tens place of the Multiplier and lastly by Hundreds place.
Step I : Multiply Multiplicand with ones place of Multiplier.
i.e, 343 x 2 = 686
Step II : Multiply Multiplicand with tens place of Multiplier.
i.e, 343 x 3 = 1029
Since the Number obtained at Step II, has a Tens value, we add a “0” or “X” at the end of the answer obtained by multiplying Multiplicand with Tens place of the Multiplier.
Step III : Multiply Multiplicand with hundreds place of Multiplier.
i.e, 343 x 7 = 2401
Since the Number obtained at Step III, has a Hundreds place value, we add two “0” or “x” at the end of the answer obtained by multiplying Multiplicand with Tens place of the Multiplier.
Step III : Add all the numbers
Hence, 343 x 732 = 251076<|endoftext|>
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At least 4 pages ,double space, using 5-8 sources is good
ANALYSIS PAPER: This paper will analyze a text. “Text” is defined as anything that is “read” or interpreted. This paper should focus on how texts are interpreted or how meaning is created. Secondary research should be used, and primary research may be used if appropriate to the topic. A successful paper will interact with multiple sources, build one’s own ideas upon those sources, and argue logically.
Summarize(THRORY): You will use your own words to explain clearly what happened. If it is an event, you will describe the situation, people and circumstances. If you are dealing with a text or a performance, you will explain the author’s thesis, purpose and audience. Your summary is intended to help your audience understand this subject clearly and thoroughly.
Analyze: Next, you will explain the meaning of this event, text or performance. You will evaluate what happened and explain how it is either good, bad or both. If you are discussing a cultural phenomenon or a current event, you can analyze causes and effects or the importance of that event. For a written text or a performance, you will tell how well the author conveys their intentions to the audience. Are they convincing? What are the weaknesses?
Respond(INTERPRETATION) : Often, a Critical Analysis assignment requires you to present the summary and analysis objectively. However, another way of writing a critical analysis is to include your own point of view. Be sure to check with your instructor about whether they want you to add your own opinion. If you write this paper with your own personal opinion included, it is sometimes called a Summary, Analysis Response.
Chose one from the topics:
1.The effect of speak in a euphemistic way
OR.Write an essay in which you consider the effects of doublespeak. Is it always a form of lying? Is it harmful to our society, and if so, how? How can we measure its effect?
2.Multicultural Families. Adoption and intermarriage between people of different cultures and races has created more families of mixed races and cultures in the United States and other countries. Describe this situation and analyze how this affects individuals in those families, and the cohesion of the family as a whole.
- The paper must be properly documented and formatted according to MLA, or APA style. The style chosen should be appropriate to the topic and should be consistently followed throughout. Your research may be primary (interviews, surveys) or secondary (books, articles). For your secondary research, print sources are preferred (books, periodicals, newspapers).<|endoftext|>
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Did you know that the coast redwood forest is home to the largest slug in North America and the second largest slug worldwide*? Yep, our very own banana slug (Ariolimax columbianus) grows up to 8 inches in length and can live for 7 years. Banana slugs depend upon the moist habitat provided by Pacific Northwest forests and crawl along their own slime trails in coniferous forests up through British Columbia. Banana slugs secrete slime to aid in their locomotion, as defense from predators (beetles and racoons), for reproductive purposes, and to keep from drying out. They eat a little bit of everything in the redwood forest including live plants, fallen leaves, and dead animals, but never munch on the redwoods themselves. The striking yellow color of the banana slug varies with the diet of the slug, light conditions, and general health, but it isn’t uncommon to see slugs with dark brown spots that help the slug hide on the forest floor. Banana slugs breathe through a special lung called a pulmonata that is visible as a “hole” at the back of the head.
Perhaps the most fascinating aspect of the banana slug is their mating behavior. These slugs are hermaphrodites and can actually fight quite aggressively before mating (the slug equivalent of biting). It isn’t uncommon for one of the slugs to chew the penis of its mate and render it unable to fertilize other slug eggs.
Once again, truth is stranger than fiction.
Love these facts? Check out our Fun Forest Facts and leave a comment telling us what you’d like to know about in the redwood forest!
*The world’s largest slug is found in Europe, is gray, and grows up to 12 inches in length (Limax cinereoniger).<|endoftext|>
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An arterial ischemic stroke (AIS) occurs when blood flow in an artery to the brain is blocked by narrowing of the artery, or when a blood clot forms in the artery and blocks the supply of blood to a part of the brain. Ischemia, or ischemic, comes from the Greek word to "keep back" or "stop" the supply of blood to an area.
A blockage of blood flow in an artery can happen in several different ways:
- Embolism: The blood clot can form in some other part of the body (often the heart) and travel to the brain. It gets stuck in a blood vessel of the brain. This moving clot is often called an embolus or embolism. This clot happens most of the time in a child with underlying heart disease.
- Thrombosis: The blood clot can form over time in an artery that supplies blood to the brain. It can eventually block the artery and the supply of blood to the brain. This clot is sometimes called a thrombus.
- Clotting disease: Thick blood. Some children have underlying conditions that cause their blood to "thicken" and clot.
- Narrow arteries: Other children may have arteries that are very narrow leading to a risk that even a tiny clot could get stuck in them, or causing decrease in blood flow to a low enough level to cause a stroke.
- Damaged arteries: Some children may have arteries that are damaged by trauma or inflammation leaving a rough or jagged inner lining where blood clots can get stuck. These clots can build up enough so that eventually the artery is clogged and no blood can flow through.
The blood carries oxygen and other important nutrients to the brain. The brain needs oxygen to survive. If a part of the brain does not receive oxygen from the blood for a certain period of time, the tissue in that part of the brain will become damaged or die.
For many children, it can be hard to find out exactly what caused the blockage of the artery that led to the stroke. Some of the leading causes of childhood stroke include:
- Heart disease - mostly congenital heart defects (present at birth)
- Sickle cell anemia
- Narrow or damaged blood vessels in the brain or neck
- Accidents involving the head and neck region
- Blood clotting disorders
- Moyamoya disease
- Head and neck radiation for cancer
- Serious infections, especially meningitis from bacteria
- Drug abuse
- Other childhood diseases (for example, Chicken Pox)
- Chronic metabolic disorders
In children and teenagers
Children and teenagers may have one or more symptoms when they are having an arterial ischemic stroke. Typically these symptoms occur suddenly. They include:
- Weakness or numbness of the face, arm or leg, usually on one side of the body
- Trouble walking due to weakness or trouble moving one side of the body
- Problems speaking or understanding language, including slurred speech, trouble trying to speak, inability to speak at all, or difficulty in understanding simple directions
- Severe headache especially with vomiting, sleepiness or double vision
- Trouble seeing clearly in one or both eyes
- Severe dizziness or unsteadiness that may lead to losing balance or falling
- New appearance of seizures, especially if they affect one side of the body and are followed by paralysis on the side of the seizure activity
In newborns and infants:
- Extreme sleepiness
- A tendency to use only one side of their body
Sometimes there are no symptoms with newborns and very young infants until they grow older. Then, they may have trouble with movement on one side of the body.
If your child is experiencing any of the symptoms of a stroke, you should:
- Dial 911 or go to your nearest hospital Emergency Department.
- Have your child lie flat.
- Do not give your child anything to eat or drink.
- Suggest that your local medical professional contact the CHOP Stroke Program for consultation. They may contact the CHOP operator at 215-590-1000 and ask for the Stroke team to be paged.
The Stroke team will want to find out as much information as possible to diagnose your child's stroke and the reasons why the stroke occurred.
The Stroke team will perform a thorough physical exam including a neurological exam to gain more information about how your child's brain is currently working.
The Stroke team will request that blood tests be done to test to see if your child has any underlying blood problems that could cause a stroke.
The Stroke team will request that various types of tests be done that will give a better picture or "image" of your child's brain. These tests may include computed tomography (CT) scans, magnetic resonance imaging (MRI), magnetic resonance angiogram (MRA) or cerebral angiogram (also called an arteriogram). If the Stroke Program team feels more information about your child's heart may be useful, they may order an echocardiogram.
Depending on your child's symptoms, the Stroke team may feel a need to order some other tests. These may include a lumbar puncture (LP) also known as a "spinal tap" to look for signs of infection or inflammation that may have caused the stroke.
Immediately after the stroke, the treatment given to your child will focus on things that will prevent the stroke from getting worse. This will include ensuring that your child has enough fluids (given through an IV or intravenous needle typically put in a vein in the child's arm or hand) and keeping the child lying flat. Medications for any pain symptoms may be given as well.
Further treatment will vary depending on the suspected cause of the stroke. Some treatment may focus on preventing blood clots from happening in the future. In this case anticoagulant ("anti-clot") medications (such as COUMADIN, HEPARIN or LOVENOX) or an antiplatelet medication such as aspirin (that prevents blood platelets from clumping together to form a clot) may be prescribed. In addition, other treatment will focus on preventing fever and keeping the sugar levels in the blood at normal levels.
How much the stroke will affect your child's day-to-day life depends on the location and severity of the stroke. Some strokes cause mild problems. Some cause more severe problems. Some children may continue to have seizures.
Your child's Stroke team will consult with the Rehabilitation team for advice on how best to help your child recover. Rehabilitation uses a structured series of exercises to help your child recover from the effects of the stroke. Therapy usually begins within 48 hours of admission, provided your child is medically stable and is able to tolerate these activities. Members of the Rehabilitation team may include a physical therapist (PT), occupational therapist (OT), speech therapist, neuropsychologist and physiatrist (Rehab or "PM &R" Physician).
Depending on the results of their assessments, it may be recommended that your child have further rehabilitation. This rehabilitation could range from a stay in the inpatient rehabilitation unit, to Day Hospital rehabilitation, to outpatient rehabilitation appointments. Rehabilitation may last from weeks to months depending on your child's needs.
It's important to keep in mind that children recover more quickly from a stroke as compared to adults. Members of the Stroke team and the Rehabilitation team will continue to assist you and your child in the recovering process and help your child return to a more normal lifestyle as soon as possible. If your child is school age, we will work with your child's school to recommend any needed changes to your child's curriculum and school day.<|endoftext|>
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Cell division is the process by which a parent cell divides into two or more daughter cells. Cell division is usually a small segment of a larger cell cycle. This type of cell division in eukaryotes is known as mitosis, and leaves the daughter cell capable of dividing again. The corresponding sort of cell division in prokaryotes is known as binary fission. In another type of cell division present only in eukaryotes, called meiosis, a cell is permanently transformed into a gamete and may not divide again until fertilization. Right before the parent cell splits, it undergoes DNA replication.
Asexual reproduction is a mode of reproduction by which offspring arise from a single parent, and inherit the genes of that parent only; it is reproduction which does not involve meiosis, ploidy reduction, or fertilization. The offspring will be exact genetic copies of the parent.
While all prokaryotes reproduce asexually (without the formation and fusion of gametes), mechanisms for lateral gene transfer such as conjugation, transformation andtransduction are sometimes likened to sexual reproduction<|endoftext|>
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Trapezoids and Kites
# Trapezoids and Kites
## Trapezoids and Kites
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##### Presentation Transcript
1. Trapezoids and Kites Notes 24 – Section 6.6
2. Essential Learnings • Students will understand and be able to recognize and apply the properties of trapezoids and kites.
3. Trapezoid • A trapezoid is a quadrilateral with exactly one pair of parallel sides. Base Leg Leg Base
4. Isosceles Trapezoid • If the legs of a trapezoid are congruent then it is an isosceles trapezoid. Base Leg Leg Base
5. Isosceles Trapezoids • If a trapezoid is isosceles, then each pair of base angles is congruent. A B D C
6. Isosceles Trapezoids • If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid. A B D C
7. Isosceles Trapezoids • A trapezoid is isosceles if and only if its diagonals are congruent. A B D C
8. Example 1 JKLM is an isosceles trapezoid. Find m∠K and x. J K 2x 14 112º L M
9. Midsegment of a Trapezoid • The midsegment of a trapezoid is a segment that connects the midpoints of the legs of the trapezoid.
10. Trapezoid Midsegment Theorem • The midsegment of a trapezoid is parallel to each base and its measure is one half the sum of the lengths of the bases. x m y
11. Example 2 For trapezoid JKLM, P and Q are midpoints of the legs. If JK = 18 and PQ = 28, find LM. J K P Q M L
12. Kite • A kite is a quadrilateral with exactly two pairs of consecutive congruent sides.
13. Kites • If a quadrilateral is a kite, then its diagonals are perpendicular. • One diagonal is bisected. B A C D
14. Kites • If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent.
15. Example 3 If WXYZ is a kite, find m∠XYZ. W Z 48º 110º X Y
16. Example 4 If MNPQ is a kite, find NP. P 4 N Q 7 11 M
17. Example 5 If m∠WXY = 72, m∠WZY = 4x and m∠ZWX = 10x, find m∠ZYX. W U Z X Y
18. Assignment Pages 440-442: 8-11, 17, 19, 21, 26, 27, 35-39, 41, 43, 49, 50, 53-57 Unit Study Guide 4 – Due Monday (12/10)<|endoftext|>
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### During a day, the hour hand and the minute hand of a clock form a right angle, at multiple times. For example, the two hands form a right angle at 9 am. How many times during a day (24 hours) will the two hands form a right angle?
44
Step by Step Explanation:
1. Let us look at the clock at 9 am:
We can see that the hour hand and the minute hand are making a right angle.
2. In a clock, while the minute hand moves, the hour hand also moves, although a lot slowly.
It is easy to see that in a 12 hour period, the minute hand make 12 revolutions while, the hour hand makes one.
We can visualize the above statement this way:
If we hold the clock in our hands and always keep on rotating it slowly, such that the hour hand always stay on the same position, the minute hand will make 12 - 1 = 11 revolutions.
In other words, the minute hand makes 11 revolutions around the hour hand in a 12 hour period.
3. For each revolution around the hour hand, the minute hand makes a right angle twice with it. The total number of times we see the two hands making a right angle is 11 × 2 = 22.
4. In 12 hours, the number of times the two hands make a right angle = 22
In 24 hours, the number of times the two hands make a right angle = 22 × 2 = 44<|endoftext|>
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# squiz1 - teams we get the same game so the total number of...
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Solutions to quiz # 1 (January 12) 1. a) We can choose a committee of 3 from a group of 10 people in ± 10 3 ² = 10! 3!7! ways and then we can choose a committee of 4 from the group of remaining 7 people in ± 7 4 ² = 7! 4!3! ways. Altogether, there are 10! 3!7! · 7! 4!3! = 10! 4!3!3! = 4200 ways to choose the committees. Answer: the two committees can be chosen in 10! 4!3!3! ways. b) We can choose the first team in ± 10 4 ² = 10! 4!6! ways, after which we can choose the second team in ± 6 4 ² = 6! 4!2! ways. Hence the total number of ways to choose first the first team and then the second team is 10! 4!6! · 6! 4!2! = 10! 4!4!2! ways. However, the order in which the teams are chosen does not matter (if we switch the
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Unformatted text preview: teams we get the same game), so the total number of diļ¬erent games is 1 2 Ā· 10! 4!4!2! = 1575 . Answer: there are 10! 4!4!2!2! diļ¬erent games. 2. The sample space is { AB,AC,BC } . We have P ( { AB,AC } ) = P ( AB ) + P ( AC ) = 5 8 , P ( { AB,BC } ) = P ( AB ) + P ( BC ) = 5 8 and P ( AB ) + P ( AC ) + P ( BC ) =1 . Therefore, P ( { AC,BC } ) = P ( AC ) + P ( BC ) =2 Ā³ P ( AB ) + P ( AC ) + P ( BC ) Ā“-Ā³ P ( AB ) + P ( AC ) Ā“-Ā³ P ( AB ) + P ( BC ) Ā“ =2-5 8-5 8 = 3 4 . Answer: The probability that one of the chosen courses is Communications is 3 / 4. 1...
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Introduction to R for non-programmers using gapminder data.
The goal of this lesson is to teach novice programmers to write modular code and best practices for using R for data analysis. R is commonly used in many scientific disciplines for statistical analysis and its array of third-party packages. We find that many scientists who come to Software Carpentry workshops use R and want to learn more. The emphasis of these materials is to give attendees a strong foundation in the fundamentals of R, and to teach best practices for scientific computing: breaking down analyses into modular units, task automation, and encapsulation.
Note that this workshop will focus on teaching the fundamentals of the programming language R, and will not teach statistical analysis.
The lesson contains more material than can be taught in a day. The instructor notes page has some suggested lesson plans suitable for a one or half day workshop.
A variety of third party packages are used throughout this workshop. These are not necessarily the best, nor are they comprehensive, but they are packages we find useful, and have been chosen primarily for their usability.
Understand that computers store data and instructions (programs, scripts etc.) in files. Files are organised in directories (folders). Know how to access files not in the working directory by specifying the path.
|Setup||Download files required for the lesson|
|00:00||1. Introduction to R and RStudio||
How to find your way around RStudio?
How to interact with R?
How to manage your environment?
How to install packages?
|00:55||2. Project Management With RStudio||How can I manage my projects in R?|
|01:25||3. Seeking Help||How can I get help in R?|
|01:45||4. Data Structures||
How can I read data in R?
What are the basic data types in R?
How do I represent categorical information in R?
|02:40||5. Exploring Data Frames||How can I manipulate a data frame?|
|03:10||6. Subsetting Data||How can I work with subsets of data in R?|
|04:00||7. Control Flow||
How can I make data-dependent choices in R?
How can I repeat operations in R?
|05:05||8. Creating Publication-Quality Graphics with ggplot2||How can I create publication-quality graphics in R?|
|06:25||9. Vectorization||How can I operate on all the elements of a vector at once?|
|06:50||10. Functions Explained||How can I write a new function in R?|
|07:50||11. Writing Data||How can I save plots and data created in R?|
|08:10||12. Splitting and Combining Data Frames with plyr||How can I do different calculations on different sets of data?|
|09:10||13. Dataframe Manipulation with dplyr||How can I manipulate dataframes without repeating myself?|
|10:05||14. Dataframe Manipulation with tidyr||How can I change the format of dataframes?|
|10:50||15. Producing Reports With knitr||How can I integrate software and reports?|
|12:05||16. Writing Good Software||How can I write software that other people can use?|
The actual schedule may vary slightly depending on the topics and exercises chosen by the instructor.<|endoftext|>
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# Homogeneous Functions
We start with equivalence of two integrals $\displaystyle\int\frac{dx}{1+x^2}$ and $\displaystyle\int\frac{dx}{p^{2}+x^2}.$ I assume here that $p$ is a real nonzero number. For $p = 1,$ the second integral reduces to the first and, therefore, is a clear generalization. However, once we know how to compute the more special integral, we may, using a standard substitution, compute the more general one.
Indeed, let $u = x/p.$ Then successively $x = pu,$ $dx = pdu,$ and
$\displaystyle\int\frac{dx}{p^{2}+x^2}=\frac{1}{p^2}\int\frac{p\space du}{1+u^2}.$
Since
$\displaystyle\int\frac{dx}{1+x^2}=\arctan(x)+C$
where $C$ is an arbitrary constant, this leads to
$\displaystyle\int\frac{dx}{p^{2}+x^2}=\frac{1}{p}\arctan(\frac{x}{p})+C.$
The situation, where one parameter appears superfluous, is quite common to the class of homogeneous functions.
## Definition
A function $f(x, y)$ such that, for $t\gt 0,$ $f(tx, ty) = t^{a}f(x, y),$ is called (positively) homogeneous of order (or degree) $a.$
## Examples
1. $f(x, y) = x^{2} + y^{2}$ is homogeneous of order 2. Indeed, $f(tx,ty) = (tx)^{2} + (ty)^{2} = t^{2}(x^{2} + y^{2}) = t^{2}f(x,y).$
2. $f(x, y) = (x^{2} + y^{2})/x$ is homogeneous of order $1$
3. $f(x, y) = \sqrt{x^{2} + y^{2}}$ is also homogeneous of order $1$
4. $f(x, y) = (x^{2} - y^{2})/(x^{2} + y^{2})$ has order $0.$
Functions in examples 1 and 3 are also symmetric because they do not change when one swaps values of $x$ and $y.$
The famous Binomial Theorem (discovered and repeatedly used but not proved by I. Newton) may be written in two equivalent forms:
1. $\displaystyle (1+x)^{a}=1+\frac{a}{1}x+\frac{a(a-1)}{1\cdot 2}x^{2}+\frac{a(a-1)(a-2)}{1\cdot 2\cdot 3}x^{3}+\ldots$
2. $\displaystyle (x+y)^{a}=x^{a}+\frac{a}{1}x^{a-1}y+\frac{a(a-1)}{1\cdot 2}x^{a-2}y^{2}+\frac{a(a-1)(a-2)}{1\cdot 2\cdot 3}x^{a-3}y^{3}+\ldots$
All terms in the latter are homogeneous of order $a.$
The notion of homogeneity extends to functions of more than $2$ variables. For example, all kinds of means are symmetric and naturally homogeneous of order $1.$ For $N$ variables,
1. Arithmetic mean, $a(x, y, z, ...) = (x + y + z + ...)/N$
2. Geometric mean, $g(x, y, z, ...) = (x\cdot y\cdot z\cdot ...)^{1/N}$
3. Harmonic mean, $h(x, y, z, ...) = N/(1/x + 1/y + 1/z + ...)$
A famous inequality relates arithmetic and geometric means of nonnegative numbers: $a(x, y, z, ...) ≥ g(x, y, z, ...).$ If we can prove this inequality for powers of $2,$ $N = 2^{n},$ then from a previous result it will follow for all integer $N.$ Let's use induction.
Let $n = 1.$ The inequality is then equivalent to $(x + y)^{2} \ge 4xy$ which is true because, by rearranging terms, it reduces to $(x - y)^{2} \ge 0$ which is obviously true.
For the inductive step, assume the inequality has been proven for $N = 2^{k},$ and let there be given $N = 2^{k+1}.$ Note that $N = 2^{k+1} = 2\cdot 2^{k} = 2M.$ Thus we may split the given set of $N$ numbers into two groups of $M$ elements each. Let these be $x, y, z, \ldots$ and $u, v, w, \ldots.$ We have
\begin{align} a(x, y, z, ..., u, v, w, \ldots ) &= (x + y + z + \ldots + u + v + w + \ldots )/N\\ &= [(x + y + z + \ldots )/M + (u + v + w + \ldots )/M]/2\\ &= (a(x, y, z, \ldots ) + a(u, v, w, \ldots ))/2\\ &= a(a(x, y, z, \ldots ), a(u, v, w, \ldots )). \end{align}
Quite similarly $g(x, y, z, \ldots , u, v, w, \ldots ) = g(g(x, y, z, \ldots ),g(u, v, w, \ldots )).$ Since, by the inductive assumption,
$a(x, y, z, \ldots ) \ge g(x, y, z, \ldots )$ and $a(u, v, w, \ldots ) \ge g(u, v, w, \ldots )$
we finally have
\begin{align} a(x, y, z, \ldots , u, v, w,\ldots ) &= a(a(x, y, z, \ldots ), a(u, v, w, \ldots ))\\ & \ge a(g(x, y, z, \ldots ), g(u, v, w, \ldots ))\\ & \ge g(g(x, y, z, \ldots ), g(u, v, w, \ldots ))\\ &= g(x, y, z, \ldots , u, v, w, \ldots ).\\ \end{align}
Q.E.D.
### References
1. G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge University Press (2nd edition) 1988.
2. G. Polya, Mathematical Discovery, John Wiley & Sons, 1981<|endoftext|>
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# Solving algebraic equations
How may I make n and k in terms of x and y in this equation?:
$$\frac{1-n-i k}{1+n+i k}=\sqrt{x} e^{i y}$$
I have tried separating the real and imaginary terms on the left hand side using complex expand so that I can have 2 sets of equations now but I have no idea how to continue?
ComplexExpand[((1 - n - I k)/(1 + n + I k))]
by hand:
\begin{align*} \frac{1-n-ik}{1+n+ik} & =\sqrt{x}e^{iy}\\ \frac{\left( 1-n-ik\right) \left( 1+n-ik\right) }{\left( 1+n+ik\right) \left( 1+n-ik\right) } & =\sqrt{x}\left( \cos y+i\sin y\right) \\ \frac{-k^{2}-2ik-n^{2}+1}{k^{2}+n^{2}+2n+1} & =\sqrt{x}\cos y+i\sqrt{x}\sin y\\ i\left( \frac{-2k}{k^{2}+n^{2}+2n+1}\right) +\frac{-k^{2}-n^{2}+1} {k^{2}+n^{2}+2n+1} & =\sqrt{x}\cos y+i\sqrt{x}\sin y \end{align*} Hence \begin{align*} \sqrt{x}\cos y & =\frac{-k^{2}-n^{2}+1}{k^{2}+n^{2}+2n+1}\\ \sqrt{x}\sin y & =\frac{-2k}{k^{2}+n^{2}+2n+1}% \end{align*}
Use Mathematica to help solve the last part
Clear[x, y, n, k]
lhs = (1 - n - I k)/(1 + n + I k);
rhs = Sqrt[x] Exp[I y];
lhsReal = ComplexExpand[Re[lhs]];
lhsIm = ComplexExpand[Im[lhs]];
rhsReal = ComplexExpand[Re[rhs]];
rhsIm = ComplexExpand[Im[rhs]];
eq1 = Assuming[Element[{x, y}, Reals] && x > 0 && y > 0, Simplify[lhsReal == rhsReal]];
eq2 = Assuming[Element[{x, y}, Reals] && x > 0 && y > 0, Simplify[lhsIm == rhsIm]];
sol=Solve[{eq1, eq2}, {n, k}]
Update:
let me simplify the solution so verify it is the same solution obtained by the nice method below by b.gatessucks by applying Simplify :
Simplify[k /. sol]
Simplify[n /. sol]
This might be what you want.
With[{ee = ((1 - n - I k)/(1 + n + I k))},
Solve[ComplexExpand[{Re[x*Exp[I*y] - ee], Im[x*Exp[I*y] - ee]},
TargetFunctions -> {Re, Im}] == 0, {k, n}]]
Out[376]= {{k -> -((2 x Sin[y])/(
1 + 2 x Cos[y] + x^2 Cos[y]^2 + x^2 Sin[y]^2)),
n -> (1 - x^2 Cos[y]^2 - x^2 Sin[y]^2)/(
1 + 2 x Cos[y] + x^2 Cos[y]^2 + x^2 Sin[y]^2)}}
I'd set z = n + I k, solve for z and then use ComplexExpand to read off real and imaginary parts (n and k respectively) :
z /. First@Solve[(1 - z)/(1 + z) == Sqrt[x] Exp[I y], z]
(* (1 - E^(I y) Sqrt[x])/(1 + E^(I y) Sqrt[x]) *)
sol=ComplexExpand[(1 - E^(I y) Sqrt[x])/(1 + E^(I y) Sqrt[x]), TargetFunctions -> {Re, Im}] ;
(* n *)
Simplify[sol /. I -> 0, Assumptions -> {x > 0}]
(* (1 - x)/(1 + x + 2 Sqrt[x] Cos[y]) *)
(* I k *)
Simplify[sol - (sol /. I -> 0), Assumptions -> {x > 0}]
(* -((2 I Sqrt[x] Sin[y])/(1 + x + 2 Sqrt[x] Cos[y])) *)<|endoftext|>
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# Radius of convergence and interval of convergence
Printable View
• May 30th 2013, 06:39 PM
Educated
Radius of convergence and interval of convergence
Quote:
(a) Write down the Maclaurin series for $\dfrac{1}{1-x}$. What it it's radius of convergence?
(b) Write down the Maclaurin series for $\dfrac{1}{1+x}$
(c) Use (b) to find the Maclaurin series for f(x) = ln (1+x)
(d) What is the radius of convergence found in (c)?
(e) What is the interval of convergence found in (c)?
I can do a, b, c and d alright, but I'm stuck on e.
(a) $\dfrac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + x^5 ...$
The radius of convergence is 1 (It's a geometric series where x is the ratio).
(b) $\dfrac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 ...$
(c) $\displaystyle \int \dfrac{1}{1+x} = \ln (1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \dfrac{x^5}{5} - \dfrac{x^6}{6} ...$
(d) The radius of convergence is still 1, integrating does not alter the radius of convergence.
Now how do I go about checking the interval of convergence? Is it just (-1, 1)?
• May 30th 2013, 08:31 PM
Prove It
Re: Radius of convergence and interval of convergence
Have you tried substituting in x = -1 and x = 1? Obviously the series must be divergent when x = -1 because the logarithm is undefined at 0. As for when x = 1, we then have \displaystyle \begin{align*} 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots - \dots \end{align*}, which is the alternating harmonic series and is known to be convergent.
• May 30th 2013, 08:57 PM
Educated
Re: Radius of convergence and interval of convergence
Oh right, haha, so the interval of convergence is (-1, 1]
How would I know that the interval of convergence is around 0? Is there any series (can you give an example?) where the interval of convergence is centered around something else?
• May 30th 2013, 09:56 PM
Prove It
Re: Radius of convergence and interval of convergence
A MacLaurin Series is centred at 0 and so therefore is its radius of convergence, a general Taylor series centred at x = c will have a radius of convergence centred at x = c.<|endoftext|>
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History of Littmann Stethoscope
The term stethoscope is derived from two Greek words; stethos (meaning chest) and scopos (meaning examination). Apart from being used by medical practitioners to hear the heart and chest sounds, the stethoscope is also used to listen to bowel sounds and blood flow noises within the veins and arteries.
The first stethoscope was invented in 1816 by a French physician known as Rene Laennec who used a long, rolled paper tube to funnel the sound. The stethoscope was somewhat simple, but very effective. Three years later, Laennec improved his paper stethoscope into a wooden monaural stethoscope that had a large diameter base that funneled sound through a small opening. But, unlike modern scopes, this early version only transmitted sound to one ear and had ivory ear-tips.
It wasn’t until 1843, that George P. Cammann of New York developed the first binaural (sound into both ears) stethoscope. This is the model that would be used for over 100 years with very few modifications, one of which was the introduction of an ebony chestpiece with tubing made of spirals of wire that was covered in layers of silk dipped in gum elastic. Another modification to the original Cammann stethoscope was done in 1890 when the device was refined with a steel spring between the metal ear tubes.
In 1961, Dr. David Littmann, M.D., (1906-1981), a noted cardiologist, distinguished professor at Harvard Medical School, researcher, inventor, and a recognized international authority on electrocardiography, patented a revolutionary new stethoscope with greatly improved acoustics.
Considered as the most significant milestone in the development of the stethoscope, Dr. Littmann’s invention was a streamlined, lightweight stethoscope, with a single tube binaural, which was available in both stainless steel and light alloy.
The device was described by Dr. Littmann, in the November issue of the AMA journal, as the ideal stethoscope. It included a closed chestpiece with a stiff plastic diaphragm to filter-out low-pitched sounds. It also had an open chestpiece to hear low frequency as well as high frequency sounds.
Other innovative features of Dr. Littmann’s stethoscope included firm tubing with a single lumen bore, a spring with precise tension to hold the ear tubes apart with firm and short tubing and had the shortest practical overall length. The device was generally light and convenient to carry and use.
In a bid to promote his invention, Dr. Littmann founded the company Cardiosonics. This company was meant to produce two distinct stethoscopes, one for doctors and another for nurses. On April, 1st 1967, 3M, upon recognizing the acoustics and unique design of the Littmann nurse stethoscopes, decided to acquire Cardiosonics. Dr. Littmann then worked for 3M as an adviser where he helped to further innovate this essential diagnostic tool.
Throughout the years until the late 1970′s, 3M had developed a total of 40 “combination stethoscope” models. The devices were mainly in stainless steel and aluminum featherweight. They had pediatric size, gold-plated chestpieces. The tubing color was still limited to black and grey with lengths of 22 and 28 inches.
It is during this period that the ultimate 3M Littmann Cardiology Stethoscope was born. Commonly known as the “cardiologists’ dream stethoscope” because it was developed by a team of 50 world’s leading cardiologists, this product had a revolutionary two-tubes-in-one design, softer ear-tips and deeper bell for improved response to low frequency. Improvements to the product continued henceforth and in 1987, Tom Packard, a 3M engineer, invented the 3M Littmann Master Cardiology StethoscopeFeature Articles, a unique single-sided chestpiece stethoscope which features 3M’s tunable diaphragm technology.
ABOUT THE AUTHOR
Jason L Godwin is a content marketer. A writer by day and a reader by night, he is loathe to discuss himself in the third person, but can be persuaded to do so from time to time. visit this site for best stethoscope review<|endoftext|>
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German idealism (also known as post-Kantian idealism, post-Kantian philosophy, or simply post-Kantianism) was a philosophical movement that emerged in Germany in the late 18th and early 19th centuries. It began as a reaction to Immanuel Kant's Critique of Pure Reason. German idealism was closely linked with both Romanticism and the revolutionary politics of the Enlightenment.
The most notable thinkers in the movement were Johann Gottlieb Fichte, Friedrich Wilhelm Joseph Schelling, Georg Wilhelm Friedrich Hegel, and the proponents of Jena Romanticism (Friedrich Hölderlin, Novalis, and Karl Wilhelm Friedrich Schlegel). Friedrich Heinrich Jacobi, Gottlob Ernst Schulze, Karl Leonhard Reinhold, Salomon Maimon and Friedrich Schleiermacher also made major contributions.
Meaning of idealismEdit
The word "idealism" has multiple meanings. The philosophical meaning of idealism is that the properties we discover in objects depend on the way that those objects appear to us, as perceiving subjects. These properties only belong to the perceived appearance of the objects, and not something they possess "in themselves". The notion of a "thing in itself" should be understood here as an option of a set of functions for an operating mind, such that we consider something that appears without respect to the specific manner in which it appears. The term "idea-ism" is closer to this intended meaning than the common notion of idealism. The question of what properties a thing might have "independently of the mind" is thus unknowable and a moot point, within the idealist tradition.
Kant's work purported to bridge the two dominant philosophical schools in the 18th century: 1) rationalism, which held that knowledge could be attained by reason alone a priori (prior to experience), and 2) empiricism, which held that knowledge could be arrived at only through the senses a posteriori (after experience), as expressed by philosopher David Hume, whom Kant sought to rebut. Kant's solution was to propose that, while we depend on objects of experience to know anything about the world, we can investigate a priori the form that our thoughts can take, determining the boundaries of possible experience. Kant called his mode of philosophising "critical philosophy", in that it was supposedly less concerned with setting out positive doctrine than with critiquing the limits to the theories we can set out. The conclusion he presented, as above, he called "transcendental idealism". This distinguished it from classical idealism and subjective idealism such as George Berkeley's, which held that external objects have actual being or real existence only when they are perceived by an observer. Kant said that there are things-in-themselves, noumena, that is, things that exist other than being merely sensations and ideas in our minds. Kant held in the Critique of Pure Reason that the world of appearances (phenomena) is empirically real and transcendentally ideal. The mind plays a central role in influencing the way that the world is experienced: we perceive phenomena through time, space and the categories of the understanding. It is this notion that was taken to heart by Kant's philosophical successors.
Arthur Schopenhauer considered himself to be a transcendental idealist. In his major work The World as Will and Representation he discusses his indebtedness to Kant, and the work includes Schopenhauer's extensive analysis of the Critique. The Young Hegelians, a number of philosophers who developed Hegel's work in various directions, were in some cases idealists. On the other hand, Karl Marx, who was numbered among them, had professed himself to be a materialist, in opposition to idealism. Another member of the Young Hegelians, Ludwig Feuerbach, advocated for materialism, and his thought was influential in the development of historical materialism, where he is often recognized as a bridge between Hegel and Marx.
Kant's transcendental idealism consisted of taking a point of view outside and above oneself (transcendentally) and understanding that the mind directly knows only phenomena or ideas. Whatever exists other than mental phenomena, or ideas that appear to the mind, is a thing-in-itself and cannot be directly and immediately known.
Kant had criticized pure reason. He wanted to restrict reasoning, judging, and speaking only to objects of possible experience. The main German Idealists, who had been theology students, reacted against Kant's stringent limits. "It was Kant’s criticism of all attempts to prove the existence of God which led to the romantic reaction of Fichte, Schelling, and Hegel." "Kant sets out to smash not only the proofs of God but the very foundations of Christian metaphysics, then turns around and 'postulates' God and the immortality of the soul, preparing the way for Fichte and idealism."
In 1787, Friedrich Heinrich Jacobi addressed, in his book On Faith, or Idealism and Realism, Kant's concept of "thing-in-itself". Jacobi agreed that the objective thing-in-itself cannot be directly known. However, he stated, it must be taken on belief. A subject must believe that there is a real object in the external world that is related to the representation or mental idea that is directly known. This belief is a result of revelation or immediately known, but logically unproved, truth. The real existence of a thing-in-itself is revealed or disclosed to the observing subject. In this way, the subject directly knows the ideal, subjective representations that appear in the mind, and strongly believes in the real, objective thing-in-itself that exists outside the mind. By presenting the external world as an object of belief, Jacobi legitimized belief. "…[B]y reducing the external world to a matter of faith, he wanted merely to open a little door for faith in general..."
Karl Leonhard Reinhold published two volumes of Letters Concerning the Kantian Philosophy in 1790 and 1792. They provided a clear explication of Kant's thoughts, which were previously inaccessible due to Kant's use of complex or technical language.
Reinhold also tried to prove Kant's assertion that humans and other animals can know only images that appear in their minds, never "things-in-themselves" (things that are not mere appearances in a mind). In order to establish his proof, Reinhold stated an axiom that could not possibly be doubted. From this axiom, all knowledge of consciousness could be deduced. His axiom was: "Representation is distinguished in consciousness by the subject from the subject and object, and is referred to both."
He thereby started, not from definitions, but, from a principle that referred to mental images or representations in a conscious mind. In this way, he analyzed knowledge into (1) the knowing subject, or observer, (2) the known object, and (3) the image or representation in the subject's mind. In order to understand transcendental idealism, it is necessary to reflect deeply enough to distinguish experience as consisting of these three components: subject, subject's representation of object, and object.
Kant noted that a mental idea or representation must be a representation of something, and deduced that it is of something external to the mind. He gave the name of Ding an sich, or thing-in-itself to that which is represented. However, Gottlob Ernst Schulze wrote, anonymously, that the law of cause and effect only applies to the phenomena within the mind, not between those phenomena and any things-in-themselves outside the mind. That is, a thing-in-itself cannot be the cause of an idea or image of a thing in the mind. In this way, he discredited Kant's philosophy by using Kant's own reasoning to disprove the existence of a thing-in-itself.
After Schulze had seriously criticized the notion of a thing-in-itself, Johann Gottlieb Fichte (1762–1814) produced a philosophy similar to Kant's, but without a thing-in-itself. Fichte asserted that our representations, ideas, or mental images are merely the productions of our ego, or knowing subject. For him, there is no external thing-in-itself that produces the ideas. On the contrary, the knowing subject, or ego, is the cause of the external thing, object, or non-ego.
Fichte's style was a challenging exaggeration of Kant's already difficult writing. Also, Fichte claimed that his truths were apparent to intellectual, non-perceptual, intuition. That is, the truth can be immediately seen by the use of reason.
Schopenhauer, a student of Fichte's, wrote of him:
...Fichte who, because the thing-in-itself had just been discredited, at once prepared a system without any thing-in-itself. Consequently, he rejected the assumption of anything that was not through and through merely our representation, and therefore let the knowing subject be all in all or at any rate produce everything from its own resources. For this purpose, he at once did away with the essential and most meritorious part of the Kantian doctrine, the distinction between a priori and a posteriori and thus that between the phenomenon and the thing-in-itself. For he declared everything to be a priori, naturally without any proofs for such a monstrous assertion; instead of these, he gave sophisms and even crazy sham demonstrations whose absurdity was concealed under the mask of profundity and of the incomprehensibility ostensibly arising therefrom. Moreover, he appealed boldly and openly to intellectual intuition, that is, really to inspiration.— Schopenhauer, Parerga and Paralipomena, Vol. I, §13
Schelling attempted to rescue theism from Kant's refutation of the proofs for God's existence. "Now the philosophy of Schelling from the first admitted the possibility of a knowledge of God, although it likewise started from the philosophy of Kant, which denies such knowledge."
With regard to the experience of objects, Friedrich Wilhelm Joseph Schelling (1775–1854) claimed that the Fichte's "I" needs the Not-I, because there is no subject without object, and vice versa. So the ideas or mental images in the mind are identical to the extended objects which are external to the mind. According to Schelling's "absolute identity" or "indifferentism", there is no difference between the subjective and the objective, that is, the ideal and the real.
In 1851, Arthur Schopenhauer criticized Schelling's absolute identity of the subjective and the objective, or of the ideal and the real. "...[E]verything that rare minds like Locke and Kant had separated after an incredible amount of reflection and judgment, was to be again poured into the pap of that absolute identity. For the teaching of those two thinkers [Locke and Kant] may be very appropriately described as the doctrine of the absolute diversity of the ideal and the real, or of the subjective and the objective."
Friedrich Schleiermacher was a theologian who asserted that the ideal and the real are united in God. He understood the ideal as the subjective mental activities of thought, intellect, and reason. The real was, for him, the objective area of nature and physical being. Schleiermacher declared that the unity of the ideal and the real is manifested in God. The two divisions do not have a productive or causal effect on each other. Rather, they are both equally existent in the absolute transcendental entity which is God.
Salomon Maimon influenced German idealism by criticizing Kant's dichotomies, claiming that Kant did not explain how opposites such as sensibility and understanding could relate to each other.
Maimon claimed that the dualism between these faculties was analogous to the old Cartesian dualism between the mind and body, and that all the problems of the older dualism should hold mutatis mutandis for the new one. Such was the heterogeneity between understanding and sensibility, Maimon further argued, that there could be no criterion to determine how the concepts of the understanding apply to the intuitions of sensibility. By thus pointing out these problematic dualisms, Maimon and the neo-Humean critics left a foothold open for skepticism within the framework of Kant’s own philosophy. For now the question arose how two such heterogeneous realms as the intellectual and the sensible could be known to correspond with one another. The problem was no longer how we know that our representations correspond with things in themselves but how we know that a priori concepts apply to a posteriori intuitions.
Schelling and Hegel, however, tried to solve this problem by claiming that opposites are absolutely identical. Maimon's concept of an infinite mind as the basis of all opposites was similar to the German idealistic attempt to rescue theism by positing an Absolute Mind or Spirit.
Maimon's metaphysical concept of "infinite mind" was similar to Fichte's "Ich" and Hegel's "Geist." Maimon ignored the results of Kant's criticism and returned to pre-Kantian transcendent speculation.
What characterizes Fichte’s, Schelling’s, and Hegel’s speculative idealism in contrast to Kant's critical idealism is the recurrence of metaphysical ideas from the rationalist tradition. What Kant forbade as a violation of the limits of human knowledge, Fichte, Schelling, and Hegel saw as a necessity of the critical philosophy itself. Now Maimon was the crucial figure behind this transformation. By reviving metaphysical ideas from within the problematic of the critical philosophy, he gave them a new legitimacy and opened up the possibility for a critical resurrection of metaphysics.
Maimon is said to have Influenced Hegel's writing on Spinoza. "[T]here seems to be a striking similarity between Maimon’s discussion of Spinoza in the Lebensgeschichte (Maimon's autobiography) and Hegel’s discussion of Spinoza in the Lectures in the History of Philosophy."
Georg Wilhelm Friedrich Hegel (1770–1831) was a German philosopher born in Stuttgart, Württemberg, in present-day southwest Germany. Hegel responded to Kant's philosophy by suggesting that the unsolvable contradictions given by Kant in his Antinomies of Pure Reason applied not only to the four areas Kant gave (world as infinite vs. finite, material as composite vs. atomic, etc.) but in all objects and conceptions, notions and ideas. To know this he suggested makes a "vital part in a philosophical theory." Given that abstract thought is thus limited, he went on to consider how historical formations give rise to different philosophies and ways of thinking. For Hegel, thought fails when it is only given as an abstraction and is not united with considerations of historical reality. In his major work The Phenomenology of Spirit he went on to trace the formation of self-consciousness through history and the importance of other people in the awakening of self-consciousness (see master-slave dialectic). Thus Hegel introduces two important ideas to metaphysics and philosophy: the integral importance of history and of the Other person. His work is theological in that it replaces the traditional concept of God with that of an Absolute Spirit. Spinoza, who changed the anthropomorphic concept of God into that of an abstract, vague, underlying Substance, was praised by Hegel whose concept of Absolute fulfilled a similar function. Hegel claimed that "You are either a Spinozist or not a philosopher at all". Reality results from God's thinking, according to Hegel. Objects that appear to a spectator originate in God's mind.
Neo-Kantianism refers broadly to a revived type of philosophy along the lines of that laid down by Immanuel Kant in the 18th century, or more specifically by Schopenhauer's criticism of the Kantian philosophy in his work The World as Will and Representation (1818), as well as by other post-Kantian philosophers such as Jakob Friedrich Fries and Johann Friedrich Herbart. It has some more specific reference in later German philosophy.
Hegel was hugely influential throughout the nineteenth century; by its end, according to Bertrand Russell, "the leading academic philosophers, both in America and Britain, were largely Hegelian". His influence has continued in contemporary philosophy but mainly in Continental philosophy.
Arthur Schopenhauer contended that Spinoza had a great influence on post-Kantian German idealists. Schopenhauer wrote: "In consequence of Kant's criticism of all speculative theology, almost all the philosophizers in Germany cast themselves back on to Spinoza, so that the whole series of unsuccessful attempts known by the name of post-Kantian philosophy is simply Spinozism tastelessly got up, veiled in all kinds of unintelligible language, and otherwise twisted and distorted."
According to Schopenhauer, Kant's original philosophy, with its refutation of all speculative theology, had been transformed by the German Idealists. Through the use of his technical terms, such as "transcendental," "transcendent," "reason," "intelligibility," and "thing-in-itself" they attempted to speak of what exists beyond experience and, in this way, to revive the notions of God, free will, and immortality of soul. Kant had effectively relegated these ineffable notions to faith and belief.
In England, during the nineteenth century, philosopher Thomas Hill Green embraced German Idealism in order to salvage Christian monotheism as a basis for morality. His philosophy attempted to account for an eternal consciousness or mind that was similar to Berkeley's concept of God and Hegel's Absolute. John Rodman, in the introduction to his book on Thomas Hill Green's political theory, wrote: "Green is best seen as an exponent of German idealism as an answer to the dilemma posed by the discrediting of Christianity…."
"German idealism was initially introduced to the broader community of American literati through a Vermont intellectual, James Marsh. Studying theology with Moses Stuart at Andover Seminary in the early 1820s, Marsh sought a Christian theology that would 'keep alive the heart in the head.' " Some American theologians and churchmen found value in German Idealism's theological concept of the infinite Absolute Ideal or Geist [Spirit]. It provided a religious alternative to the traditional Christian concept of the Deity. "…[P]ost–Kantian idealism can certainly be viewed as a religious school of thought…." The Absolute Ideal Weltgeist [World Spirit] was invoked by American ministers as they "turned to German idealism in the hope of finding comfort against English positivism and empiricism." German idealism was a substitute for religion after the Civil War when "Americans were drawn to German idealism because of a 'loss of faith in traditional cosmic explanations.' " "By the early 1870s, the infiltration of German idealism was so pronounced that Walt Whitman declared in his personal notes that 'Only Hegel is fit for America — is large enough and free enough.' "
Ortega y GassetEdit
According to José Ortega y Gasset, with Post-Kantian German Idealism, "…never before has a lack of truthfulness played such a large and important role in philosophy." "They did whatever they felt like doing with concepts. As if by magic they changed anything into any other thing." According to Ortega y Gasset, "…the basic force behind their work was not strictly and exclusively the desire for truth…." Ortega y Gasset quoted Schopenhauer's Parerga and Paralipomena, Volume II, in which Schopenhauer wrote that Fichte, Schelling, and Hegel forgot "the fact that one can feel an authentic and bitter seriousness" for philosophy. Schopenhauer, in Ortega y Gasset's quote, hoped that philosophers like those three men could learn "true and fruitful seriousness, such that the problem of existence would capture the thinker and bestir his innermost being."
George Santayana had strongly-held opinions regarding this attempt to overcome the effects of Kant's transcendental idealism.
German Idealism, when we study it as a product of its own age and country, is a most engaging phenomenon; it is full of afflatus, sweep, and deep searchings of the heart; but it is essentially romantic and egoistical, and all in it that is not soliloquy is mere system-making and sophistry. Therefore when it is taught by unromantic people ex cathedra, in stentorian tones, and represented as the rational foundation of science and religion, with neither of which it has any honest sympathy, it becomes positively odious – one of the worst impostures and blights to which a youthful imagination could be subjected.— George Santayana, Winds of Doctrine, IV, i.
G. E. MooreEdit
In the first sentence of his The Refutation of Idealism, G. E. Moore wrote: "Modern Idealism, if it asserts any general conclusion about the universe at all, asserts that it is spiritual," by which he means "that the whole universe possesses all the qualities the possession of which is held to make us so superior to things which seem to be inanimate." He does not directly confront this conclusion, and instead focuses on what he considers the distinctively Idealist premise that "esse is percipere" or that to be is to be perceived. He analyzes this idea and considers it to conflate ideas or be contradictory.
Slavoj Žižek sees German Idealism as the pinnacle of modern philosophy, and as a tradition that contemporary philosophy must recapture: "[T]here is a unique philosophical moment in which philosophy appears 'as such' and which serves as a key—as the only key—to reading the entire preceding and following tradition as philosophy... This moment is the moment of German Idealism...":7–8
Hannah Arendt stated that Immanuel Kant distinguished between Vernunft ("reason") and Verstand ("intellect"): these two categories are equivalents of "the urgent need of" reason, and the "mere quest and desire for knowledge". Differentiating between reason and intellect, or the need to reason and the quest for knowledge, as Kant has done, according to Arendt "coincides with a distinction between two altogether different mental activities, thinking and knowing, and two altogether different concerns, meaning, in the first category, and cognition, in the second". These ideas were also developed by Kantian philosopher, Wilhelm Windelband, in his discussion of the approaches to knowledge named "nomothetic" and "idiographic".
Kant's insight to start differentiating between approaches to knowledge that attempt to understand meaning (derived from reason), on the one hand, and to derive laws (on which knowledge is based), on the other, started to make room for "speculative thought" (which in this case, is not seen as a negative aspect, but rather an indication that knowledge and the effort to derive laws to explain objective phenomena has been separated from thinking). This new-found room for "speculative thought" (reason, or thinking) touched-off the rise of German idealism. However, the new-found "speculative thought", reason or thinking of German idealism "again became a field for a new brand of specialists committed to the notion that philosophy's 'subject proper' is 'the actual knowledge of what truly is'. Liberated by Kant from the old school of dogmatism and its sterile exercises, they erected not only new systems but a new 'science' - the original title of the greatest of their works, Hegel's Phenomenology of the mind, was Science of the experience of consciousness - eagerly blurring Kant's distinction between reason's concern with the unknowable and the intellect's concern with cognition. Pursuing the Cartesian ideal of certainty as though Kant had never existed, they believed in all earnest that the results of their speculations possessed the same kind of validity as the results of cognitive processes".
- People associated with the movement
- Terry Pinkard, German Philosophy 1760-1860: The Legacy of Idealism, Cambridge University Press, 2002, p. 217.
- Frederick C. Beiser, German Idealism: The Struggle Against Subjectivism, 1781-1801, Harvard University Press, 2002, p. viii: "the young romantics—Hölderlin, Schlegel, Novalis—[were] crucial figures in the development of German idealism."
- Dudley, Will. Understanding German Idealism. pp. 3–6. ISBN 9781844653935.
- The German Idealists did not take "…Kant’s advice that we should not engage with concepts of which we can have no experience (instances of this are Fichte’s Absolute I, Schelling’s Absolute, and Hegel’s Geist)…." ("Fichte: Kantian or Spinozian? Three Interpretations of the Absolute I," Alexandre Guilherme, South African Journal of Philosophy, 2010, vol. 29 number 1, p. 14)
- Nicholas Churchich, Marxism and Alienation, Fairleigh Dickinson University Press, 1990, p. 57: "Although Marx has rejected Feuerbach's abstract materialism," Lenin says that Feuerbach's views "are consistently materialist," implying that Feuerbach's conception of causality is entirely in line with dialectical materialism."
- Harvey, Van A., "Ludwig Andreas Feuerbach", The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/archives/win2008/entries/ludwig-feuerbach/.
- "[Fichte], like both Schelling and Hegel, the other leading Idealist philosophers,...began as a student of theology…." Green, Garrett. "Introduction," Attempt at a Critique of All Revelation, by J.G. Fichte, Cambridge: Cambridge University Press, 1978, p. i, Note.
- "Fichte (and the other absolute Idealists) have disregarded Kant’s advice that we should not engage with concepts of which we can have no experience (instances of this are Fichte’s Absolute I, Schelling’s Absolute, and Hegel’s Geist)…." "Fichte: Kantian or Spinozian? Three Interpretations of the Absolute I" by Alexandre Guilherme, Durham University, South African Journal of Philosophy, (2010), Volume 29, Number 1, p. 14.
- Karl Popper (1945), The Open Society and Its Enemies, Volume 2, Chapter 11, II, p. 21.
- The Portable Nietzsche, translated with an introduction by Walter Kaufmann, "Introduction," V, p. 17, Penguin Books, New York, (1982).
- Schopenhauer, The World as Will and Representation, Vol. 2, Ch. I
- Hegel, Lectures on the History of Philosophy, Section Three: "Recent German Philosophy," D. "Schelling"
- Parerga and Paralipomena, Vol. I, "Fragments for the History of Philosophy," § 13
- The Cambridge Companion to German Idealism, Edited by Karl Ameriks (2000), Chapter I, Frederick C. Beiser, "The Enlightenment and idealism," , Section V, "The meta-critical campaign," page 28
- Frederick C. Beiser, The Fate of Reason: German Philosophy from Kant to Fichte, Chapter 10, "Maimon’s Critical Philosophy," page 287, Harvard University Press, 1987.
- "Salomon Maimon and the Rise of Spinozism in German Idealism," Yitzhaky Melamed, Journal of the History of Philosophy, vol. 42, no. 1 (2004) 67–96
- Hegel, "The Science of Logic" in The Encyclopedia of Philosophical Sciences (1817-1830)
- "[T]he task that touches the interest of philosophy most nearly at the present moment: to put God back at the peak of philosophy, absolutely prior to all else as the one and only ground of everything." (Hegel, "How the Ordinary Human Understanding Takes Philosophy as displayed in the works of Mr. Krug," Kritisches Journal der Philosophie, I, no. 1, 1802, pages 91-115)
- "The Hegelian philosophy is the last grand attempt to restore a lost and defunct Christianity through philosophy…. [Die Hegelsche Philosophie ist der letzte großartige Versuch, das verlorene, untergegangene Christentum durch die Philosophie wieder herzustellen]" (Ludwig Feuerbach, Principles of the Philosophy of the Future [Grundsätze der Philosophie der Zukunft (1843)], § 21)
- Hegel's Lectures on the History of Philosophy, Section 2, Chapter 1, A2. Spinoza. General Criticism of Spinoza's Philosophy, Second Point of View (cf. paragraph beginning with "The second point to be considered…")
- "…the deepest fact about the nature of reality is that it is a product of God’s thought.… Hegel even goes so far as to claim that the fact that objects appear to human beings in a particular way, as phenomena, is a reflection of the essential nature of those objects and of their origin in a divine intelligence rather than in our own." (The Cambridge Companion to German Idealism, edited by Karl Ameriks: Chapter 2, "Absolute idealism and the rejection of Kantian dualism" by Paul Guyer, Section I, "Hegel on the sources of Kantian dualism")
- Bertrand Russell, A History of Western Philosophy.
- "Spinoza’s influence on German Idealism was remarkable. He was both a challenge and inspiration for the three major figures of this movement (footnote: A very detailed examination of Spinoza’s influence on German Idealism is given in Jean-Marie Vaysse’s Totalité et Subjectivité: Spinoza dans l’Idéalisme Allemand. ). Hegel, Schelling and Fichte all sought to define their own philosophical positions in relation to his." (Bela Egyed, "Spinoza, Schopenhauer and the Standpoint of Affirmation," PhaenEx 2, no. 1 (spring/summer 2007): 110-131)
- Schopenhauer, The World as Will and Representation, Vol. 2, Ch. 50
- "In order to have insight into the existence of God, freedom, and immortality, speculative reason must use principles that are intended merely for objects of possible experience. If the principles are applied to God, freedom, and immortality, which cannot be objects of experience, the principles would always treat these three notions as though they were mere phenomena [appearances]. This would render the practicality of pure reason impossible. Therefore, I had to abandon knowledge in order to make room for faith." Kant, Critique of Pure Reason, B xxx.
- John Rodman, The Political Theory of T. H. Green, New York: Appleton Century–Crofts, 1964, "Introduction"
- James Marsh, as quoted by James A. Good (2002) in volume 2 of his The early American reception of German idealism, p. 43.
- “The Absolute or World Spirit was easily identified with the God of Christianity….”, (Morton White (Ed.) The Mentor Philosophers: The Age of Analysis: twentieth century philosophers, Houghton Mifflin, 1955, Chapter 1, “The Decline and Fall of the Absolute”)
- James Allan Good, A search for unity in diversity, in James Allan Good (editor), The Early American Reception of German Idealism (Volume 2 of 5), Bristol: Thoemmes Press 2002, ISBN 1-85506-992-X, p. 83
- Herbert Schneider, History of American philosophy (2nd edition), New York: Columbia University Press, 1963, p. 376.
- Lawrence Dowler, The New Idealism, Ph.D. dissertation, University of Maryland, 1974, p. 13, as quoted in James Allan Good, A search for unity in diversity, p. 83.
- Walt Whitman, The complete writings, vol. 9, p. 170, as quoted in James A. Good (2005), A search for unity in diversity, ch. 2, p. 57
- José Ortega y Gasset, Phenomenology and Art, New York: W. W. Norton & Co., 1975, ISBN 0-393-08714-X, "Preface for Germans," p. 48 ff.
- Žižek, Slavoj (2012). Less Than Nothing: Hegel and the Shadow of Dialectical Materialism. Verso. ISBN 9781844678976.
- Arendt, Hannah (1978). The life of the mind. One / thinking. Harcourt Brace Jovanovich. p. 14.
- Arendt, Hannah (1978). The life of the mind. One / thinking. Harcourt Brace Jovanovich. pp. 15 to 16.
- Karl Ameriks (ed.), The Cambridge Companion to German Idealism. Cambridge: Cambridge University Press, 2000. ISBN 978-0-521-65695-5.
- Frederick C. Beiser, German Idealism. The Struggle Against Subjectivism, 1781-1801. Cambridge: Harvard University Press, 2002.
- James Allan Good, A search for unity in diversity: The "permanent Hegelian deposit" in the philosophy of John Dewey. Lanham: Lexington Books 2006. ISBN 0-7391-1360-7.
- Pinkard, Terry (2002). German Philosophy 1760–1860: The Legacy of Idealism. Cambridge University Press. ISBN 9780521663816.
- Josiah Royce, Lectures on Modern Idealism. New Haven: Yale University Press 1967.
- Solomon, R., and K. Higgins, (eds). 1993. Routledge History of Philosophy, Vol. VI: The Age of German Idealism. New York: Routledge.
- Tommaso Valentini, I fondamenti della libertà in J.G. Fichte. Studi sul primato del pratico, Presentazione di Armando Rigobello, Editori Riuniti University Press, Roma 2012. ISBN 978-88-6473-072-1.
- The London Philosophy Study Guide offers many suggestions on what to read, depending on the student's familiarity with the subject: Nineteenth-Century German Philosophy
- Stanford Encyclopedia of Philosophy articles on Fichte, Reinhold, Kant, Hegel, and Schelling.
- German Idealism from the Internet Encyclopedia of Philosophy<|endoftext|>
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If you’re updating your kitchen, bathroom, outdoor living space, or font foyer, you may need to determine the amount of tile you need to purchase for your project. Whether you’re installing the tile yourself, or you simply need to get a good estimate on what you expect to spend, you can easily determine how many tiles you will need to purchase.
It’s helpful to tile installers if you can give them an estimated amount of tile up front. This helps them estimate the cost of installing the tile for you, as well as gives them a basis for helping you price the type of tile you want to use.
In this article, Transworld Tile is going to share with you a simple way to estimate how much tile you actually need for your project.
## Calculate The Square Footage
Calculating the amount of tile you need for your project is determined by the square footage of the space. You’ll need some paper, a pencil (with an eraser), measuring tape, and a calculator (if you don’t want to do long-hand math).
### Step #1. Measure the Room
First measure one wall for total length from one end of the room to the other end. Then, measure the adjacent wall from one end of the room to the other end. This will give you two different measurements. Most of the time, these measurements are different, but if you have a perfect square for a room, the measurements may be the same.
### Step #2. Do Some Math
Now that you’ve measured the two walls, it’s time to do some math. Multiply the two numbers and record the total. For example, you may have a room that’s 25 feet by 10 feet. You would multiply these two numbers and record the total: 25 x 10 = 250. Now you have the square footage of the room. In this example, the square footage is 250 square feet.
### Step #3. Translate Square Footage Into Tile Amount
Typically, tile comes in a box, and you have to buy the entire box. This means that you will need to determine how many boxes you will need to fit your square footage needs. First, determine the square footage of tile in the box. Then, divide your total room square footage by the square footage of the tile in the box. For example, you have 250 square feet of floor to fill. The tile that you’ve chosen comes in 10 square feet per box: 250/10 = 25. In this example, you will need a minimum of 25 boxes of tile.
### Step #4. Determine 10 Percent Extra
Take the total square footage of your floor and multiply it by 10. Then, add the number you get to the total square footage. Then redo the math for step three:
• 10 x 250 = 25
• 25 + 250 = 275
• 275 / 10 = 27.5
• Round up to 28
At the end of your math, for this example, you will want to purchase 28 boxes of tile. The extra 3 boxes make up a little over 10 percent extra so that if mistakes during installation are made, you have extra.
### Step #5. Will Your Tile Be Discontinued?
A great reason to add a box or two extra to your order is in the event that the tile you choose gets discontinued. If you cannot get your type of tile anymore in the event of damage, you’re out of luck. It’s best to be safe than purchase an entirely new floor because one piece of tile gets damaged.
#### Transworld Tile: Your Local Tile Experts
Whether you want to shop online or visit our showroom, we are here to help you find the best tile for your project. If you bring in the estimated square footage of your space, we can help you find the best deal on tile. Give us a call or stop by today!<|endoftext|>
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## Introductory Algebra for College Students (7th Edition)
The solution is $(3, 0)$.
The coefficients of the $y$ term differ only in sign, so if we add these equations without modification, we will cancel out the $y$ term and can then solve for $x$: Let us add the equations. First, we cancel out the $y$ term: $2x + 3y = 6$ $2x - 3y = 6$ _____________ $2x = 6$ $2x = 6$ Now we add both sides of the two equations to get: $$4x = 12$$ Divide by $4$ to solve for $x$: $$x = 3$$ Now that we have the value for $x$, we can plug this into one of the equations to solve for $y$. Let's use the first equation: $$2(3) + 3y = 6$$ Multiply: $$6 + 3y = 6$$ Subtract $6$ from both sides to isolate the variable on one side and constants on the other: $$3y = 0$$ Divide both sides by $3$ to solve for $y$: $$y = 0$$ The solution is $(3, 0)$.<|endoftext|>
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GDP A Economy Strength
WHAT IS GDP AND WHY IT IS IMPORTANT
The gross domestic product (GDP) is one the primary indicators used to gauge the health of a country’s economy. It represents the total dollar value of all goods and services produced over a specific time period – you can think of it as the size of the economy. Usually, GDP is expressed as a comparison to the previous quarter or year. For example, if the year-to-year GDP is up 3%, this is thought to mean that the economy has grown by 3% over the last year.
Measuring GDP is complicated (which is why we leave it to the economists), but at its most basic, the calculation can be done in one of two ways: either by adding up what everyone earned in a year (income approach), or by adding up what everyone spent (expenditure method). Logically, both measures should arrive at roughly the same total.
The income approach, which is sometimes referred to as GDP (I), is calculated by adding up total compensation to employees, gross profits for incorporated and non-incorporated firms, and taxes less any subsidies. The expenditure method is the more common approach and is calculated by adding total consumption, investment, government spending and net exports.
As one can imagine, economic production and growth, what GDP represents, has a large impact on nearly everyone within that economy. For example, when the economy is healthy, you will typically see low unemployment and wage increases as businesses demand labor to meet the growing economy. A significant change in GDP, whether up or down, usually has a significant effect on the stock market. It’s not hard to understand why: a bad economy usually means lower profits for companies, which in turn means lower stock prices. Investors really worry about negative GDP growth, which is one of the factors economists use to determine whether an economy is in a recession.<|endoftext|>
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President Wilson’s words at the commemoration of the first Armistice Day in 1919 ring true to this day: “To us in America, the reflections of Armistice Day will be filled with solemn pride in the heroism of those who died in the country’s service and with gratitude for the victory, both because of the thing from which it has freed us and because of the opportunity it has given America to show her sympathy with peace and justice in the councils of the nations.” Germany and the allied nations signed a peace treaty known as the Treaty of Versailles, ending World War I on November 11th. That day was declared a day to not only remember World War I veterans, but to observe and maintain world peace as well.
Armistice Day gave birth to Veteran’s Day in 1954. Though many hoped and even proclaimed that World War I would be “the end of all wars,” World War II followed and brought that hope to a tragic end. Furthermore, a great number of soldiers, airmen, sailors, and several other military personnel were deployed for this war, leading the 83rd Congress to replace the word “Armistice” with “Veterans” to honor veterans of all wars.
Veteran’s Day is often easily confused with Memorial Day. While Veteran’s Day is a day to honor the living veterans of all wars, Memorial Day was established a year after the Civil War to honor those who fell in active duty. Both holidays are celebrated in a similar way and can even be interchangeable, but what they each stand for possesses distinct uniqueness. Moreover, Veteran’s Day honors men and women who have served in the military, regardless of whether it was in combat or not. Several people go out of their way to celebrate Veteran’s Day; from decorations and gatherings, to free goods and services for veterans, there are many ways people choose to express their gratitude and appreciation.
Most people agree that war is brutal and ugly, but when a nation is faced with the question “why are we going to war?” the answers vary, which makes it a rather controversial topic that garners some serious reactions from people across the political spectrum. Though Veteran’s Day is a national holiday, those who served aren’t always treated with the honor and respect they deserve. For instance, veterans of the Vietnam War were not fortunate enough to get a festive reception. This was due to the fact that the US neither had an objective or declared war beforehand, which caused the rise of an extremely contentious political climate during the war.
Students on university campuses and in the academic society started an anti-Vietnam War movement; soon, protests became more prominent and drew people’s attention to the reality of the war. When the soldiers that fought in the Vietnam War returned home, they didn’t get a hero’s welcome. Several veterans testified about being mistreated, insulted, and in some cases, assaulted. Another war that was ethically, morally, and politically controversial was the Iraq War. In 2003, the US invaded Iraq due to their alleged possession of “weapons of mass destruction,” an idea that was not clearly verified. Both wars took so many lives and nearly destroyed nations for reasons that are not clear to this day, which is why many felt the need to protest and oppose.
Some might begin to wonder if there is an instance where one should or shouldn’t honor a veteran. Most of the people that decide to either protest or refrain from celebrating holidays like Veteran’s Day have probably wondered the same thing as well. The answer to the question depends on the individual, but there are some factors we can consider to help guide us towards it. Sometimes, men and women in armed forces can decided whether or not they wish to serve; however, like in the Vietnam War, many had no choice but to serve and were simply following orders from their superiors. Therefore, despising them and blaming them for everything doesn’t change the situation.
Everyone has the right to agree or disagree with the government’s decisions on wars, and they have the freedom of speech to express that too. However, political criticism should be the last thing veterans get considering the several challenges they face that are often unique to their circumstances and background as former military personnel. On Veteran’s Day, the focus should be on the bravery and will of the human spirit displayed through ordinary men and women who exhibit extraordinary courage. The least we can do is to put our politics aside, take time out of our day to give back, and help make them feel appreciated.
Written by Kenean
Office of Public and Intergovernmental Affairs. “Office of Public and Intergovernmental Affairs.” Learn to Communicate Assertively at Work, 20 Mar. 2006, www.va.gov/opa/vetsday/vetdayhistory.asp.
IowaPublicTelevision. “Experiences of Vietnam Veterans Returning Home from War.” YouTube, YouTube, 21 Oct. 2015, http://www.youtube.com/watch?v=b6t9jchhVRg.<|endoftext|>
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|Belize Table of Contents
Because the political parties contesting the March 1961 elections had declared their intent to seek full independence, another constitutional conference was held in London in 1963. The conference led to the establishment of full internal selfgovernment under a constitution that took force on January 1, 1964.
The changes introduced by this constitution significantly reduced the powers of the governor, transformed the Executive Council into a cabinet headed by a premier, and established a bicameral National Assembly, composed of a House of Representatives and a Senate. The House of Representatives had eighteen members, all of whom were elected. The Senate had eight members, all appointed by the governor after consultation with majority and minority party leaders and other "suitable persons." The Senate's powers were limited to ratifying bills passed by the House or delaying, for up to six months, bills with which it disagreed (but for only one month on financial bills). General elections had to be held at least every five years on a date determined by the prime minister. The governor was still appointed by the crown but was now bound by the recommendations of the cabinet in executive matters. The leader of the majority party in the House of Representatives was to be appointed premier by the governor. Members of both the House and the Senate were eligible for appointment to the cabinet.
The constitution of 1964 established internal self-rule, and Britain had conceded the readiness of the colony for independence as early as 1961. But Guatemalan territorial claims against Belize delayed full independence until 1981.
More about the Government and Politics of Belize.
Source: U.S. Library of Congress<|endoftext|>
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Northern and southern China
Northern China and southern China are two approximate regions within China. The exact boundary between these two regions has never been precisely defined. Nevertheless, the self-perception of Chinese people, especially regional stereotypes, has often been dominated by these two concepts, given that regional differences in culture and language have historically fostered strong regional identities (simplified Chinese: 乡土; traditional Chinese: 鄉土; pinyin: xiāngtǔ) of the Chinese people.
Often used as the geographical dividing line between northern and southern China is the Huai River–Qin Mountains Line. This line approximates the 0 °C January isotherm and the 800 millimetres (31 in) isohyet in China.
Culturally, however, the division is more ambiguous. In the eastern provinces like Jiangsu and Anhui, the Yangtze River may instead be perceived as the north–south boundary instead of the Huai River, but this is a recent development.
There is an ambiguous area, the region around Nanyang, Henan, that lies in the gap where the Qin has ended and the Huai River has not yet begun; in addition, central Anhui and Jiangsu lie south of the Huai River but north of the Yangtze, making their classification somewhat ambiguous as well. As such, the boundary between northern and southern China does not follow provincial boundaries; it cuts through Shaanxi, Henan, Anhui, and Jiangsu, and creates areas such as Hanzhong (Shaanxi), Xinyang (Henan), Bengbu (Anhui) and Xuzhou (Jiangsu) that lie on an opposite half of China from the rest of their respective provinces. This may have been deliberate; the Mongol Yuan Dynasty and Han Chinese Ming Dynasty established many of these boundaries intentionally to discourage regionalist separatism.
The Northeast (Manchuria) and Inner Mongolia, areas that are often thought of as being outside "China proper", are also conceived to belong to northern China according to the framework above. Historically, Xinjiang, Tibet and Qinghai were not usually conceived of as being part of either the north or south. However, Xinjiang is now regarded as being part of the north due to the spread of north Chinese culture and the use of Mandarin.
The concepts of northern and southern China originate from differences in climate, geography, culture, and physical traits; as well as several periods of actual political division in history. Northern China is too cold and dry for rice cultivation (though rice is grown there today with the aid of modern technology) and consists largely of flat plains, grasslands, and desert; while Southern China is warm and rainy enough for rice and consists of lush mountains cut by river valleys. Historically, these differences have led to differences in warfare during the pre-modern era, as cavalry could easily dominate the northern plains but encountered difficulties against river navies fielded in the south. There are also major differences in language, cuisine, culture, and popular entertainment forms.
Episodes of division into North and South include:
- Three Kingdoms (220–280)
- Sixteen Kingdoms (317–420) and Southern and Northern Dynasties (420–589)
- Five Dynasties and Ten Kingdoms period (907–960)
- Southern Song Dynasty (1127–1279) and Jin dynasty (1115–1234)
- Warlord era (1916–1928) of the Republic of China
The Northern and Southern Dynasties showed such a high level of polarization between North and South that northerners and southerners referred to each other as barbarians; the Mongol Yuan Dynasty also made use of the concept: Yuan subjects were divided into four castes, with northern Han Chinese occupying the third-caste and southern Han Chinese occupying the lowest one.
For a large part of Chinese history, northern China was economically more advanced than southern China . The Jurchen and Mongol invasion caused a massive migration to southern China, and the Emperor shifted the Song Dynasty capital city from Kaifeng in northern China to Hangzhou, located south of the Yangtze river. The population of Shanghai increased from 12,000 households to over 250,000 inhabitants after Kaifeng was sacked by invading armies. This began a shift of political, economic and cultural power from northern China to southern China. The east coast of southern China remained a leading economic and cultural center of China until the Republic of China. Today, southern China remains economically more prosperous than northern China.
During the Qing dynasty, regional differences and identification in China fostered the growth of regional stereotypes. Such stereotypes often appeared in historic chronicles and gazetteers and were based on geographic circumstances, historical and literary associations (e.g. people from Shandong, were considered upright and honest) and Chinese cosmology (as the south was associated with the fire element, Southerners were considered hot-tempered). These differences were reflected in Qing dynasty policies, such as the prohibition on local officials to serve their home areas, as well as conduct of personal and commercial relations. In 1730, the Kangxi Emperor made the observation in the Tingxun Geyan (《庭訓格言》):
The people of the North are strong; they must not copy the fancy diets of the Southerners, who are physically frail, live in a different environment, and have different stomachs and bowels.— the Kangxi Emperor, Tingxun Geyan (《庭訓格言》)
According to my observation, Northerners are sincere and honest; Southerners are skilled and quick-minded. These are their respective virtues. Yet sincerity and honesty lead to stupidity, whereas skillfulness and quick-mindedness lead to duplicity.— Lu Xun, Lu Xun Quanji (《魯迅全集》), pp. 493–495
In modern times, North and South is merely one of the ways that Chinese people identify themselves, and the divide between northern and southern China has been complicated both by a unified Chinese nationalism and as well as by local loyalties to province, county and village which prevent a coherent Northern or Southern identity from forming.
During the Deng Xiaoping reforms of the 1980s, South China developed much more quickly than North China leading some scholars to wonder whether the economic fault line would create political tension between north and south. Some of this was based on the idea that there would be conflict between the bureaucratic north and the commercial south. This has not occurred to the degree feared in part because the economic fault lines eventually created divisions between coastal China and the interior, as well as urban and rural China, which run in different directions from the north–south division, and in part because neither north or south has any type of obvious advantage within the Chinese central government. In addition there are other cultural divisions that exist within and across the north–south dichotomy.
Stereotypes and differences
Nevertheless, the concepts of North and South continue to play an important role in regional stereotypes.
"Northerners" are seen as:
- taller According to the 2014 census, the average male height between the age of 20-24 was 173.4 cm in Beijing, 173.4 cm in Jiling and 171.9 cm in Shanxi.
- speaking Mandarin Chinese with a northern (rhotic) accent
- more likely to eat noodles, dumplings and wheat-based foods (rather than rice-based foods)
while "Southerners" are seen as:
- shorter According to 2014 census, the average male height between the age of 20-24 was 173.3 cm in Shanghai, 171.6 cm in Zhejiang and 171.9 cm in Fujian
- speaking Mandarin Chinese with a southern (non-rhotic) accent or speaking a southern Chinese language such as Cantonese, Wu, Hakka, Xiang, Min, or Gan
- more likely to eat rice-based foods (rather than wheat-based food)
It should be noted that these are only rough and approximate stereotypes among a large and greatly varied population.
- Eastern China (disambiguation)
- List of regions of China
- Nanquan (Southern Fist)
- Northern and southern Vietnam
- Northern China (disambiguation)
- South China (disambiguation)
- Wushu (Kung Fu)
- Zhonghua minzu
- simplified Chinese: 华北; traditional Chinese: 華北; pinyin: Huáběi (northern China), and simplified Chinese: 华南; traditional Chinese: 華南; pinyin: Huánán (southern China), also referred to in China as simply (Chinese: 北方; pinyin: Běifāng) the north and (Chinese: 南方; pinyin: Nánfāng the south
- Source: United States Central Intelligence Agency, 1990. The map shows the distribution of linguistic groups according to the historical majority ethnic groups by region. Note this is different from the current distribution due to age-long internal migration and assimilation.
- Smith, Richard Joseph (1994). China's cultural heritage: the Qing dynasty, 1644–1912 (2 ed.). Westview Press. ISBN 978-0-8133-1347-4.
- Hanson, Marta E. (July 2007). "Jesuits and Medicine in the Kangxi Court (1662–1722)" (PDF). Pacific Rim Report. San Francisco: Center for the Pacific Rim, University of San Francisco (43): 7, 10.
- Young, Lung-Chang (Summer 1988). "Regional Stereotypes in China". Chinese Studies in History. 21 (4): 32–57. doi:10.2753/csh0009-4633210432.
- Zhang, Xuan; Huang, Ze (July–August 1988). "The Second National Growth and Development Survey of Children in China, 1985: children 0 to 7 years". Annals of Human Biology. Informa Healthcare. 15 (4): 289–305. doi:10.1080/03014468800009761. PMID 3408235.
- Fodor's (2009). Kelly, Margaret, ed. Fodor's China. Random House. p. 135. ISBN 978-1-4000-0825-4.
- Eberhard, Wolfram (December 1965). "Chinese Regional Stereotypes". Asian Survey. University of California Press. 5 (12): 596–608. doi:10.2307/2642652. JSTOR 2642652.
- 北京市2014年国民体质监测结果公报 , 北京市体育局
- 2014年吉林省第三次国民体质监测公报, 吉林省体育局
- 2014年山西省国民体质监测统计数据, 山西省体育局
- Regions of Chinese food-styles/flavours of cooking, University of Kansas
- 2014年上海市第四次国民体质监测公报, 上海市体育局
- 2010年贵州省第三次国民体质监测公报, 贵州省体育局
- 福建省2014年国民体质监测公报 , 福建省体育局
- Brues, Alice Mossie (1977). People and Races. Macmillan series in physical anthropology. New York: Macmillan. ISBN 978-0-02-315670-0.
- Lamprey, J. (1868). "A Contribution to the Ethnology of the Chinese". Transactions of the Ethnological Society of London. Royal Anthropological Institute of Great Britain and Ireland. 6: 101–108. JSTOR 3014248.
- Morgan, Stephen L. (July 2000). "Richer and Taller: Stature and Living Standards in China, 1979–1995". The China Journal. Contemporary China Center, Australian National University. 44: 1–39. doi:10.2307/2667475. JSTOR 2667475.
- Muensterberger, Warner (1951). "Orality and Dependence: Characteristics of Southern Chinese." In Psychoanalysis and the Social Sciences, (3), ed. Geza Roheim (New York: International Universities Press).
- Ebrey, Patricia Buckley; Liu, Kwang-chang. (1999). The Cambridge Illustrated History of China. Cambridge University Press. ISBN 978-0-521-66991-7 (ch. 4, 5)
- Lewis, Mark Edwards. (2009). China Between Empires: The Northern and Southern Dynasties. Harvard University Press. ISBN 978-0-674-02605-6
- Tu, Jo-fu. (1992). Chinese Surnames and the Genetic Differences Between North and South China. Project on Linguistic Analysis, University of California, Berkeley.<|endoftext|>
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The soil system is a dynamic ecosystem that has inputs, outputs, storages and flows.
The quality of soil influences the primary productivity of an area.
Knowledge and understanding:
The soil system may be illustrated by a soil profile that has a layered structure (horizons).
Soil system storages include organic matter, organisms, nutrients, minerals, air and water.
Transfers of material within the soil, including biological mixing and leaching (minerals dissolved in water moving through soil), contribute to the organization of the soil.
There are inputs of organic material including leaf litter and inorganic matter from parent material, precipitation and energy. Outputs include uptake by plants and soil erosion.
Transformations include decomposition, weathering and nutrient cycling.
The structure and properties of sand, clay and loam soils differ in many ways, including mineral and nutrient content, drainage, water-holding capacity, air spaces, biota and potential to hold organic matter. Each of these variables is linked to the ability of the soil to promote primary productivity.
A soil texture triangle illustrates the differences in composition of soils.
Applications and skills:
Outline the transfers, transformations, inputs, outputs, flows and storages within soil systems.
Explain how soil can be viewed as an ecosystem.
Compare and contrast the structure and properties of sand, clay and loam soils, with reference to a soil texture diagram, including their effect on primary productivity.
Significant differences exist in arable (potential to promote primary productivity) soil availability around the world. These differences have socio-political, economic and ecological influences.
Theory of knowledge:
The soil system may be represented by a soil profile—since a model is, strictly speaking, not real, how can it lead to knowledge?
Communities and ecosystems (2.2)
Flows of energy and matter (2.3)
Investigating ecosystems (2.5)
Biomes, zonation and succession (2.4)
Introduction to water systems (4.1)
Terrestrial food production systems and food choices (5.2)<|endoftext|>
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HISTORY OF MEDIA BIAS
It is not possible to be impartial or neutral without making judgments
No matter how intelligent, open minded, and logical you believe your opinion to be, there will always be other smart, impartial, and honest people who believe the exact opposite
Few occasions render a person more confident about their fairness and open-mindedness than when they describe Bias in others
The majority of news and documentary media are quite sure their reports are impartial and objective. Wikipedia has pride in their principle of neutrality and stresses that the content of their site be written from a neutral point-of-view. Academia is confident that any biases that creep into their work will be weeded out by consultations with experts, their well-founded procedures, and the peer review process. Many now think racism, sexism, and all the other “isms” and phobias can be clearly identified with fair reprisals handed out to offenders. Today, a lot of folk accept that impartiality in both the media and the public arena has been accomplished; and that it is only an intransigent and ignorant minority who fail to accept those facts.
So are we really living in a super-enlightened age, essentially free of ignorance, myth, and superstition? To address this question it is helpful to have some background about the history of public bias through the prism of the media.
Until recent times the idea that bias could be avoided by the academic world was assumed to be in direct conflict with human nature. Virtually every philosophical thinker over the history of human civilization unreservedly accepted that it was impossible to attain total objectivity. Similarly, the idea that there was a fair and detached approach to interpret news events or documentary topics was considered to be absurd.
During the nineteenth century this deep-rooted suspicion of neutrality gave rise to newspapers (in both the US and the UK) that were openly and unashamedly biased - often towards the publisher’s interests. Near the end of that century newspaper editors realized they could substantively influence the voting behaviours of readers. And they argued this was for the benefit of citizens, as straightforward facts were said to be too difficult for the average person to fully grasp. Gradually the media assumed the role of adjudicators for the public interest. As there was a broad selection of newspapers cheaply and readily available, with each providing their own biases, or appealing to different interest groups, consumers could access the points-of-view that best suited their needs.
During this time newspapers also promoted the idea of investigative reporting. While biases dictated where and how the reporters would dig out their stories, a sense of balance was attained due to the broad spectrum of interests available for the public.
By the middle of the twentieth century - joined now by radio and television news and documentary programs - this accepted system began to change. Firstly, journalism transformed from a trade into a profession which required a university education. A little later these professionals shifted their perspective of the news from local, cultural, and national interests to international and multicultural perspectives. And at the same time their viewpoints converged with the beliefs of their university mentors; while the influence of individuals, religions, and businesses became secondary to the perspectives of academia. It was at this time reporters and columnists assumed the position of unbiased advocates for providing the solutions to the many challenges existing in the world. As the media could, to some degree present their opinions as unbiased facts, investigative reporting tended to diminish in importance.
The science media followed a similar path once it was introduced. Science reporters often lacked the time or technical knowledge to dig into stories so they did not critically evaluate the quality of the studies they covered. While today there are a few internet-based investigative science journalists, the science media essentially acts as a champion for the opinions, or biases, held in other parts of the media.
This history, if it is known at all, is now often regarded as one more example of the mistakes of the ignorant past and, as observed above, a good many people today are confident impartiality in the public arena has now been achieved; and accept that truth and fairness can be determined by unbiased, informed and caring leaders in academia and the media.
In contrast, the social sciences still support the traditional view: that people can never free themselves from bias. Quotations of leading scientists that support this are given on the page: Science of Bias. Over a hundred different ways of being biased have been identified and characterized.
It has been observed that many people cite specific media outlets that are biased. But this implies that others in the media are not noticeably biased. This is especially the case with many of today’s young and educated. This cohort does however possess some very positive characteristics: for example they are open-minded, passionate about solving the world’s problems, and they believe their ideas are important. In addition they mistrust non-conformists, have high expectations and seem to need approval of their peers. But these ideals do come at a price. Each of them has what can be called a “high bias potential”. The open-minded have the tendency to accept certain notions too readily, as compared to skeptics who probe for errors of fact, logic, or perspective. Passion in solving world issues brings with it emotions and powerful needs; and each conceal personal biases. Looking for approval, perhaps by the adoption of culturally acceptable ways of thinking, inhibits creative ideas and encourages the bias of Group-think.
We can conclude the idea that impartiality has been accomplished by the academic world is no different to the culturally agreed superstitions of the past. Just like other historical myths it is reassuring to believe there are educated and honest people who can dispense unbiased truths. Reality reveals that to be neutral and impartial one needs judgement; and human judgement will be biased. What the media actually imply when they claim neutrality is: our judgments are more judicious than your judgments. The question becomes: should you hand over the authority to others to adjudicate the legitimacy of your opinions and ideas? To do so is to give away your thoughts, your free speech and your freedom of expression.
Few people are capable of expressing (…) opinions which differ from the prejudices of their social environment - most people are incapable of even forming such opinions
Albert Einstein – Theoretical Physicist
Bias is not the real problem: It is the stubborn belief "other" people are biased while you, and those who agree with you, are not
We are all influenced by our biases: the big difference is some people are aware of this, while others are not<|endoftext|>
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# How does Cramer's rule work?
I know Cramer's rule works for 3 linear equations. I know all steps to get solutions. But I don't know why (how) Cramer's rule gives us solutions?
Why do we get $x=\frac{\Delta_1}\Delta$ and $y$ and $z$ in the same way?
I want to know how these steps give us solutions?
• Just read the proof here en.wikipedia.org/wiki/Cramer%27s_rule#Proof Commented Sep 26, 2016 at 1:29
• @JosePaternina it's out of my reach can you explain it in simplest manner?. Commented Sep 26, 2016 at 1:32
• Well, for the general case (even for the case of $3$ inear equations) the proof is like in wikipedia. Try to do it for the case of $2$ linear equations with $2$ variables. Take $ax+by=r_1$ and $cx+dy=r_2$. Multiply first equation by $d$ and the second equation by $b$, you get $dax+dby=dr_1$ and $bcx+bdy=br_2$. If you substract these equations, you get $(da-bc)x=dr_1-br_2$, so $x=\frac{dr_1-br_2}{da-bc}=\frac{\Delta_1}{\Delta}$ (do you see why?). Commented Sep 26, 2016 at 1:40
• Did you mean to write linear equations rather than polynomial equations? Commented Sep 27, 2016 at 3:12
## 8 Answers
It's actually simple; I explain it here in two variables, but the principle is the same.
Say you have an equation $$\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}p\\q \end{pmatrix}$$
Now you can see that the following holds
$$\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x&0\\y&1\end{pmatrix}=\begin{pmatrix}p&b\\q &d\end{pmatrix}$$
Finally just take the determinant of this last equation; $\det$ is multiplicative so you get $$\Delta x=\Delta_1$$
• This is nice, I'd never seen this proof before. Commented Sep 26, 2016 at 1:51
• @Wojowu Sorry, I came up with it on my own. Commented Sep 26, 2016 at 12:47
• I suspect it does exist somewhere in a book. Commented Sep 26, 2016 at 12:59
• This is cute! :D Just to give other people an idea on how this generalizes, for the three-variable case, we need the matrices $$\begin{pmatrix}x & 0 & 0 \\y & 1 & 0 \\z & 0 & 1\end{pmatrix},\begin{pmatrix}x & 0 & 1\\y & 0 & 0\\z & 1 & 0\end{pmatrix}, \begin{pmatrix}x & 1 & 0\\y & 0 & 1\\z & 0 & 0\end{pmatrix}$$ and note which columns of the appropriately-dimensioned identity matrix are being used to construct these matrices. Commented Oct 9, 2016 at 6:45
• Thats one way but actually much more transparent is \begin{pmatrix}x & 0 & 0 \\y & 1 & 0 \\z & 0 & 1\end{pmatrix},\begin{pmatrix}1& x & 0\\0& y & 0\\0 & z & 1\end{pmatrix}, \begin{pmatrix}1 & 0 & x\\0& 1& y\\0 & 0 & z\end{pmatrix} Commented Oct 9, 2016 at 13:19
Cramer's rule is very easy to discover because if you solve the linear system of equations \begin{align*} a_{11} x_1 + a_{12} x_2 + a_{13} x_3 &= b_1 \\ a_{21} x_1 + a_{22} x_2 + a_{23} x_3 &= b_2 \\ a_{31} x_1 + a_{32} x_2 + a_{33} x_3 &= b_3 \\ \end{align*} by hand, just using a standard high school approach of eliminating variables, then out pops Cramer's rule! In my opinion, this is the most likely way that a mathematician would discover the determinant in the first place, and Cramer's rule is discovered simultaneously.
I remember thinking that it must be quite difficult to prove Cramer's rule for an $n \times n$ matrix, but it turns out to be surprisingly easy (once you take the right approach). We'll prove it below.
The most useful way of looking at the determinant, in my opinion, is this: the function $M \mapsto \det M$ is an alternating multilinear function of the columns of $M$ which satisfies $\det(I) = 1$. This characterization of the determinant gives us a quick, simple proof of Cramer's rule.
For simplicity, I'll assume $A$ is a $3 \times 3$ matrix with columns $a_1, a_2, a_3$. Suppose that $$b = Ax = x_1 a_1 + x_2 a_2 + x_3 a_3.$$ Then \begin{align*} \begin{vmatrix} b & a_2 & a_3 \end{vmatrix} &= \begin{vmatrix} x_1 a_1 + x_2 a_2 + x_3 a_3 & a_2 & a_3 \end{vmatrix} \\ &= x_1 \begin{vmatrix} a_1 & a_2 & a_3 \end{vmatrix} + x_2 \begin{vmatrix} a_2 & a_2 & a_3 \end{vmatrix} + x_3 \begin{vmatrix} a_3 & a_2 & a_3 \end{vmatrix} \\ &= x_1 \det A. \end{align*} If $\det A \neq 0$, it follows that $$x_1 = \frac{\begin{vmatrix} b & a_2 & a_3 \end{vmatrix}}{\det A}.$$
I learned this proof in section 4.4, problem 16 ("Quick proof of Cramer's rule") in Gilbert Strang's book Linear Algebra and its Applications.
• Why det(x_1a_1 + x_2a_2 + x_3a_3 , a_2 , a_3) = det(x_1a_1 , a_2 , a_3) + det(x_2a_2 , a_2 , a_3) + det(x_3a_3 , a_2 , a_3) = x_1det(a_1 , a_2 , a_3) + x_2det(a_2, a_2, a_3) + x_3det(a_3, a_2, a_3) ? Commented Jul 29, 2020 at 10:52
• Why det(a_2 a_2 a_3) = 0 ? Why det(a+b c d) = det(a c d) + det(b c d) ? Commented Jul 29, 2020 at 10:52
• @kevin Those facts follow from the assumption that the determinant is an alternating multilinear function of the columns of $M$ (which satisfies $det(I) = 1$). This fundamental characterization of the determinant is often the best way or the easiest way to think about the determinant, in my opinion. Commented Jul 29, 2020 at 14:24
Here is a very simple solution that only uses some properties of determinants. Consider the following system:
$$\left\{ \begin{array}{c} a_1x+b_1y+c_1z=d_1 \\ a_2x+b_2y+c_2z=d_2 \\ a_3x+b_3y+c_3z=d_3 \end{array} \right.$$ Assume $\Delta\neq0$, then, \require{action}\begin{align} \Delta_1& \mathtip{=\left| \matrix{d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3}\right|}{\text{by definition of }\Delta_1} \\& \\ &\mathtip{=\left| \matrix{(a_1x+b_1y+c_1z) & b_1 & c_1 \\ (a_2x+b_2y+c_2z) & b_2 & c_2 \\ (a_3x+b_3y+c_3z) & b_3 & c_3}\right|}{\text{by the system of equations}} \\& \\ &\mathtip{=\left| \matrix{(a_1x+b_1y+c_1z)-(b_1y+c_1z) & b_1 & c_1 \\ (a_2x+b_2y+c_2z)-(b_2y+c_2z) & b_2 & c_2 \\ (a_3x+b_3y+c_3z)-(b_3y+c_3z) & b_3 & c_3}\right|}{\text{If a multiple of one column is added to another column, the value of the determinant is not changed.}} \\ & \\ &\mathtip{=\left| \matrix{a_1x & b_1 & c_1 \\ a_2x & b_2 & c_2 \\ a_3x & b_3 & c_3}\right|}{\text{Simplifying}} \\& \\ &\mathtip{= x\left| \matrix{a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3}\right|}{\text{If each entry in a given row is multiplied by k, then the value of the determinant is multiplied by k.}} \\& \\ &\mathtip{= x\Delta}{\text{by definition of }\Delta} \end{align} Thus $x=\dfrac{\Delta_1}{\Delta}$. The proof is due to D. E. Whitford and M. S. Klamkin. (“On an Elementary Derivation of Cramer's Rule”, American Mathematical Monthly, vol. 60 (1953), pp.186–7).
• I didn't get that 3rd step for $\Delta_1$ Commented Sep 26, 2016 at 13:04
• It seems to me it also uses that a solution exists when the determinant is non-zero.
– quid
Commented Sep 26, 2016 at 13:42
• @Ramanujan If a multiple of one column is added to another column, the value of the determinant is not changed. Commented Sep 26, 2016 at 15:37
This is another way to look at it. $Ax = b$ where A is invertible. First we define the following matrix:
$$I_i(x) = [e_1\,\,e_2\,\, e_{i-1}\,\,x\,\,e_{i+1}\,\, ...\,\, e_n]$$ By the defenition of matrix multiplication: $$AI_i(x) = [Ae_1\,\,Ae_2\,\, Ae_{i-1}\,\,Ax\,\,Ae_{i+1}\,\, ...\,\, Ae_n]$$ $$AI_i(x) = A_i(b)$$ $$det\,AI_i(x) = det\,A_i(b)$$
The determinant of the product of two matrices is the product of the determinants, so:
$$det\,A \,\cdot\, det\,I_i(x) = det\,A_i(b)$$
Let's now look at $det\,I_i(x)$, note that the determinants are the same because you can get rid of all the $x_j$ where $j\ne i$ by row reduction. Also $x$ is in the $i^{th}$ column: $$det\,I_i(x) = det \begin{bmatrix} 1 & x_1 & \cdots & 0 \\ 0 & x_2 & 0 & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & x_n & 0 & 1 \\ \end{bmatrix} = det \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & x_i & 0 & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} = 1 \cdot 1 \cdot\cdot\cdot 1\cdot x_i \cdot 1\cdot\cdot\cdot1=x_i$$
$det\,I_i(x)$ is simply $x_i$, So we can say that: $$x_i = \frac{det\,A_i(b)}{det\,A}$$
• This is essentially the same as Rene's answer, but this is a bit more explicit. Commented Oct 14, 2016 at 10:26
I encountered a proof of Cramer's rule with $$2$$ unknowns in my textbook. It involves application of the method of elimination.
Consider $$a_1x+b_1y+c_1=0 \tag1a_2x+b_2y+c_2=0 \tag2$$
Define $$\Delta_1=\left|\begin{matrix}c_1 & b_1 \\ c_2 & b_2\end{matrix}\right|, \Delta_2=\left|\begin{matrix}a_1 & c_1 \\ a_2 & c_2\end{matrix}\right|, \Delta=\left|\begin{matrix}a_1 & b_1 \\ a_2 & b_2\end{matrix}\right|$$.
1. Multiplying equation $$(1)$$ by $$b_2$$
2. Multiplying equation $$(2)$$ by $$b_1$$
3. Subtracting equations
\begin{align}b_2(a_1x+b_1y+c_1=0) \\-b_1(a_2x+b_2y+c_2=0) \end{align}\\ \rule{10cm}{0.5pt}\\ x(a_1b_2-a_2b_1)+(b_2c_1-b_1c_2)=0 \implies x=\dfrac{\Delta_1}{\Delta}
1. Multiplying equation $$(1)$$ by $$a_2$$
2. Multiplying equation $$(2)$$ by $$a_1$$
3. Subtracting equations
\begin{align} a_2(a_1x+b_1y+c_1=0) \\ -a_1(a_2x+b_2y+c_2=0)\end{align} \\\rule{10cm}{0.5pt} \\ y(a_2b_1-a_1b_2)+(a_2c_1-a_1c_2)=0\implies y=\dfrac{\Delta_2}{\Delta}
Note that this is valid provided $$\Delta=(a_1b_2-a_2b_1)\ne0$$.
Given a linear system $\rm A x = b$, where $\mathrm A \in \mathbb R^{n \times n}$ is invertible and $\mathrm b \in \mathbb R^n$, let $\rm \bar{x} := A^{-1} b$ denote the (unique) solution. Replacing the $k$-th column of matrix $\rm A$ (denoted by $\mathrm a_k$) with $\rm b$ and computing the determinant of this new matrix using the matrix determinant lemma,
\begin{aligned} \det \left( \mathrm A + (\mathrm b - \mathrm a_k) \mathrm e_k^\top \right) &= \det (\mathrm A) \cdot \left( 1 + \mathrm e_k^\top \mathrm A^{-1} (\mathrm b - \mathrm a_k) \right)\\ &= \det (\mathrm A) \cdot \left( 1 + \mathrm e_k^\top \mathrm A^{-1} \mathrm b - \mathrm e_k^\top \mathrm A^{-1} \mathrm a_k \right)\\ &= \det (\mathrm A) \cdot \left( 1 + \mathrm e_k^\top \mathrm{\bar{x}} - \mathrm e_k^\top \mathrm e_k \right)\\ &= \det (\mathrm A) \cdot \bar{x}_k\end{aligned}
and, thus, we obtain Cramer's rule
$$\bar{x}_k = \frac{\det \left( \mathrm A + (\mathrm b - \mathrm a_k) \mathrm e_k^\top \right)}{\det (\mathrm A)}$$
Let $$A$$ be an $$n \times n$$ invertible matrix, and $$b \in \mathbb{R}^{n}$$. We're looking for the $$x \in \mathbb{R}^n$$ such that $$Ax = b$$.
Let's write columns of $$A$$ as $$A_1, \ldots, A_n$$ (since $$A$$ is invertible, these form a basis of $$\mathbb{R}^n$$). Now $$Ax = b$$ becomes $$x_1 A_1 + \ldots + x_n A_n = b$$. Notice $$x_k$$ is that scalar $$t$$ such that $$b - t A_k$$ lies in the span of $$\{ A_i : i \neq k \}$$ [ We can visualise this for $$n \leq 3$$ ].
So $$\det(A_1, \ldots, b - x_k A_k, \ldots, A_n) = 0$$, from which $$x_k = \frac{\det(A_1, \, \ldots \, , \, b \, , \, \ldots \, , A_n)}{\det(A_1, \, \ldots \, , A_n)}$$ [ Here the $$b$$ in numerator appears in $$k^{th}$$ position ].
Here is a more geometric proof, which I'll illustrate for case of a two variable system.
$$c_1 = a_1 x + b_1 y$$
$$c_2 = a_2 x + b_2 y$$
The above is equivalent to:
$$\begin{bmatrix} c_1 \\ c_2 \\ 0 \end{bmatrix} = x \begin{bmatrix} a_1 \\ a_2 \\ 0\end{bmatrix} + y \begin{bmatrix} b_1 \\ b_2 \\ 0\end{bmatrix}$$
To solve for $$x$$ , cross product both sides with $$\begin{bmatrix} b_1 \\ b_2 \\ 0\end{bmatrix}$$ :
$$\begin{bmatrix} c_1 \\ c_2 \\ 0\end{bmatrix} \times \begin{bmatrix} b_1 \\ b_2 \\ 0\end{bmatrix}=x\begin{bmatrix} a_1 \\ a_2 \\ 0\end{bmatrix} \times \begin{bmatrix} b_1 \\ b_2 \\ 0\end{bmatrix}$$
After this equate the 3rd component of third entry in each column vector and do some algebra:
$$\frac{c_1 b_2 - b_1 c_2}{a_1 b_2 -b_1 a_2} = x$$
Done.<|endoftext|>
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Dengue virus is a viral disease that is spread by mosquitoes. It is a problem in many tropical and subtropical parts of the world, including Africa, Asia, South America and some parts of northern Queensland. Estimates suggest that around 100 million cases occur each year.
Dengue virus ranges in severity from a mild flu-like illness through to a severe disease. There is no specific treatment and no vaccine. The best way to protect against dengue virus and other mosquito-borne diseases is to avoid mosquito bites. Seek medical attention immediately if you think you may have contracted dengue virus. Early diagnosis and management of symptoms is critical to reduce the risk of complications and avoid further spread of the virus.
Cause of dengue virus
Dengue virus is caused by infection with one of four closely related viruses known as DEN-1, DEN-2, DEN-3 and DEN-4. Infection with one type gives you lifelong immunity to that particular dengue virus. However, the infection does not offer immunity to the other three types, so it is possible to contract dengue virus again. A person who has had dengue virus once is at increased risk of experiencing more severe dengue virus symptoms if they get infected again.
Symptoms of dengue virus
The typical signs and symptoms of dengue virus may include:
- high temperature
- severe headache
- pain behind the eyes
- joint and muscle aches
- appetite loss
- nausea and vomiting
- generally feeling unwell (malaise)
- skin rash
In most cases, symptoms resolve within one to two weeks.
Symptoms of severe dengue virus
Although rare in Australia, certain people can develop severe dengue virus infection. Babies, young children, and people who have had dengue more than once are at increased risk of this complication.
Warning signs of more severe dengue virus include the typical signs and symptoms in additional to some or all of the following:
- severe abdominal pain
- restlessness and fatigue
- persistent vomiting (which may include blood)
- shortness of breath
- nose bleeds and bleeding gums.
Most people who experience these symptoms recover fully. A small number of people who experience these symptoms will go on to have severe dengue which can include:
- severe bleeding
- extremely low blood pressure caused by blood loss (shock)
Where dengue virus commonly occurs
Dengue virus is common throughout tropical and subtropical areas of:
- the Caribbean
- Central America
- Central Pacific
- the Middle East
- South America
- Southeast Asia
- the South Pacific.
Dengue virus in Australia
Cases of dengue virus occur in northern Queensland from time to time when travellers who have been infected overseas return and introduce the virus to the local mosquito population. To date, it isn’t as common as in other subtropical regions.
How dengue virus is spread
Dengue virus is not transmitted (spread) from person to person. Only infected mosquitoes transmit dengue virus. It is thought that the mosquito contracts the virus when it bites an infected person. The mosquito is then infective for the rest of its life and can spread the virus every time it bites someone.
At least three different kinds of mosquito in Australia are suspected to be dengue carriers. They are Aedes aegypti, Aedes scutellaris and Aedes katherinensis. These mosquitoes are found in northern Queensland, the Northern Territory and northern Western Australia. They are not found in Victoria.
Avoid mosquito bites and avoid dengue virus in commonly affected areas
Protect yourself against mosquito bites to avoid dengue virus (and other mosquito-borne diseases) in dengue-affected areas. Suggestions include:
- Wear socks, long pants and long-sleeved shirts. Loose fitting clothing makes it harder for mosquitoes to bite you through your clothes.
- Wear mosquito repellent that contains the active constituents DEET (N,N-Diethyl-m-toluamide) or picaridin. Reapply regularly and make sure you follow directions for safe use on the label. (For kids, it can be safer to spray insect repellent on their clothes rather than their skin.)
- Apply insect repellent first thing in the morning because dengue mosquitoes bite during the day, both outdoors and inside homes and buildings.
- Apply a product, such as permethrin, to your clothes or bedding.
- Use a bed net (mosquito net).
- Stay in air-conditioned accommodation with flyscreens on the windows.
Diagnosis of dengue virus
See a doctor immediately if you think you may have dengue virus. Early diagnosis is important to reduce the risk of complications and avoid further spread of the virus.
Your doctor will ask about your medical history, including any travel, and will do a physical examination. Blood tests are required to diagnose dengue.
Treatment for dengue virus
There is no specific treatment for dengue virus. Medical care aims to manage the symptoms and reduce the risk of complications while the person recovers. Most cases of uncomplicated dengue virus resolve fully within one to two weeks.
During this time, your doctor may advise:
- bed rest
- plenty of fluids
- medication to reduce fever, such as paracetamol (do not take aspirin because of its blood-thinning properties).
Hospital admission is usually required if the person develops warning signs of more severe dengue. Treatment for these complications may include intravenous fluids and replacement of lost electrolytes.
Where to get help
This page has been produced in consultation with and approved by:
Department of Health and Human Services - RHP&R - Health Protection - Communicable Disease Prevention and Control Unit
Content on this website is provided for information purposes only. Information about a therapy, service, product or treatment does not in any way endorse or support such therapy, service, product or treatment and is not intended to replace advice from your doctor or other registered health professional. The information and materials contained on this website are not intended to constitute a comprehensive guide concerning all aspects of the therapy, product or treatment described on the website. All users are urged to always seek advice from a registered health care professional for diagnosis and answers to their medical questions and to ascertain whether the particular therapy, service, product or treatment described on the website is suitable in their circumstances. The State of Victoria and the Department of Health & Human Services shall not bear any liability for reliance by any user on the materials contained on this website.<|endoftext|>
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## Laundry Sorting
In this lesson, children will sort "laundry" by color, size, shape and clothing attribute.
### Lesson for:
Toddlers/Preschoolers
(See Step 5: Adapt lesson for toddlers or preschoolers.)
### Content Area:
Algebra
Data Analysis and Probability
Measurement
### Learning Goals:
This lesson will help toddlers and preschoolers meet the following educational standards:
• Understand measurable attributes of objects and the units, systems and processes of measurement
• Formulate questions that can be addressed with data and collect, organize and display relevant data to answer these questions
• Understand patterns, relations and functions
### Learning Targets:
After this lesson, toddlers and preschoolers should be more proficient at:
• Sorting and classifying objects according to their attributes and organizing data about the objects
• Recognizing the attributes of length, volume, weight, area and time and then comparing and ordering objects according to these attributes
• Formulating questions that can be addressed with data and collecting, organizing and displaying relevant data to answer these questions
## Laundry Sorting
### Lesson plan for toddlers/preschoolers
#### Step 1: Gather materials.
• Doll clothing from your dramatic play area or the children’s own clothing (Using the children’s own clothing works particularly well during the winter season, when children have hats, scarves, mittens and other outdoor gear. To make sure that the children are able to sort by colors, sizes and shapes, it is best to organize the materials beforehand. You can also create different clothing items (shirts, pants, socks, skirts, shorts, dresses, hats, etc.) using stencils and colored construction paper (blue, red, green, yellow). Keep the cutouts simple (e.g., a basic shirt shape in several colors and in small and large sizes).
• Small plastic produce baskets such as the ones that berries come in or large laundry baskets (the size of the basket will be determined by the size of the objects that you are using)
Note: Small parts pose a choking hazard and are not appropriate for children age five or under. Be sure to choose lesson materials that meet safety requirements.
#### Step 2: Introduce activity.
1. Ask the children if they have ever helped their family members or friends with the laundry. Ask them what they need to do before they actually wash the laundry. Explain that, before people do laundry, they often need to sort their laundry into specific piles according to specific attributesSay: People often sort their laundry by color. I sort my laundry into light colors like white and pale pink and darks like black and dark blue. Sometimes, people sort their laundry by types of clothing. I sort my laundry into jeans, towels and delicates such as underwear, t-shirts and fancy shirts. Sometimes, people separate their laundry by size. I sort my laundry by large items such as blankets and comforters and small items such as socks, mittens and hand towels.”
2. Explain that today the children are going to be given piles of laundry. Tell them that their job is to sort the laundry according to various attributes.
3. Give the children a working definition of the word attribute: “A characteristic like size, shape or color.”
#### Step 3: Engage children in lesson activities.
1. Give each child the laundry cutouts and several baskets. Make sure there are enough baskets to accommodate the attributes that the children will be sorting.
2. Start sorting with just two attributesSay: “Our first sort will be by size. You will have two baskets and you will be sorting your laundry into big and little piles. Who can give me an example of a piece of laundry that is big?” (Coat, towel, pants) “Yes. Go ahead and put your big laundry into one basket and the other basket will be for small laundry. Who can give me an example of small laundry?” (Mittens, hand towels, underwear) Once their piles of laundry have been correctly sorted, tell the children to put all of their laundry back into a pile and get ready for the next sort.
3. Increase the attributes and also give the children more baskets. Give the children one basket for each of the attributes that they will be sorting by. Say: “Next we are going to sort by color. What different colors do you see in your laundry pile?” (Blue, red, green, yellow) “So, all of the yellow pieces of laundry will go into one basket and all of your green pieces of laundry will go into another basket and so on.” Once their piles of laundry have been correctly sorted, have them put all of their laundry back into a pile and get ready for the last sort.
4. Say: “Our last sort is going to be by type of clothing.” Again, make sure the children have the correct number of baskets to match the number of sort topics. Say: “Let’s start by sorting our laundry into piles of clothing that we wear on the top parts of our bodies and clothing that we wear on the bottom parts of our bodies. Who can give me an example of clothing that we wear on the top parts of our bodies?” (Hats, shirts, jackets) “What is an example of clothing that we wear on the bottom parts of our bodies?” (Pants, socks, underwear) You can suggest a number of sorts within this category:
• Pieces of laundry that you don’t wear but that still need to be washed (towels, blankets) and pieces of laundry that you do wear
• Clothing that you wear inside and clothing that you wear outside
• Clothing that you wear when it is cold outside and clothing that you wear when it is hot outside
• Clothing that you wear to sleep, to play sports, to go to school or to attend a fancy event
• Clothing that you wear to swim, to play soccer or to sled
• Create piles of laundry pieces with the same attributes. Have the children try to figure out the common attribute of all of the items in the pile.
• Have the children come up with their own attributes for laundry sorting. Ask a child if he/she can come up with a way to sort the laundry.
• Create a questionnaire that the children can use to collect their data. (On the long side of a sheet of paper, list all of the children’s first names. At the top of the sheet, list the various categories, such as red shirts, blue pants, white socks and yellow sweaters. Make a grid of each child’s name and the listed category, so that there is a box under each category on the same line that the child’s name is listed. Tell the children to interview their classmates and record their data on the sheets.)
#### Step 4: Vocabulary.
• Attribute: A characteristic like size, shape, or color (e.g., “Many times, before people do laundry, they need to sort their laundry into specific piles according to specific attributes.”)
• Sort: Separate items according to a given attribute (e.g., “Many times, before people do laundry, they need to sort their laundry into specific piles according to specific attributes.”)
#### Step 5: Adapt lesson for toddlers or preschoolers.
###### Toddlers may:
• Have difficulty recognizing the category sorts
###### Child care providers may:
• Brainstorm various category ideas
• Have the children respond to questions that have many right answers
• Incorporate these questions into the attributes that make up the various sorts (for example, ask the children what types of clothes they wear to go outside and then ask them to find those clothes in the laundry pile)
###### Preschoolers may:
• Easily identify laundry items possessing like attributes
###### Child care providers may:
• Create piles of laundry pieces with the same attributes and ask the children to try to figure out the common attribute of all of the items in the pile
• Have the children come up with their own laundry sorting attributes or ask a child if he/she can come up with a way to sort the laundry
• Create a questionnaire that the children can use to collect their data. (On the long side of a sheet of paper, list all of the children’s first names. At the top of the sheet, list the various categories, such as red shirts, blue pants, white socks and yellow sweaters. Make a grid of each child’s name and the listed category, so that there is a box under each category on the same line that the child’s name is listed. Tell the children to interview their classmates and record their data on the sheets.)
### Suggested Books
• Sort it Out! by Barbara Mariconda (Mt. Pleasant, SC: Sylvan Dell Publishing, 2008)
• Sorting by Henry Arthur Pluckrose. (New York: Children’s Press, 1995)
• Grandma’s Button Box by Linda Williams Aber (Minneapolis, MN: Kane Press Paperback, 2002)
### Outdoor Connections
• A great fall activity rich with opportunity: The children can collect different types of autumn leaves and then sort and categorize the leaves by shape, size and color.<|endoftext|>
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# How Do I Calculate How Much Concrete I Need?
Calculating how much concrete you need for a project is simply a matter of math. You just need to know some basic geometry formulas and then plug your measurements into the formula. The only tools you need for this part of the project are a measuring tape, calculator, pencil and paper.
## Preparations
You can take the measurements in inches, feet or yards, but all of them will have to be converted to the same type of measurement to get an accurate count. You are dealing with volume, and a large amount of it. Because of this, you want your final measurement to be in cubic feet or cubic yards. To convert inches to feet, divide the inches by 12. To convert inches to yards, divide the inches by 36. To convert feet to yards, divide the number of feet by 3. Finally, if you want to convert cubic feet into cubic yards, divide the number of cubic feet by 27.
## Slabs and Footers
For a concrete slab, you need to know three measurements: thickness, length and width. Make sure all of the measurements are in the same units and then multiply them together. This works best for a rectangular slab or a footer. If you have an odd-shaped slab, then you should break the shape into smaller rectangular shapes. Then add the totals of the smaller shapes together to get the concrete needed for the entire slab. For example, you need a slab of concrete that is 12 feet wide, 10 feet long and 4 inches thick. The math is: 12 ft. x 10 ft. x (4 in./12 in. per ft.) ft. = 12 x 10 x .333 = 40 cubic feet.
## Columns
The area of rectangular columns can be calculated the same way as a slab, but since most columns are circular, you need to use a different formula: h?r2. In this formula, h is the height of the column, ? is 3.14, r is the radius of the column circle (half of the circle's diameter). For example, you need concrete for a column that is 12 feet high and is circular with a diameter of 1 foot. The math is: 12 ft. x 3.14 x (1 ft./2)2 = 12 ft. x 3.14 x (.5 ft.)2 = 12 ft. x 3.14 x .25 ft.2 = 9.42 cubic feet.
## Steps
Steps are actually a series of smaller concrete slabs stacked on each other. This is the way you should calculate the volume. You need the ultimate height of the stairs, the ultimate depth, the height of each stair and width of the stairs. Consider each step a slab, calculate its volume and then add it to the volume of the other stairs. For example, you need concrete for three stairs that are 8 inches high and 8 inches deep and the total set of stairs will be 1 yard deep and 2 feet across. Since there are three steps, we'll make three measurements. The first step is 1 yd. x 2 ft. x 8 in. The second step is (1 yd. – 8 in.) x 2 ft. x 8 in. The third step is (1 yd. – 16 in.) x 2 ft. x 8 in. The first thing to do is to convert everything to the same units. In this case we'll use feet. So the equation is: (3 ft. x 2 ft. x .67 ft.) + (.78 ft. x 2 ft. x .67 ft.) + (.56 ft. x 2 ft. x .67 ft.) = 4.02 cubic ft. + 1.05 cubic ft. + .75 cubic ft. = 5.82 cubic ft.
## Overage
When dealing with concrete, you need to allow extra for spillage or variations that might occur in the surfaces. Many people simply add 10 percent to their final numbers. Simply multiply your final concrete needs by 1.1 to add the 10 percent overage.<|endoftext|>
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# Application of Derivatives Class 12 Mathematics Important Questions
Please refer to Application of Derivatives Class 12 Mathematics Important Questions with solutions provided below. These questions and answers have been provided for Class 12 Mathematics based on the latest syllabus and examination guidelines issued by CBSE, NCERT, and KVS. Students should learn these problem solutions as it will help them to gain more marks in examinations. We have provided Important Questions for Class 12 Mathematics for all chapters in your book. These Board exam questions have been designed by expert teachers of Standard 12.
## Class 12 Mathematics Important Questions Application of Derivatives
Question. If the function f given by f(x) = x3 – 3(a – 2)x2 + 3ax + 7, for some a∈R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, ( f(x) − 14) / (x − 1)2 = (x ≠ 1) is
(a) –7
(b) 5
(c) 7
(d) 6
Question. Let
x ∈ R where a, b and d are non-zero real constants. Then :
(a) f is an increasing function of x
(b) f is a decreasing function of x
(c) f’ is not a continuous function of x
(d) f is neither increasing nor decreasing function of x
Question. The tangent at the point (2, –2) to the curve, x2y2 – 2x = 4 (1–y) does not pass through the point :
(a) (4, 1/3)
(b) (8, 5)
(c) (–4, –9)
(d) (–2, –7)
Question. Consider
normal to y = f(x) at x = π/6 also passes through the point:
(a) (π/6, 0)
(a) (π/4, 0)
(c) (0, 0)
(d) (0, 2π/3)
Question. If f(x) = xex(1 – x), x ∈ R , then f(x) is
(a) decreasing on [–1/2, 1]
(b) decreasing on R
(c) increasing on [–1/2, 1]
(d) increasing on R
Question. Angle between the tangents to the curve y = x2 – 5x + 6 at the points (2, 0) and (3, 0) is
(a) π
(b) π/2
(c) π/6
(d) π/4
Question. The normal to the curve x = a (cos θ + θ sin θ ), y = a (sin θ – θ cos θ ) at any point θ is such that
(a) it passes through the origin
(b) it makes an angle π/2 + θ with the x- axis
(c) it passes through (a + π/2, −a)
(d) It is at a constant distance from the origin
Question. For real x, let f (x) = x3 + 5x + 1, then
(a) f is onto R but not one-one
(b) f is one-one and onto R
(c) f is neither one-one nor onto R
(d) f is one-one but not onto R
Question. If b is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points (3cosθ, √3 sinθ) and (– 3sin θ, √3cos θ); ∈ (0, π/2); then (2 cotβ)/(sin 2θ) is equal to
(a) √2
(b) 2/√3
(c) 1/√3
(d) √3/4
Question. A normal to the hyperbola, 4x2 – 9y2 = 36 meets the coordinate axes x and y at A and B, respectively. If the parallelogram OABP (O being the origin) is formed, then the locus of P is
(a) 4x2 – 9y2 = 121
(b) 4x2 + 9y2 = 121
(c) 9x2 – 4y2 = 169
(d) 9x2 + 4y2 = 169
Question. A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec., then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the hori ontal ground when the top of the ladder is 1 m above the ground is:
(a) 25√3
(b) 25/√3
(c) 25/3
(d) 25
Question. If the volume of a spherical ball is increasing at the rate of 4π cc/sec, then the rate of increase of its radius (in cm/sec), when the volume is 288 π cc,
(a) 1/6
(b) 1/9
(c) 1/36
(d) 1/24
Question. The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 – x2 such that the rectangle lies inside the parabola, is:
(a) 36
(b) 20√2
(c) 32
(d) 18√3
Question. The tangent to the curve y = x2 – 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point :
(a) (7/2, 1/4)
(b) (1/8, -7)
(c) (- 1/8, 7)
(d) (1/4, 7/2)
Question. If the surface area of a sphere of radius r is increasing uniformly at the rate 8 cm2/s, then the rate of change of its volume is :
(a) constant
(b) proportional to √r
(c) proportional to r2
(d) proportional to r
Question. The weight W of a certain stock of fish is given by W = nw, where n is the si e of stock and w is the average weight of a fish. If n and w change with time t as n = 2t2 + 3 and w = t2 – t + 2, then the rate of change of W with respect to t at t = 1 is
(a) 1
(b) 8
(c) 13
(d) 5
Question. Two points A and B move from rest along a straight line with constant acceleration f and f ‘ respectively. If A takes m sec. more than B and describes ‘n’units more than B in acquiring the same speed then
(a) ( f – f ‘)m2 = ff ‘n
(b) ( f + f ‘)m2 = ff ‘n
(c) 1/2 ( f – f ‘)m = ff ‘n2
(d) ( f + f ‘)n = 1/2 ff ‘m2
Question. A li ard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of 2 cm/s2 and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. Then the lizard will catch the insect after
(a) 20 s
(b) 1 s
(c) 21 s
(d) 24 s
Question. If θ denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection, then |tan θ| is equal to:
(a) 4/9
(b) 8/15
(c) 7/17
(d) 8/17
Question. If the curves y2 = 6x,9x2 + by2 =16 intersect each other at right angles, then the value of b is :
(a) 7/2
(b) 4
(c) 9/2
(d) 6
Question. The function, f(x) = (3x – 7)x2/3 , x ∈ R, is increasing for all x lying in :
Ans : A
Question. Let f(x) = ex – x and g(x) = x2 – x, x ∈ R. Then the set of all x ∈ R, where the function h(x) = (fog) (x) is increasing, is :
Ans : B
Question. If the function f : R – {1, –1} → A defined by f(x) = x2/(1−x2), is surjective, then A is equal to:
(a) R – {–1}
(b) [0, “)
(c) R – [–1, 0)
(d) R – (–1, 0)
Question. The position of a moving car at time t is given by f (t) = at2 + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average speed of the car over the time interval [t1, t2] is attained at the point :
(a) (t2 – t1)/2
(b) a(t2 – t1) + b
(c) (t1 + t2)/2
(d) 2a(t1 + t2) + b
Question. If the surface area of a cube is increasing at a rate of 3.6 cm2/sec, retaining its shape; then the rate of change of its volume (in cm3/sec.), when the length of a side of the cube is 10 cm, is :
(a) 18
(b) 10
(c) 20
(d) 9
Question. Let f: [0 : 2] → R be a twice differentiable function such that f”(x) > 0, for all x∈(0, 2). If Φ(x) = f(x) + f(2 – x), then Φ is :
(a) increasing on (0, 1) and decreasing on (1, 2).
(b) decreasing on (0, 2)
(c) decreasing on (0, 1) and increasing on (1, 2).
(d) increasing on (0, 2)
Question. The function f defined by f(x) = x3 – 3x2 + 5x + 7, is :
(a) increasing in R.
(b) decreasing in R.
(c) decreasing in (0, ∞) and increasing in (– ∞ , 0).
(d) increasing in (0, ∞) and decreasing in (– ∞, 0).
Question. Let f(x) = sin4x + cos4x. Then f is an increasing function in the interval :
(a) ]5π/8, 3π/4]
(b) ]π/2, 5π/8]
(c) ]π/4, π/2]
(d) ]0, π/4]
Question. The maximum value of the function f(x) = 3x3 – 18x2 + 27 x – 40 on the set S = {x ∈ R : x+ 30 ≤ 11x} is :
(a) – 122
(b) – 222
(c) 122
(d) 222
Question. Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression (xmyn) / ( (1+x2m)(1+y2n) ) is :
(a) 1
(b) 1/2
(c) 1/4
(d) (m + n)/6mn
Question. Let f and g be two differentiable functions on R such that f'(x) > 0 and g'(x) < 0 for all x ∈ R . Then for all x:
(a) f(g (x)) > f (g (x – 1))
(b) f(g (x)) > f (g (x + 1))
(c) g(f (x)) > g (f (x – 1))
(d) g(f (x)) < g (f (x + 1))
Question.Statement-1: The equation x log x = 2 – x is satisfied by at least one value of x lying between 1 and 2.
Statement-2: The function f (x) = x log x is an increasing function in [l, 2] and g (x) = 2 – x is a decreasing function in [1, 2] and the graphs represented by these functions intersect at a point in [1, 2]
(a) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true; Statement-2 is true; Statement-2 is not correct explanation for Statement-1.
(c) Statement-1 is false, Statement-2 is true.
(d) Statement-1 is true, Statement-2 is false.
Question. Two ships A and B are sailing straight away from a fixed point O along routes such that ∠AOB is always 120° . At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/hr):
(a) 260/√37
(b) 260/37
(c) 80/√37
(d) 80/37
Question. A spherical balloon is being inflated at the rate of 35cc/ min. The rate of increase in the surface area (in cm2/min.) of the balloon when its diameter is 14 cm, is :
(a) 10
(b) √10
(c) 100
(d) 10√10
Question. How many real solutions does the equation x7 + 14x5 + 16x3 + 30x – 560 = 0 have?
(a) 7
(b) 1
(c) 3
(d) 5
Question. If the tangent to the curve, y = f (x) = xlogex, (x > 0) at a point (c, f(c)) is parallel to the line segement oining the points (1, 0) and (e, e), then c is equal to:
(a) (e – 1)/e
(b) e(1/(e-1))
(c) e(1/(1-e))
(d) e/(e -1)
Question. The function f(x) = x/2 + x/2 has a local minimum at
(a) x = 2
(b) x = -2
(c) x = 0
(d) x = 1
Question. The real number x when added to its inverse gives the minimum value of the sum at x equal to
(a) –2
(b) 2
(c) 1
(d) –1
Question. Which of the following points lies on the tangent to the curve x4ey + 2√(y+1) =3 at the point (1, 0)?
(a) (2, 2)
(b) (2, 6)
(c) (– 2, 6)
(d) (– 2, 4)
Question. If
then dy/dx at x = 0 is ___________.
Ans : 91
Very Short Answer Type Questions
Question. The total cost C(x) associated with provision of free mid-day meals to x students of a school in primary classes is given by C(x) = 0.005x3 – 0.02x2 + 30x + 50 If the marginal cost is given by rate of change dC/dx of total cost, write the marginal cost of food for 300 students. What value is shown here?
Answer. Given, C(x) = 0.005x3 – 0.02x2 + 30x + 50
= 1350 – 12 + 30 = 1368.
The value indicated here is that a kind of care and concern is shown towards the health of students of primary classes by providing free mid-day meal to them.
Question. The total expenditure (in ₹) required for providing the cheap edition of a book for poor and deserving students is given by R(x) = 3x2 + 36x where x is the number of sets of books. If the marginal expenditure is defined as dR/dx, write the marginal expenditure required for 1200 such sets. What value is reflected in this question?
Answer. Here, R(x) = 3x2 + 36x,
The value indicated here is that a kind of help is provided to poor and deserving students who want to study but they don’t have sources to purchase books.
Question. For the curve y = 3x2 + 4x, Find the slope of the tangent to the curve at the point whose x-coordinate is –2.
Answer. The given curve is y = 3x2 + 4x.
Short Answer Type Questions
Question. Find the intervals in which the following function is (a) increasing (b) decreasing :
f(x) = 2x3 – 9x2 + 12x + 20
Answer. Given : f(x) = 2x3 + 9x2 + 12x + 20
⇒ f’ (x) = 6x2 + 18x + 12
= 6(x2 + 3x + 2) = 6(x + 1)(x + 2)
Question. Find the point on the curve 9y2 = x3, where the normal to the curve makes equal intercepts on the axes.
Answer. We have, 9y2 = x3 … (i)
Differentiating (i) w.r.t. x, we get
Question. Find the equation of the tangent and normal to the curve x = a sin3θ and y = a cos3θ at θ = π/4
Answer. We have, x = a sin3θ; y = a cos3θ
Question. Find the equations of the tangent and normal to the curve
Question. Show that the equation of normal at any point t on the curve x = 3 cost – cos3t and y = 3 sint – sin3t is 4(y cos3t – x sin3t)= 3 sin 4t.
Answer. x = 3 cost – cos3t and y = 3 sint – sin3 t
Now,
Question. The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x –5. Find the values of a and b
Answer. We have, y2 = ax3 + b
Differentiating w.r.t. x, we get
So, equation of tangent at the point (2, 3) is
y – 3 = 2a (x – 2)
⇒ y = 2ax – 4a + 3 …(i)
But we are given that equation of tangent at (2, 3) is
y = 4x – 5 …(ii)
∴ On comparing (i) and (ii), we get
2a = 4 ⇒ a = 2
Point (2, 3) lies on the curve y2 = ax3 + b,
∴ (3)2 = (2)3 a + b ⇒ 9 = 8a + b
⇒ 9 = 8 × 2 + b ⇒ b = –7
Question. Find the angle of intersection of the curves y2 = 4ax and x2 = 4by
Answer. The given curves are
y2 = 4ax … (i)
x2 = 4by …(ii)
Solving (i) and (ii), we get
Question. Show that the equation of tangent to the parabola y2 = 4ax at (x1, y1) is yy1 = 2a(x + x1)
Answer. The Given parabola is
y2 = 4ax … (i)
From (ii) and (iii), we get
yy1 = 2ax + 2ax1 = 2a(x + x1)
This is the equation of the tangent to (i) at (x1, y1).
Question. Find the points on the curve x2 + y2 – 2x – 3 = 0 at which the tangents are parallel to x-axis.
Answer. The given curve is
x2 + y2 – 2x – 3 = 0. …(i)
Differentiating with respect to x, we get
Putting the value of x = 1 in (i), we get
(1)2 + y2 – 2(1) – 3 = 0 ⇒ y2 = 4
⇒ y = ±2
∴ The points are (1, 2) and (1, –2).
Question. Find the point on the curve y = x3 – 11x + 5 at which the equation of tangent is y = x – 11.
Answer. y = x3 – 11x + 5 …(i)
Differentiating (i) w.r.t. x, we get
Also, equation of tangent is y = x – 11
∴ its slope = 1.
So 3x2 – 11 = 1 ⇒ x2 = 4
∴ x = ± 2
Putting the values of x in (i), we get
y = 23 – 11(2) + 5 = 8 – 22 + 5 = – 9
y = (– 2)3 – 11(–2) + 5 = – 8 + 22 + 5 = 19
So points are (2, –9) and (–2, 19).
But only (2, –9) satisfies the equation of tangent.
So required point is (2, –9).
Question. Find the intervals in which the function f(x) = (x – 1)3(x – 2)2 is.
(a) increasing
(b) decreasing.
Answer. Here, f(x) = (x – 1)3 · (x – 2)2
⇒ f’ (x) = 3(x – 1)2(x – 2)2 + (x – 1)3 · 2(x – 2)
= (x – 1)2(x – 2)[3(x – 2) + 2(x – 1)]
= (x – 1)2 (x – 2) (5x – 8)
(a) For f to be an increasing function,
f’ (x) > 0
⇒ (x – 1)2 (x – 2)(5x – 8) > 0
⇒ (x – 2)(5x – 8) > 0 [… (x – 1)2 > 0 ∀ x ∈ R]
⇒ x – 2 > 0, 5x – 8 > 0 or x – 2 < 0, 5x – 8 < 0
Question. Find the intervals in which the following function is (a) increasing (b) decreasing :
f (x) = 2x3 – 15x2 + 36x + 17
Answer. We have, f (x) = 2x3 – 15x2 + 36x + 17
⇒ f’ (x) = 6x2 – 30x + 36 = 6(x2 – 5x + 6)
⇒ f’ (x) = 6(x – 3)(x – 2)
Now, for critical points f’ (x) = 0
⇒ 6(x – 3)(x – 2) = 0 ⇒ x = 2, 3
The points x = 2, x = 3 divide real line into disjoint
intervals (– ∞, 2), (2, 3) and (3, ∞).
Hence, f (x) is increasing in (– ∞, 2] U [3, ∞) and decreasing in [2, 3].
Question. Find the intervals in which the function f given by
(i) increasing
(ii) decreasing
So, critical points are x = –1, x = 1.
Also f (x) is not defined for x = 0.
So, disjoint intervals are (–∞, –1), (–1, 0), (0, 1), (1, ∞).
Hence, f(x) is increasing in (–∞, – 1] U [1, ∞) and decreasing in [–1, 0) U (0, 1]
Question. Find the intervals in which the function given by f(x) = sin x + cosx, 0 ≤ x ≤ 2π is
(a) increasing,
(b) decreasing.
Answer. The given function is
f(x) = sin x + cosx; 0 < x <
⇒ f’ (x) = cos x – sin x
Now f’ (x) = 0 ⇒ cos x – sin x = 0
Question. Find the intervals in which the following function is (a) increasing (b) decreasing :
f(x) = x4 – 8x3 + 22x2 – 24x + 21
Answer. The given function is
f(x) = x4 – 8x3 + 22x2 – 24x + 21
⇒ f ‘(x) = 4x3 – 24x2 + 44x – 24
= 4(x3 – 6x2 + 11x – 6)
= 4(x – 1)(x2 – 5x + 6)
= 4(x – 1)(x – 2)(x – 3)
Thus f ‘(x) = 0 ⇒ x = 1, 2, 3.
Hence, possible disjoint intervals are
(–∞, 1), (1, 2), (2, 3) and (3, ∞).
In the interval (–∞, 1), f ‘ (x) < 0
In the interval (1, 2), f'(x) >0
In the interval (2, 3), f'(x) < 0
In the interval (3, ∞), f'(x) >0
∴ f is increasing in [1, 2] U [3, ∞) and f is
decreasing in (–∞, 1] U [2, 3].
Question.
Long Answer Type Questions
Question. Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).
Answer. the given parabola is
y2 = 4ax …(i)
and the point is P(at2, 2at).
Differentiating (i) w.r.t. x, we get
Question. Find the equations of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
Answer. The given curve is
y = x3 + 2x + 6 …(i)
Question. Find the equations of the tangent and the normal to the curve x = 1 – cos θ; y = θ – sin θ at θ = π/4
Answer. Given curves are x = 1 – cos θ and y = θ –sin θ
Question. Find the local maxima and local minima of the function f(x) = sin x – cos x, 0 < x < 2π. Also find the local maximum and local minimum values.
Answer. We have, f(x) = sinx – cosx
⇒ f’ (x) = cos x + sinx
For maxima or minima, f’ (x) = 0
⇒ cos x + sin x = 0 ⇒ tan x = –1
Question. Find the minimum value of (ax + by), where xy = c2
Answer. Let u = ax+ by, where xy = c2
Question. Find the coordinates of a point of the parabola y = x2 + 7x + 2 which is closest to the straight line y = 3x –3.
Answer. Let P (h, k) be the coordinates of the point on given parabola.
∴ k = h2 + 7h + 2 …(i)
The distance S of P from the straight line
– 3x + y + 3 = 0 is
⇒ S will be maximum or minimum according as f(h) is maximum or minimum.
Since, f(h) = h2 + 4h + 5
f’ (h) = 2h + 4
For maxima or minima, f’ (h) = 0
⇒ 2h + 4 = 0 ⇒ h = – 2
Also, f” (h) = 2 > 0 when h = – 2
S is minimum at h = –2
Putting this value in (i), we get
k = (–2)2 + 7(–2) + 2 = 4 – 14 + 2 = –8
∴ The required coordinates are (–2, –8)
Question. A tank with rectangular base and rectangular sides open at the top is to be constructed so that its depth is 3 m and volume is 75 m3. If building of tank costs ₹ 100 per square metre for the base and ₹ 50 per square metre for the sides, find the cost of least expensive tank.
Answer. Let a m and b m be the sides of the base of the tank.
Question. A point on the hypotenuse of a right triangle is at distance ‘a’ and ‘b’ from the sides of the triangle. Show that the minimum length of the hypotenuse is
Answer. Let P be any point on the hypotenuse of the given right triangle.
Let PL = a, PM = b and AM = x.
Clearly, ΔCPL and ΔPAM are similar
Question. Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
Answer. Let r and h be the radius and height of the cylindrical can respectively.
Therefore, the total surface area of the closed cylinder is given by
S = 2πrh + 2πr2 = 2πr(r + h) …(i)
Given volume of the can = 128π cm3
Also volume (V) = πr2h
Question. Show that the semi vertical angle of the cone of the maximum volume and of given slant height is cos−1 1/√3
Answer. Let θ be the semi-vertical angle of the cone, V its volume, h its height, r base radius and slant height l.
Then from ΔOAP,
r = l sin θ, h = l cos θ
Now
Question. Prove that the semi vertical angle of the right circular cone of given volume and least curved surface area is cot−1 √2.
Answer. Let r, h, l, V and S be respectively the base radius, height, slant height, volume and curved surface of the cone.
Then,
l2 = r2 + h2
Question. Prove that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3. Also find the maximum volume.
Answer. Let r and h be the base radius and height of cylinder respectively.
Question. The sum of the perimeters of a circle and a square is k, where k is some constant. Prove that the sum of their areas is least when the side of the square is equal to the diameter of the circle.
Answer. Let a be the side of the given square and r be the radius of the circle.
By hypothesis 4a + 2πr = k
Question. Show that a cylinder of a given volume which is open at the top has minimum total surface area, when its height is equal to the radius of its base.
Answer. Let r and h be the base radius and height of the cylinder respectively and volume of cylinder, V = πr2h
Now, πr3 = V ⇒ πr3 = πr2h ⇒ r = h
Hence, the cylinder of a given volume which is open at the top has minimum total surface area, when it height is equal to the radius of its base.
Question. A window is of the form of a semi-circle with a rectangle on its diameter. The total perimeter of the window is 10 m. Find the dimension of the window to admit maximum light through the whole opening.
Answer. Let ABCD be a rectangle and let the semi-circle is described on the side AB as its diameter.
Let AB = 2x and AD = 2y. Let P = 10 m be the given perimeter of window.
Therefore, 10 = 2x + 4y + πx ⇒ 4y = 10 – 2x – πx …(i)
Question. AB is a diameter of a circle and C is any point on the circle. Show that the area of ΔABC is maximum , when it is isosceles.
Answer. Here BA is a diameter of the given circle, of radius = r.
Let ∠CAB = θ
Question. Find the point P on the curve y2 = 4ax which is nearest to the point
Answer. The given parabola is
y2 = 4ax …(i)
Let Q(11a, 0).
Any point on (i) is P(at2, 2at)
∴ PQ2 = (at2 – 11a)2 + (2at – 0)2
Let l = PQ2 = a2t4 – 18 a2t2 + 121a2
∴ This corresponds to a minimum value of l i.e., of PQ2 and therefore of PQ.
Thus, there are two such points P with coordinates,
(9a, 6a)and (9a – 6a) nearest to the given point Q.
Question. If the length of three sides of a trapezium other than base is 10 cm each, then find the area of the trapezium when it is maximum.
Answer. Let ABCD be the given trapezium
Then AD = DC = CB = 10 cm
In ΔAPD and ΔBQC
DP = CQ = h
AD = BC = 10 cm
∠DPA = ∠CQB = 90°
∴ ΔAPD ≅ ΔBQC (by R.H.S. congruency)
⇒ AP = QB = x cm (Say)
∴ AB = AP + PQ + QB
= x + 10 + x = (2x + 10)cm
Also from ΔAPD,
AP2 + PD2 = AD2 ⇒ x2 + h2 = 102
Question. Find the area of the greatest rectangle that can be inscribed in an ellipse
Answer. Let ABCD be a rectangle inscribed in the ellipse,
Let AB = 2q, DA = 2p.
Then coordinates of A are (p, q).
As A lies on the ellipse so
Question. Prove that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of a radius r is 4r/3
91. Prove that of all the rectangles inscribed in a given circle, the square has the maximum area.
Answer. Let the radius of given circle = r
Let 2a and 2b are the lengths of the sides of any inscribed rectangle in the given circle.
Then from ΔOPC,
OC2 = OP2 + PC2
⇒ r2 = a2 + b2
Question. Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone, is half that of the cone.
Answer. Let R and H be the base radius and height of the given cone and r, h be the same for the inscribed cylinder. Clearly, ΔVBC is similar to ΔVOA
Question. Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 times the radius of the base.
Answer. Let r be the base radius of the cone, l be the slant height and h be its height. Let V be its volume and S be its curved surface.
Question. Show that the height of a closed right circular cylinder of given surface and maximum volume, is equal to the diameter of its base.
Answer. Let r be the radius of the circular base, h be the height and S be the total surface area of a right circular cylinder, then
S = 2πr2 + 2πrh is given to be a constant.
Let V be the volume of the cylinder, then
So, volume is maximum when the height is equal to the diameter.
Question. An open box with a square base is to be made out of a given quantity of cardboard of area c2 square units. Show that the maximum volume of the box is c36√3 cubic units.
Answer. Let h be height and x be the side of the square base of the open box. Then its area = x × x + 4 h × x = c2
(given)
Question. Prove that the area of a right angled triangle of given hypotenuse is maximum when the triangle is isosceles.
Answer. Let ΔABC be gives right angled triangle with sides a, b and hypotenuse c.
We have, a2 + b2 = c2
Question. Show that of all the rectangles with a given perimeter, the square has the largest area.
Answer. Let x and y be the length and breadth of the rectangle whose perimeter (P) is given.
Question. Show that of all the rectangles of given area, the square has the smallest perimeter.
Answer. Let x and y be the lengths and breadth of rectangle of given area A, then we have,
∴ Perimeter of rectangle is minimum, when x = √A
So, perimeter of rectangle is minimum, when y = x
i.e. rectangle is square.
Question. A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the rectangle that will produce the largest area of the window.
Answer. Let a be the breadth and b be the length of rectangle and b be the side of equilateral triangle. Total perimeter of the window
Question. Find the point on the curve y2 = 2x which is at a minimum distance from the point (1, 4).
Answer. Let the point on y2 = 2x, which is at a min.
distance from Q(1, 4), be P(x, y).
∴ We have to minimise
s = PQ2 = (x – 1)2 + (y – 4)2
Question. Show that a right circular cylinder which is open at the top and has a given surface area, will have the greatest volume, if its height is equal to the radius of its base.
Answer. Let S, V, r and h be the surface area, volume, radius and height of the cylinder. Then
Question. A manufacturer can sell x items at a price of
Answer. Let S(x) be the selling price of x items and let C(x) be the cost price of x items. Then, we have
Thus x = 240 is a point of maxima. Hence, the manufacturer can earn maximum profit, if he sells 240 items.
Question. Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height h is (1/3)h.
Answer. Let a cylinder be inscribed in a cone of radius
R and height h.
Let the cylinder’s radius be r and its height be h1
Question. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle α is 4/27 πh3 tan2 α .
Answer. Let R be the radius, H be the height of the cylinder inscribed in cone, r be the radius and h be the height of the cone.
where α is the semi-vertical angle of the cone
As α is given. ∴ It is constant.
Let V be the volume of the cylinder.
∴ V= πR2H = π[(h – H)2 tan2α]H (Using (i))
V = πH · (h – H)2 tan2α …(ii)
Differentiating (ii) w.r.t. H, we get
dV/dH = π[(h − H)2 X 1+ H · 2(h − H)(−1)]tan2 α
= π tan2 α(h2 − 4hH + 3H2 ) …(iii)
= πtan2 α(h – H)(h – 3H)
Question. Find the point on the curve x2 = 8y which is nearest to the point (2, 4).
Answer. Let P(x, y) be a point on x2 = 8y and Q ≡(2, 4).<|endoftext|>
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# What are the characteristics of isosceles triangles?
An isosceles triangle is a triangle consisting of two equal sides. Since these triangles have two equal sides, it also means that they have two equal angles. Similar to other triangles, isosceles triangles have three vertices, three edges, and their interior angles add up to 180°.
Here, we will learn about the fundamental properties of isosceles triangles. Also, we will look at its most important formulas and use them to solve some exercises.
##### GEOMETRY
Relevant for
Learning about the characteristics of isosceles triangles.
See characteristics
##### GEOMETRY
Relevant for
Learning about the characteristics of isosceles triangles.
See characteristics
## Fundamental characteristics of isosceles triangles
An isosceles triangle has the following characteristics:
• Two sides are congruent with each other, that is, two sides have the same length.
• The third side of an isosceles triangle, which is uneven to the other two sides, is called the base of the isosceles triangle.
• The two angles opposite the equal sides are congruent with each other. This means that it has two congruent base angles.
• The angle that is not congruent with the other angles is called the apex angle.
• The height from the apex angle of an isosceles triangle bisects the base into two equal parts and also bisects the apex angle into two equal angles.
• The height from the apex angle divides the triangle into two right triangles.
## Important isosceles triangle formulas
The most important formulas for isosceles triangles are the perimeter, height, and area formulas.
### Isosceles triangle perimeter formula
The perimeter of isosceles triangles is calculated by adding the lengths of all the sides of the triangle. In this case, two of the lengths are equal, so we can use the following formula:
where b is the length of the base and a is the length of the congruent sides.
### Isosceles triangle area formula
The area of any triangle can be calculated by multiplying the length of the base by the length of the height and dividing by 2. Therefore, we have:
where b is the length of the base and h is the length of the height.
### Formula for the height of isosceles triangles
We can calculate the height of isosceles triangles using the lengths of the triangle’s sides. Therefore, we have:
where a is the length of the congruent sides and b is the length of the base.
## Examples of isosceles triangle problems
### EXAMPLE 1
• What is the perimeter of an isosceles triangle with a base of length 12 m and congruent sides of length 8 m?
Solution: We have the following values:
• Base, $latex b=12$ m
• Sides, $latex a=8$ m
Using these values in the perimeter formula, we have:
$latex p=b+2a$
$latex p=12+2(8)$
$latex p=12+16$
$latex p=28$
The perimeter is 28 m.
### EXAMPLE 2
• An isosceles triangle has a base of 15 m and a height of 10 m. What is its area?
Solution: We recognize the following values:
• Base, $latex b=15$ m
• Height, $latex h=10$ m
We substitute these values in the formula for the area:
$latex A= \frac{1}{2}bh$
$latex A= \frac{1}{2}(15)(10)$
$latex A=75$
The area is 75 m².
### EXAMPLE 3
• What is the height of an isosceles triangle that has a base of length 8 m and congruent sides of length 12 m?
Solution: We have the following information:
• Base, $latex b=8$ m
• Sides, $latex a=12$ m
We substitute these values in the formula for the area:
$latex h= \sqrt{{{a}^2}- \frac{{{b}^2}}{4}}$
$latex h= \sqrt{{{12}^2}- \frac{{{8}^2}}{4}}$
$latex h= \sqrt{144- \frac{64}{4}}$
$latex h= \sqrt{128}$
$latex h=11.3$
The height of the triangle is 11.3 m.<|endoftext|>
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Humidity is the amount of water vapour in the air. In forecasting, relative humidity describes the percentage of moisture in the air in comparison to how much there is when the air is saturated. The higher the reading, the greater the likelihood of precipitation, dew and fog. Relative humidity is normally highest at dawn, when the temperature is at its lowest point of the day.
High humidity makes people feel hotter than they would on a drier day. That’s because the perspiration that occurs to cool us down cannot evaporate as readily in moist, saturated air. To better describe how hot it feels in such circumstances, Canadian meteorologists developed the humidex, a parameter that combines temperature and humidity in order to reflect the perceived temperature.
Heat and Humidity Safety
It is important to stay safe during such extreme temperatures. Avoid working or exercising intensely if it is very hot or humid outside, and head for cooler conditions if your body becomes overheated. If working outdoors is an absolute necessity, drink plenty of liquids and take frequent rest breaks. Be sure to maintain salt levels in your body and avoid high-protein foods. Also ensure that pets are protected from the heat and have plenty of water to drink. Watch for signs of serious medical conditions, such as heat exhaustion and heat stroke.
Relative humidity is the amount of moisture that the air contains compared to how much it could hold at a given temperature. A figure of 100 per cent relative humidity would mean that the air has become saturated. At this point mist, fog, dew and precipitation are likely.
Relative humidity is normally at its maximum when the temperature is at its lowest point of the day, usually at dawn. Even though the absolute humidity may remain the same throughout the day, the changing temperature causes the ratio to fluctuate.
The humidex is a Canadian innovation, that was first used in 1965. It describes how hot, humid weather feels to the average person. The humidex combines the temperature and humidity into one number to reflect the perceived temperature. Because it takes into account the two most important factors that affect summer comfort, it can be a better measure of how stifling the air feels than either temperature or humidity alone.
The humidex is widely used in Canada. However, extremely high readings are rare except in the southern regions of Ontario, Manitoba and Quebec. Generally, the humidex decreases as latitude increases. Of all Canadian cities, Windsor, Ontario has had the highest recorded humidex measurement: 52.1 on June 20, 1953. The hot, humid air masses which cause such uncomfortable weather usually originate in the Gulf of Mexico or the Caribbean.
Guide to summer comfort
Range of humidex: Degree of comfort
- Less than 29: No discomfort
- 30 to 39: Some discomfort
- 40 to 45: Great discomfort; avoid exertion
- Above 45: Dangerous; Heat stroke possible
An extremely high humidex reading can be defined as one that is over 40. In such conditions, all unnecessary activity should be curtailed. If the reading is in the mid to high 30s, then certain types of outdoor exercise should be toned down or modified, depending on the age and health of the individual, physical shape, the type of clothes worn and other weather conditions.
If working outdoors is an absolute necessity, drink plenty of liquids and take frequent rest breaks. In hot, humid conditions, there is a considerable risk of heat stroke and sun stroke.
During the dog days of summer, remember that animals also feels the heat. When the humidex is high, take special care to ensure that your pet is well-protected from the heat and has plenty of water to drink. Also remember to never leave pets in hot vehicles, even with the window down. On extremely hot days, the inside temperature of a car can be several degrees warmer than the air outside and it is therefore never safe to leave pets or children – even for a few minutes.<|endoftext|>
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You know the scenario: 65 million years ago, a big meteor crash sets off volcanoes galore, dust and smoke fill the air, dinosaurs go belly up.
One theory holds that cold, brought on by the Sun's concealment, is what did them in, but a team of paleontologists led by Pascal Godefroit, of the Royal Belgian Institute of Natural Sciences in Brussels, argues otherwise. Some dinosaurs (warm-blooded, perhaps) were surprisingly good at withstanding near-freezing temperatures, they say.
Witness the team's latest find, a diverse stash of dinosaur fossils laid down just a few million years before the big impact, along what's now the Kakanaut River of northeastern Russia. Even accounting for continental drift, the dinos lived at more than 70 degrees of latitude north, well above the Arctic Circle.
And they weren't lost wanderers, either. The fossils include dinosaur eggshells — a first at high latitudes, and evidence of a settled, breeding population.
It's true the Arctic was much warmer back then, but it wasn't any picnic. The size and shape of fossilized leaves found with the bones enabled Godefroit's team to estimate a mean annual temperature of 50 degrees Fahrenheit, with wintertime lows at freezing.
Yet there is more than one way to skin a dino. All that dust in the atmosphere must have curtailed photosynthesis everywhere, weakening the base of the food chain and inflicting starvation, and finally extinction, upon the dinosaurs.
The research was detailed in the journal Naturwissenschaften.
- Gallery: Drawing Dinosaurs
- Cool Video: High-Tech 'Robo-Saurus' Destroys a Car
- Dinosaur News, Information & Images<|endoftext|>
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# The Logarithmic Rules
In this item, I will show how the basic logarithmic rules, including the Change of Base formula, follow from this equivalency:
$\log_b m = n \Leftrightarrow b^n = m$
For the ease of reading, I’ll generally use the natural base ($$e$$) and the natural logarithm ($$\ln$$). However, everything here applies to all valid bases ($$b > 0, b \ne 1$$).
Also, note these are not fully rigorous, complete proofs.
To demonstrate: $$e^{\ln x} = x$$.
First, note $$\log_b x = a \Leftrightarrow b^a = x$$.
So $$\color{blue} b^{\color{red} {\log_b x}} = \color{green} x \Leftrightarrow \log_{\color{blue} b} \color{green} x = \color{red} {\log_b x}$$. Since the latter is always true, the former is also always true.
That’s the general case, so it’s also true for the specific case of the natural base.
To demonstrate: $$\ln(mn) = \ln m + \ln n$$.
Consider $$mn = e^{\ln(mn)}$$.
Now consider $$mn = m\cdot n = e^{\ln m}\cdot e^{\ln n}$$.
Recall that $$b^p \cdot b^q = b^{p+q}$$.
Hence $$e^{\ln m} \cdot e^{\ln n} = e^{\ln m + \ln n}$$.
Thus $$e^{\ln(mn)} = e^{\ln m + \ln n}$$. $$e^p = e^q \Leftrightarrow p = q$$, so $$\ln(mn) = \ln m + \ln n$$.
The demonstration that $$\ln(\frac{m}{n}) = \ln m – \ln n$$ is nearly identical.
To demonstrate: $$\ln(m^n) = n\ln(m)$$.
Recall that, for positive integer $$n$$, $$m^n = m\cdot m \cdot \dotsm \cdot m (n \text{ times})$$.
For instance, $$\ln(m^2) = \ln(m\cdot m) = \ln m + \ln m = 2 \ln m$$.
This can be generalized to $$\ln(m^n) = n\ln(m)$$.
We can further generalize to all real $$n$$.
To demonstrate: $$\log_b m = \frac{\log m}{\log b}$$.
First: $$\log_b m = a \Leftrightarrow b^a = m$$.
Also: $$\frac{\log m}{\log b} = a \Rightarrow \log m = a \log b \Rightarrow \log m = \log (b^a)$$.
Furthermore, $$\log m = \log (b^a) \Leftrightarrow m = b^a$$.
Hence both $$\log_b m = a$$ and $$\frac{\log m}{\log b} = a$$ imply $$b^a = m$$, and since $$a = a$$, $$\log_b m = \frac{\log m}{\log b}$$.
Clio Corvid
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# GRE Quantitative: Combinations and Permutations
Let’s do some GRE math practice. Combinations and permutations problems often leave students wondering where on earth to begin. Knowing the equation for each operation is helpful, but not enough—you also must be able to determine which formula is necessary to answer the question at hand.
### Combinations and Permutations on the GRE
The rule of thumb is that combinations are unordered and permutations are ordered, but what does that mean? We like illustrating the difference using a social club.
• Imagine the social club has 10 different members and you’re asked, “How many groups of 3 members can you choose from the social club to make a party committee?” Would you need to do combinations or permutations in order to formulate an answer? How do you know?
• Alternatively, imagine we alter the question slightly and ask, “ An officer slate consists of a President, a Vice President, and a Treasurer. How many different officer slates can you select from the social club membership?” Is this the same question? Or is it different? Would you need to use combinations or permutations?
The questions are, in fact, quite different. So how do you apply each method on the GRE?
### Solving Combinations Problems
The first question (“How many groups of 3…”) indicates that we are counting groups of 3 people, with no need to worry about which person we choose first, second, or third—i.e., order does not matter. For that reason, this is a combinations problem.
In order to answer the question, we will use the combinations formula, where n = the total number of items (10) and k = the number of items selected (3). Note that k can equal n, but can never be greater than n (we can choose all of the items in a group, but cannot choose more items than the total). Here’s the combinations formula:
Note that an exclamation point means a factorial; factorial means multiplying the number times each integer below it down to 1. For instance, 4! = 4 * 3 * 2 * 1.
Plugging our values into the equation, we get the following (make sure you reduce numbers in the extended calculations to simplify the actual multiplying you have to do):
Therefore, we could choose 120 different groups of 3 party committees.
### Solving Permutation Problems
The second question asks, “How many different ways can you select a 3-person slate of officers?” This wording tells us that we should track each selection independently, rather than by groups of 3. For example, selecting Nick as President, then Kim as Vice President, then Priyanka as Treasurer would not be the same as selecting Kim as President, then Priyanka as Vice President, then Nick as treasurer, which would not be the same as selecting Kim as President, then Nick as Vice President, then Priyanka as Treasurer, and so on—i.e., order matters. For that reason, this is a permutations problem. In order to answer this question, we will use the following permutations formula:
As you can see, the denominator is the point of difference between the combinations and permutations formulas. For any values of n and k, the number of combinations we can form will always be smaller than the number of permutations we can form. This problem is no exception. Plugging our values into the equation, then reducing as much as possible, we get:
So, when order matters and we track each selection differently, there are 720 different ways we can choose 3 officers.
### Pay Attention to Language
The GRE test-makers create challenging problems by using subtle language to indicate whether you should use a combination or permutation formula to answer the question at hand. Combination questions will indicate that you need to form groups or sets; permutation questions will have words or phrases that indicate order, such as “first, second, third” or “how many different ways.” Some really tricky problems can offer up a mixture of the two.
As the old adage says, “practice makes perfect”—the more of these problems you do (and the more corresponding explanations you read), the better prepared you will be to ace combinations and permutations questions on GRE Test Day.
Tags:<|endoftext|>
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Chapter 11 Language of Descriptive Statistics
Section 11.1 Terminology and Language
11.1.2 Rounding
The rounding of measurement values is an everyday process.
Info 11.1.3
In principle, there are three ways of rounding:
• Rounding (off) using the $\text{floor}$ function $⌊x⌋$.
• Rounding (up) using the $\text{ceil}$ function $⌈x⌉$.
• Rounding using the $\text{round}$ function (sometimes also called $\text{rnd}$ function).
The $\text{floor}$ function is defined as
$\text{floor}:\mathrm{ }ℝ\to ℝ\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }x\mathrm{ }⟼\mathrm{ }\text{floor}\left(x\right)\mathrm{ }=\mathrm{ }⌊x⌋\mathrm{ }=\mathrm{ }max\left\{k\in ℤ : k\le x\right\} .$
If $x\in ℝ$ is a real number, then $\text{floor}\left(x\right)=⌊x⌋$ is the largest integer that is smaller than or equal to $x$. It results from rounding off the value of $x$. If a positive real number $x$ is written as a decimal, then $⌊x⌋$ equals the integer on the left of the decimal point: rounding (off) cuts off the digits on the right of the decimal point. For example $⌊3.142⌋=3$ but $⌊-2.124⌋=-3$. The $\text{floor}$ function is a step function with jumps (in more mathematical terms, jump discontinuities) of height $1$ at all points $x\in ℤ$. The function values at the jumps always lie a step up. They are indicated by the small circles in the figure below, which shows the graph of the $\text{floor}$ function.
Graph of the $\text{floor}$ function
Let a real number $a\ge 0$ be given, written as a decimal number
$a\mathrm{ }=\mathrm{ }{g}_{n} {g}_{n-1} \dots {g}_{1} {g}_{0} . {a}_{1} {a}_{2} {a}_{3} \dots$
This number $a$ can be rounded to $r$ fractional digits ($r\in ℕ{}_{0}$) using the $\text{floor}$ function by
$\stackrel{~}{a}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{r}}·⌊{10}^{r}·a⌋ .$
This process of rounding cuts off the decimal after the $r$th fractional digit. Thus, rounding using the $\text{floor}$ function is in general a rounding off.
Example 11.1.4
Rounding the number ${a}_{1}=2.3727$ to 2 fractional digits using the $\text{floor}$ function results in
${\stackrel{~}{a}}_{1}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌊{10}^{2}·2.3727⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌊237.27⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·237\mathrm{ }=\mathrm{ }=2.37 .$
Alternatively, it can be rounded by cutting off the decimal after the second fractional digit (however, this is only possible if the number is given as a decimal which is rarely the case in a computer program).
Rounding the number ${a}_{2}=\sqrt{2}=1.414213562\dots$ to 4 fractional digits using the $\text{floor}$ function results in
${\stackrel{~}{a}}_{2}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·⌊{10}^{4}·\sqrt{2}⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·⌊14142.1\dots ⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·14142\mathrm{ }=\mathrm{ }1.4142 .$
Rounding the number
${a}_{3}\mathrm{ }=\mathrm{ }\pi \mathrm{ }=\mathrm{ }3,141592654\dots$
to 2 fractional digits using the $\text{floor}$ function results in
${\stackrel{~}{a}}_{3}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌊{10}^{2}·\pi ⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌊314.159\dots ⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·314\mathrm{ }=\mathrm{ }3.14 .$
The rounding method using the $\text{floor}$ function is often applied for calculating final grades in certificates ("academic rounding"). If a mathematics student has the individual grades
Subject Grade Mathematics 1 $1.3$ Mathematics 2 $2.3$ Mathematics 3 $2.0$
then the arithmetic mean of these grades is calculated by
$\frac{1.3+2.3+2.0}{3}\mathrm{ }=\mathrm{ }\frac{5.6}{3}\mathrm{ }=\mathrm{ }1.8\stackrel{‾}{6} .$
Rounding to the first fractional digit using the $\text{floor}$ function would result in the final grade of $\stackrel{~}{a}=1.8$. The rounding methods for calculating final grades always have to be described exactly in the examination regulations.
The counterpart to the $\text{floor}$ function is the $\text{ceil}$ (a.k.a. ceiling) function:
Info 11.1.5
The $\text{ceil}$ function is defined as
$\text{ceil}:\mathrm{ }ℝ\to ℝ\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }x\mathrm{ }⟼\mathrm{ }\text{ceil}\left(x\right)\mathrm{ }=\mathrm{ }⌈x⌉\mathrm{ }=\mathrm{ }min\left\{k\in ℤ : k\ge x\right\} .$
If $x\in ℝ$ is a real number, then $\text{ceil}\left(x\right)=⌈x⌉$ is the smallest integer that is greater than or equal to $x$. The $\text{ceil}$ function is a step function with jumps (jump discontinuities) of height $1$ at all points $x\in ℤ$. The function values at the jumps always lie at the bottom. They are indicated by the small circles in the figure below showing the graph of the $\text{ceil}$ function.
Graph of the $\text{ceil}$ function
Let a real number $a\ge 0$ be given as a decimal number
$a\mathrm{ }=\mathrm{ }{g}_{n} {g}_{n-1} \dots {g}_{1} {g}_{0} . {a}_{1} {a}_{2} {a}_{3} \dots$
This number $a$ can be rounded to $r$ fractional digits ($r\in ℕ{}_{0}$) using the $\text{ceil}$ function by
$\stackrel{^}{a}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{r}}·⌈{10}^{r}·a⌉ .$
Rounding using the $\text{ceil}$ function is in general a rounding up to the next decimal digit.
Example 11.1.6
Rounding the number ${a}_{1}=2.3727$ to $2$ fractional digits using the $\text{ceil}$ function results in
${\stackrel{^}{a}}_{1}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌈{10}^{2}·2.3727⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌈237.27⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·238\mathrm{ }=\mathrm{ }2.38 .$
Analogously, rounding the number ${a}_{2}=\sqrt{2}=1.414213562\dots$ to $4$ fractional digits using the $\text{ceil}$ function results in
${\stackrel{^}{a}}_{2}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·⌈{10}^{4}·\sqrt{2}⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·⌈14142.1\dots ⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·14143\mathrm{ }=\mathrm{ }1.4143 .$
Rounding the number ${a}_{3}=\pi =3.141592654\dots$ to $2$ fractional digits using the $\text{ceil}$ function results in
${\stackrel{^}{a}}_{3}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌈{10}^{2}·\pi ⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌈314.15\dots ⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·315\mathrm{ }=\mathrm{ }3.15 .$
The rounding method using the $\text{ceil}$ function is often applied, for example, in craftsmen's invoices. A craftsman is mostly paid by the hour. If a repair takes 50 minutes (i.e. $0.8\stackrel{‾}{3}$ hours as a decimal), then a craftsmen will round up and invoice a full working hour. Colloquially, rounding mostly means mathematical rounding:
Info 11.1.7
The $\text{round}$ function (or mathematical rounding) is defined as
$\text{round}:\mathrm{ }ℝ\to ℝ\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }x\mathrm{ }⟼\mathrm{ }\text{round}\left(x\right)\mathrm{ }=\mathrm{ }\text{floor}\left(x+\frac{1}{2}\right)\mathrm{ }=\mathrm{ }⌊x+\frac{1}{2}⌋ .$
In contrast to rounding up or rounding off, the maximum change to the number by this rounding is $0.5$.
The $\text{round}$ function is a step function with jumps (jump discontinuities) of height $1$ at all points $x+\frac{1}{2},\mathrm{ }x\in ℤ$. The function values at the jumps always lie a step up. They are indicated by the small circles in the figure below showing the graph of the $\text{round}$ function.
Graph of the $\text{round}$ function
Let a real number $a\ge 0$ be given as a decimal number
$a\mathrm{ }=\mathrm{ }{g}_{n} {g}_{n-1} \dots {g}_{1} {g}_{0} . {a}_{1} {a}_{2} {a}_{3} \dots$
This number $a$ can be rounded to $r$ fractional digits ($r\in ℕ{}_{0}$) using the $\text{round}$ function:
$\stackrel{‾}{a}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{r}}·\text{round}\left({10}^{r}·a\right)\mathrm{ }=\mathrm{ }\frac{1}{{10}^{r}}·⌊{10}^{r}·a+\frac{1}{2}⌋ .$
This rounding method is called mathematical rounding and corresponds to the "normal" rounding process.
Example 11.1.8
The number ${a}_{1}=1.49$ is rounded to one fractional digit using the $\text{round}$ function to
$\begin{array}{ccc}\multicolumn{1}{c}{{\stackrel{‾}{a}}_{1}}& =\hfill & \frac{1}{10}·\mathrm{round}\left(10·1.49\right)\mathrm{ }=\mathrm{ }\frac{1}{10}·⌊10·1.49+0.5⌋\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{10}·⌊14.9+0.5⌋\mathrm{ }=\mathrm{ }\frac{1}{10}·⌊15.4⌋\mathrm{ }=\mathrm{ }\frac{1}{10}·15\mathrm{ }=\mathrm{ }1.5 .\hfill \end{array}$
The number ${a}_{2}=1.52$ is rounded to one fractional digit using the $\text{round}$ function to
$\begin{array}{ccc}\multicolumn{1}{c}{{\stackrel{‾}{a}}_{2}}& =\hfill & \frac{1}{10}·\mathrm{round}\left(10·1.52\right)\mathrm{ }=\mathrm{ }\frac{1}{10}·⌊10·1.52+0.5⌋\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{10}·⌊15.2+0.5⌋\mathrm{ }=\mathrm{ }\frac{1}{10}·⌊15.7⌋\mathrm{ }=\mathrm{ }\frac{1}{10}·15\mathrm{ }=\mathrm{ }1.5 .\hfill \end{array}$
The number ${a}_{3}=2.3727$ is rounded to two fractional digits using the $\text{round}$ function to
$\begin{array}{ccc}\multicolumn{1}{c}{{\stackrel{‾}{a}}_{3}}& =\hfill & \frac{1}{{10}^{2}}·\mathrm{round}\left({10}^{2}·2.3727\right)\mathrm{ }=\mathrm{ }\frac{1}{100}·⌊100·2.3727+0.5⌋\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{100}·⌊237.27+0.5⌋\mathrm{ }=\mathrm{ }\frac{1}{100}·⌊237.77⌋\mathrm{ }=\mathrm{ }\frac{1}{100}·237\mathrm{ }=\mathrm{ }2.37 .\hfill \end{array}$
The number ${a}_{4}=\sqrt{2}=1.414213562\dots$ is rounded to seven fractional digits using the $\text{round}$ function to
$\begin{array}{ccc}\multicolumn{1}{c}{{\stackrel{‾}{a}}_{3}}& =\hfill & \frac{1}{{10}^{7}}·\mathrm{round}\left({10}^{7}·\sqrt{2}\right)\mathrm{ }=\mathrm{ }\frac{1}{{10}^{7}}·⌊{10}^{7}·1.414213562\dots +0.5⌋\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{{10}^{7}}·⌊14142135.62\dots +0.5⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{7}}·⌊14142136.12\dots ⌋\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{{10}^{7}}·14142136\mathrm{ }=\mathrm{ }1.4142136 .\hfill \end{array}$
Exercise 11.1.9
Using the $\text{round}$ function, round the number $\pi =3.141592654\dots$ to four fractional digits: $\stackrel{‾}{\pi }$$=$
.
Exercise 11.1.10
Let the numbers
$a\mathrm{ }=\mathrm{ }\frac{47}{17}\mathrm{ }\mathrm{ }\text{and}\mathrm{ }\mathrm{ }b\mathrm{ }=\mathrm{ }3.7861$
be given.
1. Round each of the numbers $a$ and $b$ to $2$ fractional digits using the $\text{floor}$ function.
The roundings result in $\stackrel{~}{a}$$=$
and $\stackrel{~}{b}$$=$ .
2. Round each of the numbers $a$ and $b$ to $2$ fractional digits using the $\text{ceil}$ function.
The roundings result in $\stackrel{^}{a}$$=$
and $\stackrel{^}{b}$$=$ .
3. Round each of the numbers $a$ and $b$ to $2$ fractional digits using the $\text{round}$ function.
The roundings result in $\stackrel{‾}{a}$$=$
and $\stackrel{‾}{b}$$=$ .<|endoftext|>
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International Day of the World's Indigenous Peoples
Indigenous peoples are inheritors and practitioners of unique cultures and ways of relating to people and the environment. They have retained social, cultural, economic and political characteristics that are distinct from those of the dominant societies in which they live. Despite their cultural differences, indigenous peoples from around the world share common problems related to the protection of their rights as distinct peoples.
Indigenous peoples today, are arguably among the most vulnerable groups of people in the world. The international community now recognizes that special measures are required to protect their rights and maintain their distinct cultures and way of life. This International Day is an opportunity to raise public awareness of their precarious situation. The focus of the International Day in 2018 is Indigenous peoples’ migration and movement.
As a result of loss of their lands, territories and resources due to development and other pressures, many indigenous peoples migrate to urban areas in search of better prospects of life, education and employment. They also migrate between countries to escape conflict, persecution and climate change impacts. Under the mandate of UNESCO, several programmes address migration and movement issues that are relevant for indigenous peoples. UNESCO focuses on inclusion and diversity to help to combat all forms of discrimination.<|endoftext|>
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NCERT Solutions: Polynomials (Exercise 2.4)
# NCERT Solutions for Class 8 Maths Chapter 2 - Polynomials (Exercise 2.4)
Exercise 2.4
Ques 1: Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i) 2x3 + x2 - 5x + 2;
(ii) x3 - 4x2 + 5x - 2; 2, 1, 1
Sol: (i) ∵ p (x) = 2x3 + x2 - 5x + 2
⇒ 1/2 is a zero of p(x).
Again,
p (1) = 2 (1)3 + (1)2 - 5 (1) + 2
= 2 + 1 - 5 + 2
= (2 + 2 + 1) - 5
= 5 - 5 = 0
⇒ 1 is a zero of p (x).
Also p (- 2) = 2 (- 2)3 + (- 2)2 - 5 (- 2) + 2
= 2(- 8) + (4) + 10 + 2
= - 16 + 4 + 10 + 2
= - 16 + 16 = 0
⇒ -2 is a zero of p (x).
Relationship
∵ p(x) = 2x3 + x2 - 5x + 2
∴ Comparing it with ax3 + bx2 + cx + d, we have :
a = 2, b = 1, c = - 5 and d = 2
Also 1/2 , 1 and - 2 are the zeroes of p(x)
Let α = 1/2 , β = 1 and γ = - 2
∴ α + β +γ =
Sum of the zeroes =
⇒ α + β + γ = -b/a
Sum of product of zeroes taken in pair:
Product of zeroes
Also
Thus, the relationship between the co-efficients and the zeroes of p (x) is verified.
(ii) Here, p(x) = x3 - 4x2 + 5x - 2
∴ p (2) = (2)3 - 4 (2)2 + 5 (2) - 2
= 8 - 16 + 10 - 2
= 18 - 18 = 0
⇒ 2 is a zero of p(x)
Again p (1) = (1)3 - 4 (1)2 + 5 (1) - 2
= 1 - 4 + 5 - 2
= 6 - 6 = 0
⇒ 1 is a zero of p(x).
∴ 2, 1, 1 are zeroes of p(x).
Now,
Comparing p (x) = x3 - 4 (x2) + 5x - 2 with ax3 + bx2 + cx + d = 0, we have
a = 1, b = - 4, c = 5 and d = - 2
∵ 2, 1, and 1 are the zeroes of p (x)
∴ Let α = 2
β = 1
γ = 1
Relationship, α + β + γ = 2 + 1 + 1 = 4
Sum of the zeroes =
⇒ α + β + γ = (-b/a)
Sum of product of zeroes taken in pair:
αβ + βγ + γα = 2 (1) + 1 (1) + 1 (2)
= 2 + 1 + 2 = 5
and = c/a = 5/1 = 5
⇒ αβ + βγ + γα = c/a
Product of zeroes = αβγ = (2) (1) (1) = 2
Thus, the relationship between the zeroes and the co-efficients of p(x) is verified.
Ques 2: Find the cubic polynomial with the sum, sum of the products of its zeroes taken two at a time and the product of its zeroes as 2, - 7, - 14 respectively.
Sol: Let the required cubic polynomial be ax3 + bx2 + cx + d and its zeroes be a, b and g.
Since,
If a = 1, then -b/a = 2 ⇒ b = - 2
c/a = - 7 ⇒ c = - 7
-d/a = - 14 ⇒ d = 14
∴ The required cubic polynomial
= 1x3 + (-2) x2 + (-7) x + 14
= x3 - 2x2 - 7x + 14
Ques 3: If the zeroes of the polynomial x3 - 3x2 + x + 1 are a - b, a and a + b then find ‘a’ and ‘b’.
Sol: We have
p (x) = x3 - 3x+ x + 1
comparing it with Ax3 + Bx2 + Cx + D.
We have
A = 1, B = - 3, C = 1 and D = 1
∵ It is given that (a - b), a and (a + b) are the zeroes of the polynomial.
∴ Let, α = (a - b)
β = a
and γ = (a + b)
∴ α + β + γ =
⇒ (a - b) + a + (a + b) = 3
⇒ 3a = 3
⇒ a = 3/3 = 1
Again, αβγ = -D/A = -1
⇒ (a - b) × a × (a + b) = - 1
⇒ (1 - b) × 1 × (1 + b) = - 1 [∵ a = 1, proved above]
⇒ 1 - b2 = - 1
⇒ b2 = 1 + 1 = 2
⇒ b = √2
Thus, a = 1 and b = .√2
Ques 4: If two zeroes of the polynomial x4 - 6x3 - 26x2 + 138x - 35 are 2 3, find other zeroes.
Sol: Here, p(x) = x4 - 6x3 - 26x2 + 138x - 35.
∵ Two of the zeroes of p (x) are :
or (x - 2)2 - (√3)2
or (x2 + 4 - 4x) - 3
or x2 - 4x + 1 is a factor of p(x).
Now, dividing p (x) by x2 - 4x + 1, we have :
∴ (x- 4x + 1) (x2 - 2x - 35) = p (x)
⇒ (x2 - 4x + 1) (x - 7) (x + 5) = p (x)
i.e., (x - 7) and (x + 5) are other factors of p(x).
∴ 7 and - 5 are other zeroes of the given polynomial.
Ques 5: If the polynomial x4 - 6x3 + 16x2 - 25x + 10 is divided by another polynomial x2 - 2x + k, the remainder comes out to be (x + α), find k and α.
Sol: Applying the division algorithm to the polynomials x4 - 6x3 + 16x2 - 25x + 10 and
x2 - 2x + k, we have:
∴ Remainder = (2k - 9) x - k (8 - k) + 10
But the remainder = x + α
Therefore, comparing them, we have :
2k - 9 = 1
⇒ 2k = 1 + 9 = 10
⇒ k = 10/2 = 5
and
α = - k (8 - k) + 10
= - 5 (8 - 5) + 10
= - 5 (3) + 10
= - 15 + 10
= - 5
Thus, k = 5 and α = - 5
The document NCERT Solutions for Class 8 Maths Chapter 2 - Polynomials (Exercise 2.4) is a part of the Class 10 Course Class 10 Mathematics by VP Classes.
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## FAQs on NCERT Solutions for Class 8 Maths Chapter 2 - Polynomials (Exercise 2.4)
1. What are polynomials?
Ans. Polynomials are algebraic expressions that consist of variables, coefficients, and exponents. They are made up of terms, which are separated by addition or subtraction operators. The highest exponent in a polynomial determines its degree.
2. How can we classify polynomials based on their degrees?
Ans. Polynomials can be classified as constant (degree 0), linear (degree 1), quadratic (degree 2), cubic (degree 3), quartic (degree 4), and so on. The degree of a polynomial is determined by the highest exponent of the variable.
3. What is the Remainder Theorem?
Ans. The Remainder Theorem states that if a polynomial P(x) is divided by a linear polynomial (x - a), then the remainder obtained is equal to P(a). In other words, if we substitute the value of 'a' into the polynomial, the remainder will be the same as the result obtained by dividing the polynomial by (x - a).
4. How can we find the zeroes of a polynomial?
Ans. The zeroes of a polynomial are the values of 'x' for which the polynomial becomes zero. We can find the zeroes by equating the polynomial to zero and solving the equation. This can be done by factorizing the polynomial or by using methods like the Remainder Factor Theorem or the synthetic division method.
5. What is the Fundamental Theorem of Algebra?
Ans. The Fundamental Theorem of Algebra states that every polynomial equation of degree 'n' has exactly 'n' complex roots, considering multiple roots as separate roots. This theorem guarantees that any polynomial equation can be solved, either exactly or approximately, by finding its roots.
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The following definitions can help teachers and other users, such as community police officers, to identify and understand the following elements within the Mirror Image game: harassment, cyber stalking, criminal harassment, child pornography and child luring.
The Ontario Curriculum, Health and Physical Education, Grades 1-8, revised (2015)) defines harassment as a form of discrimination that may include unwelcome attention and remarks, jokes, threats, name-calling, touching, or other behaviour (including the display of pictures) that insults, offends, or demeans someone because of his or her identity. Harassment involves conduct or comments that are known to be, or should reasonably be known to be, offensive, inappropriate, intimidating, and hostile.
It includes any overt, subtle, verbal or written comments or any physical conduct which places pressure on, ridicules, degrades, or expresses hatred based on a person’s sex or sexual orientation, gender identity or gender expression, race, ethnicity, cultural background, place of birth, religion, citizenship or ancestry.
Some examples are:
- unwanted, unwelcome physical contact like touching, grabbing or patting;
- sexual gossip;
- obscene phone calls;
- rude jokes or suggestive remarks of a sexual nature;
- demeaning nicknames;
- catcalls, rating, or embarrassing whistles;
- threats, abuse, or assault;
- sexually insulting remarks, including those about sex, sexual orientation, gender identity or expression, race, ethnicity, culture, place of birth, religion, citizenship, ancestry, ability or class
It is not:
- a hug between friends;
- mutual flirtation;
- sincere and personal compliments.
These types of harassment can take place in person or over the Internet. Harassment of any kind is unacceptable both in and out of the school environment. Some forms of harassment, such as threats to physically harm someone and stalking, are also against the law.
Cyberstalking is the use of the Internet or other electronic means to repeatedly harass an individual, a group of individuals, or an organization in a menacing fashion. This could take the form of rude or threatening messages, slanderous information or repeated, unwanted messages.
How Cyberstalking Occurs
Cyberstalking occurs in a variety of ways. Often cyberstalking is an online component to offline stalking. Sometimes a cyberstalker fixates on a person he or she encountered online, including via public profiles or other regular Internet activity. It may be a particularly virulent and obsessive version of “trolling”, which is nasty online commentary that takes pleasure in making other users (or a particular user) uncomfortable or upset.
a. Anybody Could Be a Victim
The Internet offers a wide variety of social media sites and applications, hundreds of thousands of websites with personal profiles and photographs, online beauty contests, dating service websites, and sites dedicated to hobbies and sports. This wealth of information can be misused by people who cyberstalk. They can potentially amass a lot of information about the target of their unwanted attentions. Note that income, education, urban or rural setting makes no difference on who a cyberstalker will select.
b. Tracking the Victim
The Internet provides instant access to addresses, maps, telephone directories, and school websites. Many cyberstalkers create detailed lists of their victim’s activities each day of the week. This enhances the victims' fear that the cyberstalker will find them offline.
c. Disappearing without a Trace.
A cyberstalker whose identity is known can be cautioned by a restraining order or a visit from police. But a cyberstalker who uses the Internet only to mask his or her identity can be difficult to trace.
Cyberstalking and the Law
The Criminal Code of Canada makes reference to stalking as criminal harassment. It states that no person shall engage in repeated conduct (such as following, stalking, engaging in unwanted direct or indirect communication, or engaging in threatening behaviour) that causes the victim to fear for their safety or the safety of people known to them. These actions are all punishable by law. The victim’s fear for their safety must be reasonable. Most forms of online harassment do not qualify as criminal harassment.
When online harassment rises to the level of cyberstalking, children must confide in their parents/guardians, a teacher or a police officer. Adults are best situated to take the next step, which may be reporting the behaviour to their Internet Service Provider, the school board, the local police or the RCMP.
The production, distribution and possession of child pornography via the Internet causes a range of harm to children. Most child pornography is a recording, in photos or videos, of a child’s offline sexual abuse. The most direct harm is suffered by the child pictured in the image, as the sexual use of that image is a repeated and grave violation of that child’s rights. Most directly it repeatedly subjects the child pictured in the image to sexual abuse, violating his or her rights to bodily integrity, human dignity and privacy.
Child pornography may also serve as a tool in the arsenal of sexual predators who are engaged in the business of "grooming" and "luring" child victims. Here, examples of child pornography may be used by the offender to assure victims that sexual activity by children, including sexual activity with adults serve as proof to victims that children engaging in sexual acts is "normal." Predators are also known to solicit photos from children and youth, which also qualify as child pornography if they feature the child’s sexual organs. These photos may then become an irretrievable part of an international library of child pornography.
Indirectly, the widespread presence of child pornography on the Internet may fuel the fantasies of people with a sexual interest in children. Online communities of people who collect child pornography affirm and encourage each other’s sexual interests. They often swap information about how to evade detection, and even how to groom a child in order to produce more child abuse images.
How Child Pornography Occurs
The creation of child pornography has always been primarily at the hands of trusted caregivers, like family members and family friends. This continues to be the case in the internet age, although advances in and distribution of child pornography, once a backroom industry, has been radically transformed by computer and digital technology, such as smart phones and webcams. These advances have made the production and distribution of child pornography ever easier and even harder to detect. The Internet has facilitated afforded easy means to distribute and access child pornography, and people engaged in these practices have found ever more sophisticated means of hiding their activities online.
Children who are sexually abused rarely come forward on their own, and evidence indicates that children who know that images were taken of their abuse are even less likely to disclose the abuse to a trusted adult. Victims of child pornography need non-offending family members, other trusted adults in the child’s life (e.g. teachers), and police to notice signs of abuse and to create a safe environment for children to disclose.
A lot of public attention has focused on sexual images of young people that they have produced themselves – often referred to a “sexting”. Often the context for the creation of such images is different from core child sexual abuse images, because the youth consented (and was old enough to consent) to the sexual activity pictured. Naked images of a person under 18 could be considered child pornography, even though they are not engaged in sexual activity. However, these images, once distributed, can also enter into the vast international marketplace for child pornography, and may suffer similar emotional and psychological consequences as victims of child pornography.
Child Pornography and the Law
The Criminal Code of Canada defines Child Pornography as:
A photographic film, video or other visual representation, whether or not it was made by electronic or mechanical means that shows a person who is or is depicted as being under the age of 18 years and is engaged in or is depicted as engaged in explicit sexual activity or the dominant characteristic of which is the depiction of a sexual organ or anal region of an underage person.
Child luring is an illegal act whereby someone communicates with a child on the Internet for the purpose of committing a sexual offense against that child. Many of the sexual offences related to luring are connected to the age of consent, which prohibits adults from having sexual relations with children under 16. In certain instances however, such as crimes related to child pornography and other forms of sexual exploitation, the age of consent increases to include any child or youth under 18 years of age. Since 2002, the Criminal Code of Canada has criminalized child luring.
Children experiencing conflict in personal relationships may be at increased risk to cyber luring. They may look online for what is missing in their own lives or as an escape from their real life situation.
The number of police-reported sexual violations against children rose 30% from 2012 to 20131, and rose again in 2014, representing one of the few categories of violent violations to increase from the previous year. In 2014 there were approximately 4,500 police-reported sexual violations against children, about 300 more than in 2013, resulting in an increase of 6%2.
The increase in sexual violations against children was primarily the result of incidents of luring a child via a computer (including the agreement or arrangement to commit a sexual offence against a child), which increased from 850 incidents in 2013 to 1,190 incidents in 2014.
Various factors could account for the increase in sexual violations against children, such as specialized units within a police service to proactively investigate this type of crime.
How Child Luring Occurs
Child luring may occur over a long period of time, or it may occur fairly quickly, depending on the individual victim's susceptibility to the advances of the offender. Over the course of any Internet friendship, familiarity, trust and affection develop between the parties; the predator luring offenders take advantage of this trust and affection. Some predators are willing to travel thousands of miles and cross international borders to connect with their online victims. Many studies show, however, that most predators reside within 100 kilometers of the victim's home.
It is common for a predator to spend time quietly observing the dialogue in youth chat rooms. Such anonymous "spying" offers a means of identifying a vulnerable child and staying in synch with a chat room's dynamic. They stay current on issues, trends and cultural references that are important to their target age group. It is this knowledge that makes it easy to join in the conversation.
Once a conversation has been initiated, thepredator will devote a great deal of time and energy to establishing "trust" and a "friendship" with the target child. At the outset, conversations may appear normal. The predator, however, soon employs strategies to exploit the vulnerabilities of youth. They will demonstrate a "genuine" interest in the child and will go to great lengths to flatter the child and convey understanding for all aspects of his or her life. This attention leads to a "friendship" where confidences and secrets are shared.
Predators soon lure their victims into increasingly intimate conversations. They may send photos and then soft porn leading to more and more sexual conversations. Sexual content can also occur relatively early in the interaction: research indicates that most victims of child luring know that the person they are conversing with is an adult and also know that he is interested in sex. The victim may think that he or she is in a romantic relationship with the offender. At this point the predator might manoeuvre the child into meeting with him. Youth suffering from a lack of intimacy and those that have needs for friendship are the most vulnerable. Predators know that these troubled adolescents are looking for self-validation and companionship; these children are vulnerable as they lack the protective networks to safeguard them.
Since 2002, Cybertip.ca, Canada’s national online child protection website, noted that luring accounted for 10% of their reports, making it the second-largest category of complaint. The majority of these incidents involved luring adolescent girls.
Cyberluring and the Law
Canadian legislation prohibits the luring of children. Since 2002, the Criminal Code of Canada has criminalized electronic communication with a person believed to be a child for the purpose of facilitating the commission of sexual offences. Internet luring of children is punishable by law. An offline meeting, or even an attempt to arrange an offline meeting, is not required for this offence: the conversation “facilitating” the future commission of a sexual offence is sufficient.
1Statistics Canada, Police-reported crime statistics in Canada, 2013 Retrieved from http://www.statcan.gc.ca/pub/85-002-x/2014001/article/14040-eng.htm
2Statistics Canada, Police-reported crime statistics in Canada, 2014. Retrieved from http://www.statcan.gc.ca/pub/85-002-x/2015001/article/14211-eng.htm<|endoftext|>
| 3.90625 |
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Question 1: Calculate the distance between the points $(6, -4)$ and $(3,2)$ correct to $2$ decimal places.
Answer:
$(6, -4)$ and $(3, 2)$
Distance $= \sqrt{(3-6)^2+(2-(-4))^2} = \sqrt{9+36} = \sqrt{45} = 6.71$
$\\$
Question 2: Find the distance between the points $(-2, -2)$ and $(1,0)$ correct to $3$ significant figures.
Answer:
$(-2, -2)$ and $(1,0)$
Distance $= \sqrt{(1-(-2))^2+(0-(-2))^2} = \sqrt{9+4} = \sqrt{13} =3.61$
$\\$
Question 3: Show that the points $P(7, 3), Q(6, 3 + \sqrt{3}) \ and \ R (5, 3)$ form an equilateral triangle.
Answer:
$P(7, 3), Q(6, 3 + \sqrt{3}) \ and \ R (5, 3)$
$PQ = \sqrt{(6-7)^2+(3 + \sqrt{3}-3)^2} = \sqrt{1+3} = \sqrt{4} = 2$
$PR = \sqrt{(5-7)^2+(3-3)^2} = \sqrt{4+0} = \sqrt{4} = 2$
$QR = \sqrt{(5-6)^2+(3-(3 + \sqrt{3})^2} = \sqrt{1+3} = \sqrt{4} = 2$
Therefore three sides $PQ, PR \ and \ QR$ are equal which makes it an equilateral triangle.
$\\$
Question 4: The circle with center $(x, y)$ passes though the points $(3, 11), (14, 0) \ and \ (12, 8)$ . Find the values of $x \ and \ y$.
Answer:
Distance of the points from the center are equal. Therefore
$\sqrt{(3-x)^2+(11-y)^2}=\sqrt{(12-x)^2+(8-y)^2}$
$9x^2-6x+121+y^2-22y=144+x^2-24x+64+y^2-16y$
$18x-6y=78$
$3x-y=13$ … … … … i)
$\sqrt{(3-x)^2+(11-y)^2}=\sqrt{(14-x)^2+(0-y)^2}$
$9x^2-6x+121+y^2-22y=196+x^2-28x+y^2$
$22x-22y=66$
$x-y=3$… … … … ii)
Solving i) and ii), we get $x = 5$ and $y = 2$.
Therefore the center is $(5, 2)$
$\\$
Question 5: The points $A(-1, 2), B(x, y) \ and \ C = (4, 5)$ are such that $BA = BC$ . Find a linear relation between $x \ and \ y$ .
Answer:
$BA=BC$
$\sqrt{(x-(-1))^2+(y-2)^2}=\sqrt{(4-x)^2+(5-y)^2}$
$x^2+1+2x+y^2+4-4y=16+x^2-8x+25+y^2-10y$
$5+2x-4y=41-8x-10y$
$10x+6y=36$
$5x+3y=18$
$\\$
Question 6: Given a triangle $ABC$ in which $A = (4, 4), B = (0, 5) and C = (5, 10)$ . A point $P$ lies on $BC$ such that $BP : PC = 3 : 2$ . Find the length of line segment $AP$ .
Answer:
A point $P (x,y)$ lies on $BC$ such that $BP : PC = 3 : 2$ .
$x =$ $\frac{2 \times (0)+3 \times (5)}{2+3}$ $= 3$
$y =$ $\frac{2 \times (5)+3 \times (10)}{2+3}$ $= 8$
Therefore $P = (3, 8)$
Therefore
Length of $AP = \sqrt{(3-4)^2+(8-(-4))^2} = \sqrt{1+144} = \sqrt{145}$
$\\$
Question 7: $A(20, 0) \ and \ B(10, -20)$ are two fixed points. Find the co-ordinates of the point $P$ in $AB$ such that: $3PB = AB$ . Also, find the co-ordinates of some other point $Q \ in \ AB$ such that $AB=6AQ$ .
Answer:
For P When Ratio: $m_1:m_2 = 2:1$ $A(20, 0) \ and \ B(10, -20)$
Therefore
$x =$ $\frac{2 \times (10)+1 \times (20)}{2+1}$ $=$ $\frac{40}{3}$
$y =$ $\frac{2 \times (-20)+1 \times (0)}{1+2}$ $=$ $\frac{-40}{3}$
Therefore the point $P= ($ $\frac{40}{3}$ $,$ $\frac{-40}{3}$ $)$
For Q When Ratio: $m_1:m_2 = 1:5$ $A(20, 0) \ and \ B(10, -20)$
Therefore
$x =$ $\frac{1 \times (10)+5 \times (20)}{1+5}$ $=$ $\frac{55}{3}$
$y =$ $\frac{1 \times (-20)+5 \times (0)}{1+5}$ $=$ $\frac{-10}{3}$
Therefore the point $P= ($ $\frac{55}{3}$ $,$ $\frac{-10}{3}$ $)$
$\\$
Question 8: $A(-8, 0), B(0, 16) \ and \ C(0, 0)$ are the vertices of a triangle $ABC$ . Point $P$ lies on $AB \ and \ Q$ lies on $AC$ such that $AP: PB = 3: 5 \ and \ AQ: QC = 3:5$ . Show that: $PQ =$ $\frac{3}{8}$ $BC$ .
Answer:
For $P(x_1,y_1)$ When Ratio: $m_1:m_2 = 3:5$ $A(-8, 0) \ and \ B(0, 16)$
Therefore
$x_1 =$ $\frac{3 \times (0)+5 \times (-8)}{3+5}$ $= -5$
$y_1 =$ $\frac{3 \times (16)+5 \times (0)}{3+5}$ $=6$
Therefore the point $P= (-5,6)$
For $Q(x_2,y_2)$ When Ratio: $m_1:m_2 = 3:5$ $A(-8, 0) \ and \ C(0, 0)$
Therefore
$x_2 =$ $\frac{3 \times (0)+5 \times (-8)}{3+5}$ $= -5$
$y_2 =$ $\frac{3 \times (0)+5 \times (0)}{3+5}$ $=0$
Therefore the point $Q= (-5,0)$
$PQ = \sqrt{(-5-(-5))^2+(0-6)^2} = \sqrt{0+36} = 6$
$BC = \sqrt{(0-0)^2+(16-0)^2} = \sqrt{256} = 16$
Therefore $PQ =$ $\frac{3}{8}$ $\times 16=6$
Which proves that $PQ =$ $\frac{3}{8}$ $BC$
$\\$
Question 9: Find the co-ordinates of points of trisection of the line segment joining the point $(6, 9)$ and the origin.
Answer:
Let $P(x_1,y_1) \ and \ Q(x_2,y_2)$ be the two points dividing the points $(6, 9)$ and the origin in the ratio 1:2 and 2:1 respectively.
Therefore for $P$
$x_1 =$ $\frac{1 \times (0)+2 \times (6)}{1+2}$ $= 4$
$y_1 =$ $\frac{1 \times (0)+2 \times (-9)}{1+2}$ $=-6$
Hence $P(4, -6)$
Therefore for $Q$
$x_1 =$ $\frac{2 \times (0)+1 \times (6)}{2+1}$ $= 2$
$y_1 =$ $\frac{2 \times (0)+1 \times (-9)}{2+1}$ $=-3$
Hence $Q(2, -3)$
$\\$
Question 10: A line segment joining $A(-1, \frac{5}{3}) \ and \ B(a, 5)$ is divided in the ratio $1 : 3 \ at \ P$ , the point where the line segment $AB$ intersects the $y-axis$ .
(i) Calculate the value of $a$
(ii) Calculate the co-ordinates of $P$. [1994]
Answer:
Therefore for $P (0,y)$
$0 =$ $\frac{1 \times (a)+3 \times (-1)}{1+3}$ $\Rightarrow a = 3$
$y =$ $\frac{1 \times (5)+3 \times (\frac{5}{3})}{1+3}$ $=$ $\frac{5}{2}$
Hence $P(0,$ $\frac{5}{2}$ $)$
$\\$
Question 11: In what ratio is the line joining $A(0, 3) \ and \ B (4, -1)$ divided by the $x-axis$ ? Write the co-ordinates of the point where $AB$ intersects the $x-axis$ . [1993]
Answer:
Let the required ratio be $k:1$ and the point of $x-axis$ be $(x,0)$
Since $y = \frac{ky_2+y_1}{k+1}$
$\Rightarrow 0 =$ $\frac{k \times (-1) +3}{k+1}$
$\Rightarrow k=3$
$\Rightarrow m_1:m_2 = 3:1$
Therefore $x =$ $\frac{3(4)+1(0)}{3+1}$ $= 3$
Therefore $P(3,0)$
$\\$
Question 12: The mid-point of the segment $AB$ , as shown in diagram, is $C(4, -3)$ . Write down the coordinates of $A \ and \ B$ . [1996]
Answer:
Given Midpoint of $AB = (4,-3)$
Therefore
$4 =$ $\frac{1 \times x+1 \times (0)}{1+1}$ $\Rightarrow x = 8$
$-3 =$ $\frac{1 \times (0)+1 \times y}{1+1}$ $\Rightarrow y = -6$
Therefore $A = (8, 0) \ and \ B(0, -6)$
$\\$
Question 13: $AB$ is a diameter of a circle with center $C = (-2, 5)$ . If $A = (3, -7)$ , find
(i) the length of radius $AC$
(ii) the coordinates of $B$ . [2013]
Answer:
Given Midpoint of $AB = C(-2,5)$
Therefore
$-2 =$ $\frac{1 \times x+1 \times (3)}{1+1}$ $\Rightarrow x = -7$
$5 =$ $\frac{1 \times (y)+1 \times (-7)}{1+1}$ $\Rightarrow y = 17$
Therefore $B = (-7, 17)$
$AC = \sqrt{(-7-3)^2+(17-(-7))^2} = \sqrt{676} = 26$
$\\$
Question 14: Find the co-ordinates of the centroid of a triangle $ABC$ whose vertices are : $A(-1, 3), B(1, -1) \ and \ C(5, 1)$ . [2006]
Answer:
Let $O(x, y)$ be the centroid of triangle $ABC$.
Therefore
$x=$ $\frac{-1+1+5}{3}$ $=$ $\frac{5}{3}$
$y =$ $\frac{3-1+1}{3}=1$
Hence the coordinates of the centroid are $($ $\frac{5}{3}$ $, 1)$
$\\$
Question 15: The mid-point of the line segment joining $(4a, 2b-3) \ and \ (-4, 3b) \ is \ (2, -2a)$ . Find the values of $a \ and \ b$ .
Answer:
Given Midpoint of $= C(2, -2a)$
Therefore
$2 =$ $\frac{1 \times (4a) + 1 \times (-4)}{1+1}$ $\Rightarrow a = 2$
$-2(2) =$ $\frac{1 \times (2b-3)+1 \times (3b)}{1+1}$ $\Rightarrow b = -1$
$\\$
Question 16: The mid-point of the line segment joining $(2a, 4) \ and \ (-2, 2b) \ is \ (1, 2a+1)$ . Find the values of $a \ and \ b$ . [2007]
Answer:
Given Midpoint of $= C(1, 2a+1)$
Therefore
$1 =$ $\frac{1 \times (2a) + 1 \times (-2)}{1+1}$ $\Rightarrow a = 2$
$2(2)+1 =$ $\frac{1 \times (4)+1 \times (2b)}{1+1}$ $\Rightarrow b = 3$
$\\$
Question 17: (i) Write down the co-ordinates of the point $P$ that divides the line joining $A(- 4, l) \ and \ B(17, 10)$ in the ratio $1 : 2$ .
(ii) Calculate the distance $OP$ , where $O$ is the origin.
(iii) In what ratio does the $y-axis$ divide the line $AB$ ? [1995]
Answer:
i) For P When Ratio: $m_1:m_2 = 1:2$ $A(- 4, l) \ and \ B(17, 10)$
Therefore
$x =$ $\frac{1 \times (17)+2 \times (-4)}{1+2}$ $= 3$
$y =$ $\frac{1 \times (10)+2 \times (1)}{1+2}$ $= 4$
Therefore the point $P= (3, 4)$
ii) $OP = \sqrt{(3-0)^2+(4-0)^2} = \sqrt{25} = 5$
iii) Let the required ratio be $k:1$ and the point be $Q(0,y)$
Since $y =$ $\frac{ky_2+y_1}{k+1}$
$\Rightarrow 0 =$ $\frac{k \times (17) -4}{k+1}$
$\Rightarrow k=$ $\frac{4}{17}$
$\Rightarrow m_1:m_2 = 4:17$
$\\$
Question 18: Prove that the points $A(-5,4); B(-1, -2) \ and \ C(5, 2)$ are the vertices of an isosceles right-angled triangle. Find the co-ordinates of $D$ so that $ABCD$ is a square. [1992]
Answer:
$AC = \sqrt{(5-(-5))^2+(2-4)^2} = \sqrt{104}$
$AB= \sqrt{(-1-(-5))^2+(-2-(-4))^2} = \sqrt{52}$
$BC = \sqrt{(-1-(-5))^2+(-2-2)^2} = \sqrt{52}$
Since $AB=BC$ (two sides are equal). Hence triangle $ABC$ is a isosceles triangle.
$\\$
Question 19: $M$ is the mid-point of the line segment joining the points $A(-3, 7) \ and \ B(9, -1)$ . Find the coordinates of point $M$ . Further, if $R(2, 2)$ divides the line segment joining $M$ and the origin in the ratio $p : q$ , find the ratio $p : q$ .
Answer:
For M When Ratio: $m_1:m_2 = 1:1$ for $A(-3, 7) \ and \ B(9, -1)$
Therefore
$x =$ $\frac{1 \times (9)+1 \times (-3)}{1+1}$ $= 3$
$y =$ $\frac{1 \times (-1)+1 \times (7)}{1+1}$ $= 3$
Therefore the point $M= (3, 3)$
Let $R(2,2)$ divide MO in the ratio $k:1$
Since $y =$ $\frac{ky_2+y_1}{k+1}$
$\Rightarrow 0 =$ $\frac{k \times (0) +3}{k+1}$
$\Rightarrow k=$ $\frac{1}{2}$
$\Rightarrow p:q=1:2$
$\\$
Question 20: Calculate the ratio in which the line joining $A(-4, 2) \ and \ B(3, 6)$ is divided by point $P(x, 3)$ . Also, find (i) $x$ (ii) length of $AP$ . [2014]
Answer:
Let $P(x,3)$ divide MO in the ratio $k:1$
Since $y =$ $\frac{ky_2+y_1}{k+1}$
$\Rightarrow 3 =$ $\frac{k \times (6) +2}{k+1}$
$\Rightarrow k=$ $\frac{1}{3}$
$\Rightarrow m_1:m_2=1:3$
Since $x =$ $\frac{kx_2+x_1}{k+1}$
$\Rightarrow x =$ $\frac{1 \times (3) +3 \times (-4)}{1+3}$ $=$ $\frac{-9}{4}$
$AP =$ $\sqrt{(-\frac{9}{4}-(-4))^2+(3-2)^2}$ $=$ $\sqrt{(\frac{7}{4})^2+1)}$ $= \sqrt{\frac{65}{16}}$
$\\$<|endoftext|>
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Mercury's elliptical orbit
Mercury has the largest orbital eccentricity of all the planets in our solar system, at its greatest distance from the sun it is 1.5 times further than at its closest distance.
That distance ratio has been recreated using an offset gear to act as a cam, moving the planet horizontally along a spring loaded slide.
Mercury's Day Night Cycle
Mercury has the second longest solar day of any planet in the solar system at 176 Earth days.
The cam is geared to precisely imitate Mercury's' long day night cycle.
The Retrograde Rotation of Venus
Venus rotates in a clockwise direction when viewed from its north pole and is the only planet in our solar system to do so.
This unique movement is replicated by use of a gear train situated on the arm carrying Venus.
Venus' Day Night Cycle
Venus has the longest solar day of any planet in the solar system at 243 days.
The compound gear train on the arm replicates this speed relative to the movement of the Orrery.
Phobos & Deimos
Phobos and Deimos orbit the red planet Mars, represented on this Orrery by a copper ball, every 7.66 hours and 30.35 hours respectively.
Those speed are to great to replicate on an Orrery that moves five Earth days per turn of the handle and so they have been slowed. Their pace is kept relative to one another.<|endoftext|>
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If you're seeing this message, it means we're having trouble loading external resources on our website.
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### Course: Arithmetic (all content)>Unit 5
Lesson 21: Multiplying fractions
# Multiplying fractions review
Review the basics of multiplying fractions, and try some practice problems.
## Multiplying fractions
To multiply fractions, we multiply the numerators and then multiply the denominators.
Example 1: Fractions
$\phantom{=}\frac{5}{6}×\frac{5}{7}$
$=\frac{5×5}{6×7}$
$=\frac{25}{42}$
Example 2: Mixed numbers
Before multiplying, we need to write the mixed numbers as improper fractions.
$2\frac{2}{3}×1\frac{3}{5}$
$\phantom{\rule{2em}{0ex}}$
$=\frac{8×8}{3×5}$
$=\frac{64}{15}$
We can also write this as $4\frac{4}{15}$ .
### Cross-reducing
Cross-reducing is a way to simplify before we multiply. This can save us from dealing with large numbers in our product.
Example
$\phantom{=}\frac{3}{10}×\frac{1}{6}$
$=\frac{3×1}{10×6}$
$=\frac{1}{20}$
Prefer a visual understanding of fraction multiplication? Check out one of these videos:
Multiplying 2 fractions: fraction model
Multiplying 2 fractions: number line
## Practice
Problem 1
$\frac{5}{8}×\frac{7}{8}$
Want to try more problems like this? Check out these exercises:
Multiply fractions
Multiply mixed numbers
## Want to join the conversation?
• How do you do Cross redusucing?
• how do you multiply negative
fractions
• Multiply the fractions normally, and ensure that the product has the correct sign.
1/2 * 1/2 = 1/4 (Two positives equal a positive.)
-1/2 * -1/2 = 1/4 (Two negatives equal a positive.)
-1/3 * 1/3 = -1/9 (A negative times a positive produces a negative .)
1/3 * -1/3 = -1/9 (The product will be negative, whether the negative number comes first or second.)
• Why do you always have to simplify?
• So the fraction reduced to lowest terms. When we reduce or simplify the fraction, we are writing a fraction in an equivalent form that may be easier to work with.
• What is GCD? thanks to whoever answers it
• greatest common denominator= GCD
• What is the difference between LCD and LCM.
• The mathematical approach to finding the LCM and LCD is the same. For both, we need to find the least common multiple of two or more numbers. The least common denominator (LCD) is actually the least common multiple (LCM) of the denominators.
(1 vote)
• How do you multiply the single number with the mixed number?
• You multiply a single number by a mixed number by putting a 1 under the single number.For example 2 + 3 1/2 your new equation would be 2/1 + 3 1/2. This is because 2/1 is the SAME exact thing a 2.<|endoftext|>
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Chemical Concepts Demonstrated: Solubility of gases in water, pH changes in water with gas dissolved in it
The water rushes up to the top flask and the solution turns pink.
The first few drops of water that enter the upper flask absorb a little ammonia gas. When this occurs, a slight vacuum that forms sucks a little more water into the flask to balance the pressure. This water, in turn, absorbs a little more ammonia. This continuous exchange creates a chain reaction that causes the solution to come rushing into the top flask.
The solution turns pink because of the basic activation of the indicator in the water.
As an alternative, HCl gas could be used (bromothymol blue or methyl violet replace the phenolphthalein). SO2 and Cl2 gases could also be used to create acidic solutions.<|endoftext|>
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Homework Help
# The quadrilateral ABCD, where A is (4,5) and C is (3,-2), is a square. Find the...
Student
Honors
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The quadrilateral ABCD, where A is (4,5) and C is (3,-2), is a square. Find the coordinates of B and D and hence or otherwise the area of the square.
Posted by nmeisu on February 10, 2013 at 1:43 AM via web and tagged with coordinate geometry, geometry, math
High School Teacher
(Level 1) Educator Emeritus
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Given a square ABCD with the coordinates of A(4,5) and the coordinates of C(3,-2) find the area of the square:
The length of the diagonal of a square is `sqrt(2)` times the length of a side.
Using the distance formula we find `AC=sqrt((4-3)^2+(5-(-2))^2)`
so `AC=sqrt(50)=5sqrt(2)` .
The length of a side of the square is 5 so the area of the square is 25 square units.
--------------------------------------------------------------------
If you really want to know the coordinates of B and D:
`bar(BD)` is the perpendicular bisector of `bar(AC)` .
(a) Since it bisects `bar(AC)` we find the midpoint to be `((4+3)/2,(5-2)/2)=(3.5,3.5)`
(b) The slope of `bar(AC)` is `m=(5-(-2))/(4-3)=7` so the slope of the perpendicular bisector is `m=-1/7`
(c) The equation of the perpendicular bisector is `y-3.5=-1/7(x-3.5)` or `y=-1/7x+4`
(d) B and D will be the intersections of the line `bar(BD)` with a circle centered at (4,5) (or (3,-2) if you prefer) with a radius of 5.
The line: `y=-1/7x+4` so `-7y=x-28==>x=-7y+28`
The circle: `(x-4)^2+(y-5)^2=25` substituting for x we get:
`(-7y+24)^2+(y-5)^2=25`
`49y^2-336y+576+y^2-10y+25=25`
`50y^2-346y+576=0`
So `y=(173+-sqrt(1129))/50`
Substituting for y to get x the approximate coordinates for B are `(8.484,2.788)` and for D `(-0.924,4.132)` if teh vertices are read clockwise.
Posted by embizze on February 10, 2013 at 2:47 AM (Answer #1)<|endoftext|>
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Traditionally, the vast majority of the world's electricity supply has been produced by raising steam through burning of fossil fuels such as coal, oil and gas and using this steam to drive a turbine generator to produce electricity.
This method currently accounts for roughly half of the world's electricity supply with the remainder provided by nuclear, renewable and other fossil fuel-fired supplies, such as diesel engine and gas turbine based power plants.
Recent deregulation and privatisation of electricity generation industries throughout the world has also led to a new breed of producer, ranging from private independent power producers, to larger integrated energy suppliers.
Many of the 'new' producers have looked to the gas turbine as a source of electrical power since it offers many advantages over the traditional fossil fuel fired power plant, namely relatively low equipment cost and lead time, high fuel to energy efficiency and low emissions. As a result, such equipment has been and continues to be developed by the manufacturers to meet customer demand for larger, more efficient machines.
At about the same time, many of the world's boiler manufacturers were working in conjunction with the turbine manufacturers to develop boilers to recover waste heat from the turbine exhaust gases. This heat could then be used to raise steam that might then be used to drive a steam turbine generator, further increasing fuel-to-energy efficiency to over 50%. This made a highly efficient power generation plant when compared to conventional coal fired stations with energy efficiencies of 35–38%.
With a gas turbine based power plant becoming a viable alternative to provide base load electricity, power generation companies began placing orders for such equipment, based principally on their benefits over conventional fuel-fired power plants.
Advanced technologies have been developed or introduced from the aircraft industry to allow such demands to be met. Though it is clear that there are distinct advantages to using a gas turbine based plant, such equipment is not without its problems. All of the leading equipment manufacturers have suffered from various failures as technological boundaries are pushed to meet new demands and these have led the insurance industry to be wary of recently developed and 'improved' machines.
Despite all the benefits provided by the use of gas turbines and the increasing normalisation of this leading edge technology achieved through numerous technical developments, the machinery breakdown exposure and ensuing business interruption impact are still causing significant concerns to both electricity generators and underwriters.
The process is a combination of Brayton Cycle of the Gas Turbine and Rankine Cycle of the Steam Turbine. The Gas Turbines are typically consists of operation of gas turbines that are fired by natural gas/naphtha. The gas turbines convert thermal energy of the fuel air mixture to mechanical energy to drive a generator, which in turn generates electricity. A diverting damper usually leads the hot combustible gases from the gas turbine after expansion in the turbine to the HRSG or boiler.
The Gas Turbine system consists of a combustion chamber where the fuel air mixture is burned. The inlet air is filtered and admitted through the air filter banks. The fuel is fed by the fuel system and atomised using air. Atomising air is supplied from auxiliary air compressors. Supplying purge air does the removal of superfluous fuel from the lines downstream of the fuel control valves.
The next important system in the Gas Turbine is the lubricating oil system. Since the Gas Turbine operates at very high speeds and high temperatures it is very important to have an efficient lubrication system. In many plants, the lube oil is stored in a reservoir and is pumped by means of a main shaft driven pump or auxiliary emergency pump. The lube oil is sent to bearing header, gears, generator and hydraulic supply system. The lubricant is cooled down using heat exchangers and then filtered. The lube oil system is fitted with a trip system whereby in case of reduction in oil pressure, fuel stop valves are operated to stop supply of fuel to gas turbines.
Starting system of the gas turbine provides necessary cranking and turning power for start up of the turbine. It is necessary that the GT is operated at about 100 to 200 RPM prior to star up and also during the cooling period after unit shut down.
Other systems forming part of the gas turbine includes the hydraulic supply system, cooling air system, compressor water wash system and leak detection system.
Heat Recovery Steam Generator
The function of the HRSG is to utilise the waste heat of the gas turbine exhaust gases for generation of high-pressure steam. In the absence of HRSG, the exhaust gases would be discharged to the atmosphere and thermal energy which otherwise could have been utilised is wasted. By using HRSG, this energy is harnessed and used to generate additional electricity which increases the unit efficiency. The water for steam generation is supplied from the DM Plant. This high pressure and high temperature steam is then used to drive the steam turbine. Exhaust steam from the turbine is then condensed in the water-cooled condenser and again fed to the boiler. Thus the cycle is a closed cycle.
The steam turbine converts the thermal energy of steam into mechanical energy, which in turn, is used to drive the Generator and produce electrical energy. Steam Turbine consists of two sections viz., high pressure and low pressure. HP steam is supplied from the common main steam header from all the three HRSG through two stop and control valves and LP steam is supplied from the common LP main steam header through one stop and control valve. Turning gear arrangement is provided to keep the steam turbine rotor always turning during the shut down period. The Steam Turbine is also provided with various systems like lube oil system, hydraulic oil system and various other control systems.
Balance of Plant
Power evacuation is done using transformers. These transformers step up the voltage from 11KV to 220 KV for transmission to the grid. Other auxiliaries used by the plant involve Raw Water System, which supplies water to the Water Pre Treatment Plant. In the pre treatment plant the water is softened prior to sending it to the condensor cooling. In the Demineralised Water Plant or DM plant, the water is treated for removing dissolved contaminants such as salts and silica and send to feed water supply to the HRSG. In the pre-treatment plants chemicals are used to improve the water condition. Apart from the above, the plant uses cooling towers for removal of heat from cooling water, heating, ventilating and air conditioning system and material handling system for loading unloading and storage for maintenance.
Within the combustion chamber the compressed air is mixed with vaporised fuel and the mixture is burned. This creates products of combustion that are at a higher temperature than the compressed air and is used to do more work than the energy used in compressing the air.
The hot, high pressure products of combustion are passed to the turbine where they are allowed to expand through several rows of alternate stationary vanes and rotating blades. Each vane/blade set is known as a turbine stage, and as the mixture accelerates past each stage, the kinetic energy within the expanding gas is converted into rotational energy using the rotor blades.
Several different types of fuel may be used to fire gas turbines. Natural gas is the most commonly used fuel though there are facilities that use other gases such as synthesis gas (syngas), a combination of carbon monoxide and hydrogen formed by gasification of residues, or other solid organic materials or liquids such as light fuel oil. Some gas turbine installations that are designed to burn syngas feature an integrated fuel gasification system. These generally use turbine exhaust gases to raise steam which may then be used within the fuel gasification plant, and are generally found within refineries or other facilities with large amounts of organic waste materials. It should be noted that turbines are specifically designed for the fuels they are to burn and as such a gas turbine that is designed to fire natural gas will not operate effectively with a syngas fuel stream and vice-versa. This is primarily due to differences in the calorific values of the fuels, meaning that different quantities of each fuel would be required to achieve the same output. Gas turbines designed to burn more than one fuel are normally optimised for the main fuel, with a trade-off in performance on the reserve fuel supply. This allows for optimum reliability in critical applications with questionable or problematic fuel supplies and many designs allow for on-load change-over of the fuel supply.
The modern plants are operated with most modern DCS systems which send out various alarms in case of equipment mal-function and also auto shut down in case of emergency. However, optimum shaft vibration monitoring is critical. Correct operation of these machines is an important issue. Many problems have been caused through poor procedures leading to failure in communications and important steps of operations being skipped. Cooling air valves have been left in the incorrect position after maintenance activities, lubrication systems have been left out of service prior to start up and pre-start checks have been carried out incorrectly. All of these issues may in themselves only seem minor failings, but it only takes moments of running a high speed turbine with little or no lubrication to cause extensive damage to bearing surfaces and overheating of equipment components.
Use of correct components and properly designed spare parts are of equal importance during maintenance activities and manufacturer's guidelines should be properly adhered to. Improperly controlled maintenance activities can also lead to problems. Bolts can be left untightened, tools can be left in the machine and auxiliary systems may not be recommissioned correctly. This should be monitored properly. Debris or foreign objects left following maintenance activities, or loose components should be clear and not get drawn into the turbine causing damages. Gas turbine internals are relatively fragile. Cleanliness is critical to prevent damage to machine blading, burners and other internal areas, as one small foreign particle can have disastrous consequences.
Quality assurance of component parts and materials is extremely important as gas turbines operate at high speed with high operating temperatures and pressures, and low tolerances between blades and veins. Failure of a relatively minor component within the machine can cause extensive damage.
Use of proper fuel is very critical. Fuel quality is of importance as rogue chemicals can cause deposits, erosion or corrosion of machine internals leading to long term damage. Fuel pulsations as a result of varying fuel quality or irregular supply systems can cause vibration in combustion systems and turbine areas leading to mechanical damage that is exacerbated as it is exposed to high temperatures and further operation. Extremely fine particles in fuel can have an effect similar to a sand blaster on turbine blading or may become embedded within or stuck to blade surfaces leading to a build up of material and subsequent machine imbalances.
The information set out in this document constitutes a set of general guidelines and should not be construed or relied upon as specialist advice. Independent legal advice should always be sought. Therefore Risktechnik accepts no responsibility towards any person relying upon these Risk Management Guides nor any liability whatsoever for the accuracy of data supplied by another party or the consequences of reliance upon it.
© Copyright 2010 All Rights Reserved Risktechnik.com<|endoftext|>
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Thread: Product + Sum of the Roots
1. Product + Sum of the Roots
Hey guys
Solve the equation$\displaystyle 8x^4-2x^3-27x^2+6x+9=0$ given that the sum of two of its roots is zero.
I have no idea how to solve this. I have never dealt with a quartic before.
2. I admit this is not an easy problem.
Let $\displaystyle f(x) = 8x^4-2x^3-27x^2+6x+9$
Suppose $\displaystyle \alpha,\beta$ are roots such that $\displaystyle \alpha+\beta = 0$. This means $\displaystyle \beta = -\alpha$.
Thus we can write $\displaystyle f(x) = q(x)(x-\alpha)(x+\alpha) = q(x)(x^2-\alpha^2)$.
If we let $\displaystyle q(x) = ax^2+bx+c$.
we have $\displaystyle 8x^4-2x^3-27x^2+6x+9 = (ax^2+bx+c)(x^2-\alpha^2)\Rightarrow a = 8, b= -2$
And since $\displaystyle -\alpha^2b = 2\alpha^2 = 6 \Rightarrow \alpha^2 = 3$
And since $\displaystyle -\alpha^2c = -3c = 9 \Rightarrow c = -3$.
Thus we found $\displaystyle f(x) = (8x^2-2x-3)(x^2-3)$
Now you can find the roots of the 2 quadratic factors yourself.
3. Hello, Sunyata!
Solve the equation: $\displaystyle f(x) \:=\: 8x^4-2x^3-27x^2+6x+9\:=\:0$
given that the sum of two of its roots is zero.
Let the two roots be: .$\displaystyle a\text{ and }-a.$
Then: .$\displaystyle \begin{array}{ccccccc}f(a) = 0\!: & a^4 - 2a^3 - 27a^2 + 6a + 9 &= & 0 & [1] \\ f(\text{-}a) = 0\!: & a^4 + 2a^3 - 27a^2 - 6a + 9 &=& 0 & [2] \end{array}$
Subtract [2] - [1]: .$\displaystyle 4a^3 - 12a \:=\:0 \quad\Rightarrow\quad 4a(a^2-3) \:=\:0$
. . and we have three roots: .$\displaystyle a \:=\:0,\:\pm\sqrt{3}$
. . but $\displaystyle a = 0$ is an exraneous root.
Hence, two of the factors are: .$\displaystyle (x - \sqrt{3})(x + \sqrt{3})$
Dividing $\displaystyle f(x)\text{ by }x^2-3$, we get: .$\displaystyle 8x^2 - 2x - 3$
The equation becomes: .$\displaystyle (x^2-3)(2x+1)(4x-3) \:=\:0$
Therefore, the roots are: .$\displaystyle x \;=\;\pm\sqrt{3},\;-\frac{1}{2},\;\frac{3}{4}$<|endoftext|>
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Critical Thinking &
Critical Thinking & Character Development
Often regarded as equivalent to higher-level thinking, critical thinking is one of the skills that we try to develop in ISMILE children as early as possible. Children are given the opportunity to express their thoughts and ideas creatively. They are encouraged to ask questions; articulate their line of thoughts and provide a logical answer to the questions they asked. The emphasis is on observation of the environment using all their senses. Components of the KEY TO LEARNING Logic module and MATAL Early Childhood Programme from Israel form the basis in developing these critical thinking skills.
"Sow an act, reap a habit; sow a habit, reap a character; sow a character, reap a destiny."
The foundation years are crucial to the character formation of a child. We believe in investing time and resources in building an unshakeable foundation that will last the trial of time and challenges.
Good character beginnings are emphasised from the infant right up to the kindergarten programmes. Age-appropriate good habits are introduced and continually encouraged and supported in a positive environment.<|endoftext|>
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# At a certain location wind is blowing steadily at 12 m/s. Determine the mechanical energy of air per unit mass and the power generation potential of a wind turbine with 60m diameter blades at that location. Take the air density to be 1.25kg/m^3.
This question aims to develop an understanding of the power generation capacity of a wind turbine generator.
A wind turbine is a mechanical device that converts the mechanical energy (kinetic energy to be precise) of the wind into electrical energy.
The energy generation potential of a wind turbine depends upon the energy per unit mass $KE_m$ of the air and mass flow rate of the air $m_{ air }$. The mathematical formula is as follows:
$PE \ = \ KE_m \times m_{ air }$
Given:
$\text{ Speed } \ = \ v \ = \ 10 \ m/s$
$\text{ Diameter } \ = \ D \ = \ 60 \ m$
$\text{ Density of Air } = \ \rho_{ air } \ = \ 1.25 \ kg/m^3$
Part (a) – Kinetic energy per unit mass is given by:
$KE_m \ = \ KE \times \dfrac{ 1 }{ m }$
$KE_m \ = \ \dfrac{ 1 }{ 2 } m v^2 \times \dfrac{ 1 }{ m }$
$\Rightarrow KE_m \ = \ \dfrac{ 1 }{ 2 } v^2$
Substituting values:
$KE_m \ = \ \dfrac{ 1 }{ 2 } ( 12 )^2$
$\Rightarrow KE_m \ = \ 72 \ J$
Part (b) – The energy generation potential of the wind turbine is given by:
$PE \ = \ KE_m \times m_{ air }$
Where $m_{ air }$ is the mass flow rate of air passing through the wind turbine blades which is given by the following formula:
$m_{ air } \ = \ \rho_{ air } \times A_{ turbine } \times v$
Since $A_{ turbine } \ = \ \dfrac{ 1 }{ 4 } \pi D^2$, the above equation becomes:
$m_{ air } \ = \ \rho_{ air } \times \dfrac{ 1 }{ 4 } \pi D^2 \times v$
Substituting this value in the $PE$ equation:
$PE \ = \ KE_m \times \rho_{ air } \times \dfrac{ 1 }{ 4 } \pi D^2 \times v$
Substituting values into this equation:
$PE \ = \ ( 72 ) \times ( 1.25 ) \times \dfrac{ 1 }{ 4 } \pi ( 60 )^2 \times ( 12 )$
$\Rightarrow PE \ = \ 3053635.2 \ W$
$\Rightarrow PE \ = \ 3053.64 \ kW$
## Numerical Result
$KE_m \ = \ 72 \ J$
$PE \ = \ 3053.64 \ kW$
## Example
Calculate the energy generation potential of a wind turbine with a blade diameter of 10 m at a wind speed of 2 m/s.
Here:
$KE_m \ = \ \dfrac{ 1 }{ 2 } v^2$
$\Rightarrow KE_m \ = \ \dfrac{ 1 }{ 2 } ( 2 )^2$
$\Rightarrow KE_m \ = \ 2 \ J$
And:
$PE \ = \ KE_m \times \rho_{ air } \times \dfrac{ 1 }{ 4 } \pi D^2 \times v$
$\Rightarrow PE \ = \ ( 2 ) \times ( 1.25 ) \times \dfrac{ 1 }{ 4 } \pi ( 10 )^2 \times ( 2 )$
$\Rightarrow PE \ = \ 392.7 \ W$<|endoftext|>
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Why Do Fish Have Scales?
Most fish are covered with a layer of flat scales, which overlap each other like the shingles on a roof. The hard scales form a protective covering for the softer body beneath. The scales in turn are covered with a thin layer of skin.
The skin gives off a slimy substance that coats the fish’s body and helps protect it against infections and parasites, if any of the fish’s scales are lost by accident, new ones grow to take their place. Fish scales reflect light, making the fish shimmer like water and, hopefully, slip by predators unnoticed.
As the fish grows, the scales also grow by adding rings of new material around their edges. Scientists tell the age of a fish by examining its scales. Scales vary enormously in size, shape, structure, and extent, ranging from strong and rigid armour plates in fishes such as shrimpfishes and boxfishes, to microscopic or absent in fishes such as eels and anglerfishes.
The morphology of a scale can be used to identify the species of fish it came from. Cartilaginous fishes (sharks and rays) are covered with placoid scales. Most bony fishes are covered with the cycloid scales of salmon and carp, or the ctenoid scales of perch, or the ganoid scales of sturgeons and gars. Some species are covered instead by scutes, and others have no outer covering on the skin.
Fish scales are part of the fish’s integumentary system, and are produced from the mesoderm layer of the dermis, which distinguishes them from reptile scales. The same genes involved in tooth and hair development in mammals are also involved in scale development.
The placoid scales of cartilaginous fishes are also called dermal denticles and are structurally homologous with vertebrate teeth. It has been suggested that the scales of bony fishes are similar in structure to teeth, but they probably originate from different tissue.<|endoftext|>
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Definitions
# Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss’s theorem (Carl Friedrich Gauss), Ostrogradsky’s theorem (Mikhail Vasilievich Ostrogradsky), or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.
More precisely, the divergence theorem states that the outward flux of a vector field through a surface is equal to the triple integral of the divergence on the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region.
The divergence theorem is an important result for the mathematics of engineering, in particular in electrostatics and fluid dynamics.
The theorem is a special case of the more general Stokes' theorem, which generalizes the fundamental theorem of calculus.
## Intuition
If a fluid is flowing in some area, and we wish to know how much fluid flows out of a certain region within that area, then we need to add up the sources inside the region and subtract the sinks. The fluid flow is represented by a vector field, and the vector field's divergence at a given point describes the strength of the source or sink there. So, integrating the field's divergence over the interior of the region should equal the integral of the vector field over the region's boundary. The divergence theorem says that this is true.
The divergence theorem is thus a conservation law which states that the volume total of all sinks and sources, the volume integral of the divergence, is equal to the net flow across the volume's boundary.
## Mathematical statement
Suppose V is a subset of Rn (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have
$iiintlimits_Vleft\left(nablacdotmathbf\left\{F\right\}right\right)dV=iintlimits_\left\{part V\right\} mathbf\left\{F\right\} cdot mathbf\left\{n\right\}, dS.$
The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. Here ∂V is quite generally the boundary of V oriented by outward-pointing normals, and n is the outward pointing unit normal field of the boundary ∂V. (dS may be used as a shorthand for ndS.) In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume V, and the right-hand side represents the total flow across the boundary ∂V.
Note that the divergence theorem is a special case of the more general Stokes' theorem which generalizes the fundamental theorem of calculus.
### Corollaries
By applying the divergence theorem in various contexts, other useful identities can be derived (cf. vector identities).
• Applying the divergence theorem to the product of a scalar function g and a vector field F, the result is
iiintlimits_Vleft(mathbf{F}cdot left(nabla gright) + g left(nablacdot mathbf{F}right)right) dV = iintlimits_{part V}g mathbf{F} cdot dmathbf{S}
A special case of this is $mathbf\left\{F\right\}=nabla f$, in which case the theorem is the basis for Green's identities.
• Applying the divergence theorem to the cross-product of two vector fields $mathbf\left\{F\right\}times mathbf\left\{G\right\}$, the result is
$iiintlimits_V left\left(mathbf\left\{G\right\}cdotleft\left(nablatimesmathbf\left\{F\right\}right\right) - mathbf\left\{F\right\}cdot left\left(nablatimesmathbf\left\{G\right\}right\right)right\right), dV = iintlimits_\left\{part V\right\}left\left(mathbf\left\{F\right\}timesmathbf\left\{G\right\}right\right)cdot dmathbf\left\{S\right\}$
• Applying the divergence theorem to the product of a scalar function f and a nonzero constant vector, the following theorem can be proven:
$iiintlimits_V nabla f, dV = iintlimits_\left\{part V\right\} f ,dmathbf\left\{S\right\}$
• Applying the divergence theorem to the cross-product of a vector field F and a nonzero constant vector, the following theorem can be proven:
$iiintlimits_V nablatimesmathbf\left\{F\right\}, dV = iintlimits_\left\{part V\right\}dmathbf\left\{S\right\} timesmathbf\left\{F\right\}.$
## Example
Suppose we wish to evaluate
$iintlimits_Smathbf\left\{F\right\}cdot mathbf\left\{n\right\}, dS,$
where S is the unit sphere defined by $x^2+y^2+z^2=1$ and F is the vector field
$mathbf\left\{F\right\} = 2 xmathbf\left\{i\right\}+y^2mathbf\left\{j\right\}+z^2mathbf\left\{k\right\}.$
The direct computation of this integral is quite difficult, but we can simplify it using the divergence theorem:
begin{align} iintlimits_S mathbf{F} cdot mathbf{n} , dS &= iiintlimits_Wleft(nablacdotmathbf{F}right) , dV
`&= 2iiintlimits_Wleft(1+y+zright) , dV`
`&= 2iiintlimits_W ,dV + 2iiintlimits_W y ,dV + 2iiintlimits_W z ,dV.`
end{align}
Since the functions $y$ and $z$ are odd on $S$ (which is a symmetric set in respect to the coordinate planes), one has
$iiintlimits_W y, dV = iiintlimits_W z, dV = 0.$
Therefore,
$iintlimits_S mathbf\left\{F\right\} cdot mathbf\left\{n\right\} , dS = 2iiintlimits_W ,dV = frac\left\{8pi\right\}\left\{3\right\}$
because the unit ball W has volume 4π/3.
## Applications
### "Differential form" and "Integral form" of physical laws
As a result of the divergence theorem, a host of physical laws can be written in both a differential form (where one quantity is the divergence of another) and an integral form (where the flux of one quantity through a closed surface is equal to another quantity). Three examples are Gauss's law (in electrostatics), Gauss's law for magnetism, and Gauss's law for gravity.
#### Continuity equations
Continuity equations offer more examples of laws with both differential and integral forms, related to each other by the divergence theorem. In fluid dynamics, electromagnetism, quantum mechanics, and a number of other fields, there are continuity equations that describe the conservation of mass, momentum, energy, probability, or other quantities. Generically, these equations state that the divergence of the flow of the conserved quantity is equal to the distribution of "sources" or "sinks" of that quantity. The divergence theorem states that any such continuity equation can be written in a differential form (in terms of a divergence) and an integral form (in terms of a flux).
### Inverse-square laws
Any "inverse-square law" can instead be written in a "Gauss's law"-type form (with a differential and integral form, as described above). Two examples are Gauss's law (in electrostatics), which follows from the inverse-square Coulomb's law, and Gauss's law for gravity, which follows from the inverse-square Newton's law of universal gravitation. The derivation of the Gauss's law-type equation from the inverse-square formulation (or vice-versa) is exactly the same in both cases; see either of those articles for details.
## History
The theorem was first discovered by Joseph Louis Lagrange in 1762, then later independently rediscovered by Carl Friedrich Gauss in 1813, by George Green in 1825 and in 1831 by Mikhail Vasilievich Ostrogradsky, who also gave the first proof of the theorem. Subsequently, variations on the Divergence theorem are called Gauss's Theorem, Green's theorem, and Ostrogradsky's theorem.
This article was originally based on the GFDL article from PlanetMath at http://planetmath.org/encyclopedia/Divergence.html<|endoftext|>
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# Normal Distribution: A Simple Introduction
3년 전
Hey Steemians, you may have realized that there are very few students getting very high and very low marks in your class. The most of the students score around average in the exam The distribution of score obtained in the exam can be modeled approximately by the Normal distribution. The normal distribution looks like following.
Source: Wikimedia commons
The marks are in discrete i.e. whole number, but if it large in number then it can be approximated by continuous curve like above.
Along with marks, the height of people in Steemstem community, the weight of people in your college, IQ scores, the life of turbine blades, error in measurement and so on are beautifully approximated by the normal distribution. It is so consistent with the nature that:
Mathematicians take it as an empirical fact and experimentalists take it as a mathematical fact.
• Henri Poincare(you may have heard about Poincare conjecture)
It was first discovered by De Moivre in 1733 AD. It is also called Gaussian distribution as Gauss used this distribution for characterizing measurement errors. It is a bell-shaped curve so it is often called bell curve.
# Preliminary Section:
You need to know certain things for the better understanding of the post. If you have already taken basic statistics course or studied on your own, You may skip this section.
# Mean:
Suppose you are given a set of data, how can you represent this data by single numbers? One idea is to represent central location by one number and the dispersion of data relative to this central location by another number. It is called central tendency. We can use mean of the data as the central location. How can you find it? Simple, add the values of a set of data and divide it by the number of data set. It is denoted by and given by;
Here,
= Addition of set of values
x= Individual elements of data set.
N = Total number of observations.
For example:
I asked the age of five random members of Steemstem community. The data set I got was:
26,21,31, 40,34
The mean of the given data =
=
= 30.6 year
It is a single value for the representation of center of data.
To give you graphical perspective, plot the values of ages in the y-axis. Then, the mean line is the horizontal line from which the sum of the square of the distance between points of data sets and the mean is minimum.
Self plotted in Excel
To give you physical perspective, let the car travel on a road with velocities v1,v2,v3,v4,v5 for an equal interval of time t. Then, the uniform velocity with which the car should travel to cover the same distance in same time is the mean of the velocities.
Mean =
There are also other measures of central values like harmonic mean, geometric mean, median, mode, etc.
# Standard deviation:
The above subsection is basically about the central location. Standard deviation gives second characteristics of data discussed earlier, i.e. dispersion. How are the data dispersed or deviated from the central location? One former idea may be to add all the deviation from the mean and divide that by the number of observations. If we do that, we will get zero. But, we should consider deviations in both above and below mean. In standard deviation, deviation of each data from mean is squared and averaged. For the same unit as in the observation, the square root of this value is taken.
It is simply root mean square deviation from the mean.
It is denoted by
and for individual series,
= http://quicklatex.com/cache3/a3/ql_201e827fd146d529b9dfa86b9a3affa3_l3.png
The data set below is the percentage I got in 6 semesters exams.
80,85,81,77,83,75
The mean of the given data = = 80.166
The standard deviation of the given data =
= 3.38
Now, we got the value of standard deviation. What can we say about the dispersion of the data? Actually, nothing. The standard deviation gives the deviation. But, it is important to compare it with mean. Now, with the comparison with mean, we can find the coefficient of variation. The coefficient of variation of two data set can be compared to know which data is more dispersed.
# Probability:
Probability is the measure of the likelihood of the event. Consider in your college, there are 30 students. Out of them, 20 are females. If you randomly select one person, what is the probability that the person is female?
Probability(female) =
# Random variable:
Consider a random process, flipping of the two coins at a time. The possible outcomes are {H,H}, {H,T},{T,H} and {T,T}. These outcomes can be mapped into numbers. Then these can be denoted by the random variable.
Let, random variable X be the number of tails we get from tossing two coins. Then,
{H, H} is mapped to 0
{H,T}/{T,H} is mapped to 1
{T, T} is mapped to 2.
P(X =0) =
P(X=1) =
P(X=2) =
In above example, the random variable X can only take values like 0,1,2, ... values. So, it is discrete random variable.
In the measurement of the height of the people, the measurement of height may not be discrete. It can take infinitely many values even between two numbers. When random variable X can take these values then, it is called continuous random variable. Instead of P(X=x) in case of the discrete random variable, we take interval in the continuous random variable. i.e.
Examples:
• If we measure the height of people, random variable X(x) = x, x is any positive real number.
• Observe the birth of 500 babies in the hospital. The number of baby girls can be a random variable, where it can take discrete value from 0 to 500.
# Probability Distribution
In the coin toss example above, the random variable i.e. number of tails can have different values. For each value, there is a probability of occurrence. If we plot, tabulate or graph the possible values of the random variable to the probability of occurrence, it is called probability distribution.
For simplicity, let's tabulate the probability;
Possible outcomesProbability
P(X=0)1/4
P(X=1)2/4
P(X=2)1/4
If we add all the possible outcomes of a random variable, we get sum 1.
# What is normal distribution?
We say that the continuous random variable(one taking real values) X has the normal distribution with parameters and http://quicklatex.com/cache3/7f/ql_7856d36b3c847891ccdc59b5b0941f7f_l3.png (explained earlier) if its probability density function(how probability is distributed) is given by;
f(x) =
• e = 2.71828.. is called Euler Number.
• = 3.141.. is the ratio of the circumference of the circle to the diameter.
• = mean , http://quicklatex.com/cache3/0b/ql_ddc53de4f1791b0930c3a60f094a500b_l3.png
• = standard deviation. http://quicklatex.com/cache3/c8/ql_b81c54d44c7926a41166826e1e48dac8_l3.png
If we simply plot the value of x in x-axis and f(x) in the y-axis, we can get the bell curve.
Here, and http://quicklatex.com/cache3/7f/ql_7856d36b3c847891ccdc59b5b0941f7f_l3.png are called the parameters of the normal distribution as they completely define the normal curve. The value of http://quicklatex.com/cache3/46/ql_f9754bb25c626ef1781bd0ef90a4d146_l3.png is the center of the distribution and http://quicklatex.com/cache3/7f/ql_7856d36b3c847891ccdc59b5b0941f7f_l3.png determines the width of the distribution.
Source: wikimedia commons
• The red and green curve has the same standard deviation, so the width is same. But, they have different means, so centers are different.
• The blue and red has same mean but different standard deviation, so their center is same but the width is different.
The function above may look strange at first. But, it is not so common but fairly used function in physics and mathematics. Even its given a special name called Gaussian function. The term apart from exponential is simply normalizing factor i.e. coefficient to make integration unity. This can be proved in few steps using gamma function.
The area enclosed by the curve and two vertical lines x=a and x=b gives the probability of finding the values in between a and b.
Source: Wikimedia commons
# Why normal distribution?
As discussed earlier, normal distribution approximates many real-world cases so well. This is normal distribution doesn't mean other distributions are abnormal. Other distribution like binomial distribution, Poisson distribution etc. do really well approximation in their realm. But, even these distribution can be approximated by the normal distribution. For large values of n, it's tedious to calculate values using the binomial distribution. In such case, the normal distribution may be employed to solve the problem easily. All the sampling distributions like t-distribution, chi-square distribution etc. converge towards normal distribution for the large values of n. By historical point of view also, this is most important distribution. Many research inferences are based on modeling using the normal distribution. In mechanical engineering(which is my study field), the control limits are set for quality control using the normal distribution. The measurement error which is integral in case of experimental physics follows the normal distribution.
# Standard normal distribution?
We now know that the shape of normal distribution depends upon mean and standard deviation. Different data from observation have different means and standard deviation, so they have different shapes. In principle, we can calculate the probability for each case using integral above but its really tedious work. Moreover, the above integral(containing pdf) cannot be integrated directly. It needs to solved using numerical integration. So, it is logical to make a standard normal distribution and have values of the area for different values of the normal random variable.
We take the special case of the normal distribution with mean zero and the standard deviation unity as standard normal distribution. Every normal distribution can be converted into standard form. In doing so, we should convert the random variable X into something called z-score.
The probability density function =
Where z is given by;
Source: wikimedia commons
# Characteristics of standard normal distribution
• Z ~ N(0,1), standard normal variable follows the normal distribution with mean zero and standard deviation unity.
• The curve is symmetric about line z =0.
• Area rule:
Area property of standard normal distribution is most important property.
![area property.jpg()
Source: wikimedia commons
a. The area under the standard normal curve in between ordinates z =-1 to z =1 is 0.686. That means this area covers 68.26% of the observations.
b. The range Z = -2 to 2 covers 95.44 % of the observations.
c. The range Z = -3 to 3 covers 99.73 % of the observations.
The value of Z can extend from negative infinity to positive infinity but for the practical purpose, we often use the area covered by -3 to 3 range as 100%.
• We have the values of the area for every z as;
F(z)
Source: wikimedia commons
• If we have to find the area in between let's say from z=a and z=b(b>a) then,
F(a) gives the area from negative infinity to z=a.
F(b) gives the area from negative infinity to z =b.
F(b) - F(a) gives the area from z=a to z = b.
Source: wikimedia commons
• Conventionally, F(a) gives the area left of z=a. To find the area of right to z=a;
F(-a) = 1 - F(a)
Source: wikimedia commons
• Mean = Median = Mode = 0
# Example
In a School Leaving Certificate level examination in Nepal(grade 10), the mean percentage scored by students is 50 % and the standard deviation of percentage obtained is 12%. If 5 lakhs of students appeared in the examination, calculate the number of students who got the distinction (at least 80%) is the examination. Assume that the distribution is normal.
Solution:
Mean percentage ( ) = 50 %
Standard deviation( ) = 12%
X = 80 %
Calculating z score;
for distinction;
P(z 2.5)
Source: wikimedia commons
= 1- P(z < 2.5)
= 1 - 0.9938
= 0.0062
No. of students securing distinction = 0.0062* 500000 = 3100 students.
In this article, I limited the content up to basics of the normal distribution. Previously, I had the mindset to cover the real applications of normal distribution too. But, this post is already very long. So, in next article, probably I will write about inference using normal distribution, use of normal distribution in my field i.e. mechanical engineering, and other applications.
# References
1. Yamane, T. (1973). Statistics: An introductory analysis.
2. Brownlee, K. A. (1965). Statistical theory and methodology in science and engineering (Vol. 150, pp. 120-131). New York: Wiley.
3. Budhathoki, T.B. (2011). Probability and Statistics for Engineers. Kathmandu: Heritage Publications.
4. Altman, D. G., & Bland, J. M. (1995). Statistics notes: the normal distribution. Bmj, 310(6975), 298.
5. https://www.mathsisfun.com/data/standard-normal-distribution.html
6. http://www.statisticshowto.com/probability-and-statistics/normal-distributions/
Reference mathematical tools
1. Microsoft Excel
2. www.quicklatex.com
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· 3년 전
· 3년 전
Your post gave an incredible post-traumatic stress to my statistics classes last year :P but you explained it all very well!
I'm a vet student so i don't usually use statistics unless i go into epidemiology, i actually just learned how to work with a program called Epiinfo which correlates data between many epidemiology studies, it shows p-value we use the null hypothesis, etc.... you know what I'm talking about
Are you also going to explain Poisson and binomial distributions? ( i think this is what they are called)
·
· 3년 전
Thank you for your encouraging comment. That is simply hypothesis testing. Normal distribution is for continuous random variable and Poisson and Binomial distribution are for a discrete random variable. They are relatively easy to explain. I think I will explain those.
In future, Maybe, I will also request volunteers from steemstem members in data collection and analyze data to find out various inferences. That would be interesting I guess.
You forgot to add to add sup and sub-scripts to your sigma. But the rest is fine :)
·
· 3년 전
Thank you !!
Hi! I am a robot. I just upvoted you! I found similar content that readers might be interested in:
https://en.wikipedia.org/wiki/Normal_distribution
·
· 3년 전
Everyone reading this post can find out the difference in content between wiki and this post. I don't know which part of the post is similar. I didn't even read wiki before posting this article. So, I didn't mentioned it as reference/source in the article(and I still won't).
·
Can confirm that this is unique from wikipedia.
·
·
· 3년 전
Thank you so much for confirming. :D<|endoftext|>
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# 2.5 Sample Spaces Having Equally Likely Outcomes
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1 Sample Spaces Having Equally Likely Outcomes 3 Sample Spaces Having Equally Likely Outcomes Recall that we had a simple example (fair dice) before on equally-likely sample spaces Since they will appear often, we will discuss them more in detail in this section Consider an experiment with N possible outcomes Let s denote its sample space by If all outcomes are equally-likely then S = {s 1,s,,s N } P({s 1 })=P({s })= = P({s N })= 1 N since P(S)=1 So if E is an event with M elements in S, such that E = {s i1,s i,,s im } then This means that P({s i1 })+P({s i })+ + P({s im })= 1 N + 1 N N = M N in an equally-like sample space S the number of elements in E the number of elements in S Example 9 A fair dice is rolled P(an even number shows)= 3 6 Notation 1 To denote the number of elements in an event E, we ll write E Example 10 A pair of dice is rolled What is the probability that the sum of two shows 7? The event is that the sum of both dice equals 7 So it is and Also the size of the sample space is E = {(1,6),(,),(3,4),(4,3),(,),(6,1)} E = = 36 E 6 36 Example 11 3 balls are randomly drawn from an urn containing 6 white and black balls What is the probability of drawing one white and two black balls? Since there are 11 balls in total, The event of interest is S = {all possible 3-ball selections} 11 = 16 3 E = {all possible 3-ball selections where 1 is white and two are black} So
2 4 Axioms of Probability Choose 1 white out of Choose black out of = 6 4 = 60 E Example 1 A committee of is to be selected from a group of 6 men and 9 women If the selection is made randomly, what is the probability that the committee consists of 3 men and women? hence The event is and S = {all possible -people groups}, 1 E = {all possible -people groups consisting of 3 men and women} E = Choose 3 men out of Choose women out of Example A poker hand consists of cards If the cards have distinct consecutive values (1,,,10,J,Q,K) and are not all of the same suit, we say that the hand is straight What is the probability that one is dealt a straight? For example, is a straight, J Q is a straight, is not a straight S = {all -card hands}
3 Sample Spaces Having Equally Likely Outcomes and so Our event is E = {straighthands} We need to count how many straight hands can be formed For this purpose, let s consider the hands consisting of only 1,,3,4, s How many straights are there in this case? Note that there are 4 of spades, 4 of hearts, 4 of diamonds and 4 of clubs Without restriction we have = 4 of 1?? 3? 4?? But among these we have 4 hands which are not straight,namely: is not a straight, is not a straight, is not a straight, is not a straight So we have 4 4 straights if we consider only 1,,3,4, cards Similarly, we have the same amount of straights for,3,4,,6 etc 1?? 3? 4??! (4 4) straights? 3? 4?? 6?! (4 4) straights 3? 4?? 6? 7?! (4 4) straights 4?? 6? 7? 8?! (4 4) straights? 6? 7? 8? 9?! (4 4) straights 6? 7? 8? 9? 10?! (4 4) straights 7? 8? 9? 10? J?! (4 4) straights 8? 9? 10? J? Q?! (4 4) straights 9? 10? J? Q? K?! (4 4) straights 10? J? Q? K? 1?! (4 4) straights
4 6 Axioms of Probability In total, there are 10(4 4) straight hands So 10(4 4) Example 14 In the game of bridge, the entire deck of cards is dealt out to 4 players What is the probability that 1 one of the players receives all of spades, each player receives one ace? 1 Consider only the "Player 1" The sample space: which leads to The event is S = {all -card selections for "Player 1"} E 1 = {all -card selections for "Player 1" where all are spades} Note that this possible only one way So E 1 = 1 P(E 1 )= 1 Since there are 4 players in the game, each have this probability to receive all spades our probability is P(one of the players receives all of spades)= 4 Now put the aces aside Then you need to divide 48 cards equally to 4 players So there are 48 ways Next distribute aces to players in 4! ways So the size of the event is 48 4! just different ways of distributing cards to 4 players equally, which can be done in,,, P(each player receives one ace)= 4! 48,,,
5 Sample Spaces Having Equally Likely Outcomes 7 Example 1 There are n people at a party What is the probability that no two people have the same birthday? (That is, no two of them were born on the same day of a year) the all possible birthdays for n people Then its size is {z 36 } n-many =36 n And the event is that no two shares the same day E = (36 {z n + 1) } n-many = 36! (36 n)! 36! (36 n)! 36 n We can ask the question, what is the probability that at least people at the party have the same birthday? Clearly it equals ! (36 n)! 36 n Interestingly, if you choose n = 0 this probability is approximately 097! Surprising?
### STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes
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| 4.625 |
2,621 |
Equcation for number 2025 factorization is: 3 * 3 * 3 * 3 * 5 * 5; It is determined that the prime factors of number 2025 are: 3, 5; Prime Factorization Of 2024; Prime Factorization Of 2026 Solution: Step 1: The given number is resolved into its prime factors. 441. 1764. 2. The prime factors of 2025 are 3 and 5. (iii) Combine the like square root terms using mathematical operations. Square Root by Prime Factorization Example Problems. Find the square root of 1764 using the prime factorization method. For example, 2 is the square root of 4, because 2x2=4. The factors of 225 = 3x3x5x5 = 3^2x5^2 Square root of 225 = [3^2x5^2]^0.5 = 3^(2*0.5)*5^(2*0.5) = 3x5 = 15. 7 $1764 = 2\times 2\times 3\times 3\times 7\times 7$ Step 2: Identical factors are paired. 2. 49. A number bigger than zero has two square roots: one is positive (bigger than zero) and the other is negative (smaller than zero). (ii) Inside the square root, for every two same numbers multiplied, one number can be taken out of the square root. Only numbers bigger than or equal to zero have real square roots. 3. Thew following steps will be useful to find square root of a number by prime factorization. The prime factorization of 2025 = 3 4 •5 2. (i) Decompose the number inside the square root into prime factors. A composite number is a positive integer that has at least one positive divisor other than one or the number itself. In other words, a composite number is any integer greater than one that is not a prime number. 147. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. 1. Solve your math problems using our free math solver with step-by-step solutions. 3. 882. Worksheet on square root using prime factorization method is useful for the students to prepare well for the exams. 7. 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| 4.71875 |
1,628 |
Persecution of Jews and Muslims by Manuel I of Portugal
Until the 15th century, some Jews occupied prominent places in Portuguese political and economic life. For example, Isaac Abrabanel was the treasurer of King Afonso V of Portugal. Many also had an active role in Portuguese culture, and they kept their reputation of diplomats and merchants. By this time, Lisbon and Évora were home to important Jewish communities.
Expulsion of JewsEdit
On 5 December 1496, King Manuel I of Portugal decreed that all Jews must convert to Catholicism or leave the country, because one clause in the contract of marriage between himself and Isabella, Princess of Asturias, stipulated he do so in order to win her hand. The King demonstrated his wish to completely and forever eradicate Judaism from Portugal by issuing two decrees. The initial edict of expulsion of 1496 was turned into an edict of forced conversion in 1497: Portuguese Jews were prevented from leaving the country and were forcibly baptized and converted to Christianity. Those Jews who refused to pay taxes in protest were deported from Portugal and abandoned to their fate in the islands of São Tomé and Príncipe, off the western coast of Africa. Hard times followed for the Portuguese conversos, with the massacre of 2,000 individuals in Lisbon in 1506 and later the establishment of the Portuguese Inquisition in 1536.
When the King allowed conversos to leave after the Lisbon massacre of 1506, many went to the Ottoman Empire, notably Salonica and Constantinople, and to the Wattasid Sultanate of Morocco. Smaller numbers went to Amsterdam, France, Brazil, Curaçao and the Antilles, Surinam and New Amsterdam. In some of these places their presence can still be perceived in the use of the Ladino language by some Jewish communities in Greece and Turkey, the Portuguese-based dialects of the Antilles, or the multiple synagogues built by those who became known as the Spanish and Portuguese Jews, such as the Amsterdam Esnoga or the Willemstad Snoa. Some of the most famous descendants of Portuguese Jews who lived outside Portugal are the philosopher Baruch Spinoza (Bento de Espinosa in Portuguese), and the classical economist David Ricardo.
Jews who converted to Christianity were known as New Christians, and were always under the constant surveillance of the Inquisition. The dreaded presence of the Holy Office in Portugal lasted for almost three hundred years, until the Portuguese Inquisition was abolished in 1821 by the "General Extraordinary and Constituent Courts of the Portuguese Nation". Many of those New Christians were crypto-Jews who continued to secretly practice their religion; they eventually left the country in the centuries to come, and openly embraced their Jewish faith again in foreign lands. Such was the case, for example, of the ancestors of Baruch Spinoza in the Netherlands. Some other Portuguese Jews, very few in number like the Belmonte Jews, opted for a different and radical solution, practicing their faith in a strictly secret manner among a rural and isolated community. Known as the "Last of the Marranos", some have survived until today (especially the Jewish community from Belmonte in Castelo Branco, plus some scattered families) by their practice of intermarriage and their very limited cultural contacts with the outside world. Only recently, in the late 20th century, have they re-established contact with the international Jewish community and openly practice their religion in a public synagogue with a formal rabbi.
Expulsion of MuslimsEdit
According to contemporary historian François Soyer, the expulsion of Muslims from Portugal has been overshadowed by the forced conversion of Jews in the country. While tolerance of Muslim minorities in Portugal was higher than in any other part of Europe, Muslims were still perceived as "alien." Anti-Muslim riots were regular in neighboring Valencia during the 1460s; however, no similar acts of violence occurred in Portugal.
In December 1496, Manuel I ordered all Muslim subjects to leave without any apparent provocation. According to 15th-century Portuguese historians Damião de Góis and Jerónimo Osório, the Portuguese government originally planned to forcibly convert or execute Muslims as they had done to Jews, but fear of retaliation from Muslim kingdoms in North Africa led the king to settle on deportations instead. Manuel I's motivation behind the order is unclear, but some contemporary historians say it was part of a greater goal of Queen Isabella and King Ferdinand (known as the "Catholic Monarchs") to rid the peninsula of Muslims and create "religious uniformity" and "monolithic Catholic Christian unity". Other historians say it was influenced by ambitions of conquering Morocco, or by the suggestion of the Dominican confessor to the king, Friar Jorge Vogado. Some Muslims found refuge in Castile, but most fled to North Africa.
Return of some Jews to PortugalEdit
In the 19th century, some affluent families of Sephardi Jewish Portuguese origin such as the Ruah and Bensaude, resettled in Portugal from Morocco. The first synagogue to be built in Portugal since the 15th century was the Lisbon Synagogue, inaugurated in 1904.
In 2014 the Portuguese parliament changed the Portuguese nationality law in order to grant Portuguese nationality to descendants of Sephardi Jews expelled from Portugal. The law is a reaction to historical events that led to their expulsion from Portugal, but also due to increased concerns over Jewish communities throughout Europe. In order to obtain Portuguese nationality, the person must have a family surname that attests to being a direct descendant of a Sephardi of Portuguese origin or family connections in a collateral line from a former Portuguese Sephardi community. Use of expressions in Portuguese in Jewish rites or Judaeo-Portuguese or Ladino can also be considered proof.
- António José Saraiva: The Marrano Factory: The Portuguese Inquisition and Its New Christians 1536-1765, BRILL, 2001, ISBN 9789004120808, p. 10-12.
- Reuven Faingold (2013). "Judeus ibéricos deportados a São Tomé entre 1492-1497". Morashá: História Judaica Moderna (79).
- "Sao Tome & Principe". Jewish Virtual Library.
- Frédéric Brenner and Stan Neumann, Les Derniers Marranes (Motion Picture), La Sept-Les Film d'Ici, 1990.
- Soyer 2007, p. 241.
- Soyer 2007, p. 258.
- Soyer 2007, p. 254, 259.
- Soyer 2007, p. 242.
- Soyer 2007, pp. 260-261.
- Soyer 2007, p. 269.
- Soyer 2007, p. 280.
- Soyer 2007, p. 273.
- Soyer 2007, p. 262.
- Soyer 2007, p. 268.
- Lusa. "Descendentes de judeus sefarditas já vão poder pedir a nacionalidade". PÚBLICO.
- DEVOS, Olivier. "Amid rising European anti-Semitism, Portugal sees Jewish renaissance". www.timesofisrael.com. Retrieved Apr 4, 2019.
- Liphshiz, Cnaan. "New citizenship law has Jews flocking to tiny Portugal city". www.timesofisrael.com. Retrieved Apr 4, 2019.
- "Portugal open to citizenship applications by descendants of Sephardic Jews". Mar 3, 2015. Retrieved Apr 4, 2019.
Soyer, François (2007). The Persecution of the Jews and Muslims of Portugal: King Manuel I and the End of Religious Tolerance (1496–7). Leiden, The Netherlands: Koninklijke Brill NV. ISBN 9789004162624. Retrieved 15 May 2017.<|endoftext|>
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# Non Homogenous Differential Equation
• shards5
In summary, a non-homogeneous differential equation is a type of differential equation that contains terms not equal to zero. To solve it, the method of undetermined coefficients is used. The difference between a homogeneous and non-homogeneous differential equation is that the latter also includes terms not involving the dependent variable and its derivatives. A non-homogeneous differential equation can have infinitely many solutions due to the constant of integration in its general solution. Real-life applications of these equations include modeling physical phenomena in various fields such as physics, engineering, and economics.
shards5
## Homework Statement
y"' - 9y" +18y' = 30ex
y(0) = 16
y'(0) = 14
y"(0) = 11
n/a
## The Attempt at a Solution
Factor Out
r(r2 - 9r +18)
r = 0; r = 6; r =3
General Equation
y(x) = c0 + c1e3x + c2e6x
y'(x) = 3c1e3x + 6c2e6x
y"(x) = 9c1e3x + 36c2e6x
c1 = (11 - 36c2)/9
y'(x) = 14 = 3c1e3x + 6c2e6x
y'(x) = 14 = ((11 - 36c2)/9)e3x + 6c2e6x
c2 = -31/30
c1 = (11 - 36*(-31/30))/9 = 5.35555556
y(0) = 16 = c0 + c1e3x + c2e6x
16 +31/30 - 5.35555556 = c0 = 11.6777778
Solve For A in Aex
yp = Aex
y'p = Aex
y"p = Aex
y"'p = Aex
Inputting into the original equation we get.
Aex - 9Aex +18Aex = 30ex
Simplifying we get.
10Aex = 30ex
Which gives A = 3 and since 3 is a root of the original equation we add an x to differentiate between the two.
So the final equation SHOULD BE 3xex + 11.6777778 + 5.35555556e3x -31/30e6x but of course its not.
So my question is, what am I doing wrong?
Last edited:
Most of your work is fine, but I believe you have made an error in your calculations for c_0, c_1, and c_2. I used matrix methods to solve for these constants and got c_2 = 25/18. I'm fairly confident of this value, but didn't check it.
Also, in your last paragraph you say something that isn't true. You got A = 3, which means that your particular solution is y_p = 3e^x. The fact that you got a value of 3 when you solved for A is irrelevant to anything else in this problem. Your general solution will include 3e^x, not 3xe^x.
When you say matrix method do you just make a matrix like this?
1 1 1 16
0 3 6 14
0 9 36 11
And then you row reduce?
shards5 said:
When you say matrix method do you just make a matrix like this?
1 1 1 16
0 3 6 14
0 9 36 11
And then you row reduce?
Yes, and yes.
## 1. What is a non-homogeneous differential equation?
A non-homogeneous differential equation is a type of differential equation where the terms involving the dependent variable and its derivatives are not equal to zero. This means that the equation is not in its simplest form, which is known as a homogeneous differential equation.
## 2. How do you solve a non-homogeneous differential equation?
To solve a non-homogeneous differential equation, we use a method called the method of undetermined coefficients. This involves finding a particular solution to the equation by assuming a general form for the solution and then substituting it into the equation to determine the coefficients. The general solution is then found by adding this particular solution to the general solution of the corresponding homogeneous equation.
## 3. What is the difference between a homogeneous and non-homogeneous differential equation?
A homogeneous differential equation only contains terms involving the dependent variable and its derivatives, while a non-homogeneous differential equation also includes terms that do not involve the dependent variable and its derivatives. This makes the process of solving non-homogeneous differential equations more complex, as we need to find a particular solution in addition to the general solution.
## 4. Can a non-homogeneous differential equation have more than one solution?
Yes, a non-homogeneous differential equation can have infinitely many solutions. This is because the general solution of a non-homogeneous differential equation contains a constant of integration, which can take on any value. This means that there are many different solutions that satisfy the given equation.
## 5. What are some real-life applications of non-homogeneous differential equations?
Non-homogeneous differential equations are used to model various physical phenomena in fields such as physics, engineering, and economics. Some examples include modeling the growth of a population, the spread of disease, and the motion of objects under the influence of external forces. They are also used in signal processing, control theory, and other areas of mathematics.
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# A 25 watt, 220 volt bulb and a 100 watt, 220 volt bulb are connected in series across 440 volt lines.(A) only 100 watt bulbs will fuse(B) only 25 watt bulbs will fuse(C) none of these bulbs will fuse(D) both bulbs will fuse.
Last updated date: 17th Apr 2024
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Verified
35.1k+ views
Hint: When bulbs are connected in such a manner that one terminal of one bulb is kept open and another terminal is connected to one terminal of another bulb then they are considered to be connected as in series combination. Formula used to solve the problem: $P = \dfrac{{{V^2}}}{R}$ and $P = {I^2}R$ where P is the power, R is resistance and V is the Voltage or potential applied across the bulb.
Complete step by step solution:
Given
Calculate the resistance of each bulb
Since, Power $P = \dfrac{{{V^2}}}{R}$
$\Rightarrow R = \dfrac{{{V^2}}}{P}$
So, ${R_1} = \dfrac{{{V^2}}}{{{P_1}}} = \dfrac{{{V^2}}}{{25}}$
and ${R_2} = \dfrac{{{V^2}}}{{{P_2}}} = \dfrac{{{V^2}}}{{100}}$
Calculate the equivalent resistance
$R{}_{eq} = {R_1} + {R_2}$
$\Rightarrow R{}_{eq} = {V^2}\left( {\dfrac{1}{2} + \dfrac{1}{{100}}} \right)$
$\therefore R{}_{eq} = \dfrac{{{V^2}}}{{20}}$
Since, In series current is constant
So, when connected to supply of voltage 440 v the current in the circuit would be
$\Rightarrow I = \dfrac{{V'}}{{R{}_{eq}}}$
Since, V’ = 2V
$\Rightarrow I = \dfrac{2V}{\dfrac{V^2}{20}}$
$\therefore I = \dfrac{{40}}{{{V_0}}}$
Now
Power generated in the bulb 1 will be
$\Rightarrow P{}_1 = {I^2}{R_1} = {\left( {\dfrac{{40}}{V}} \right)^2} \times \left( {\dfrac{{{V^2}}}{{25}}} \right)$
$\therefore P{}_1 = 64W$
Power generated in the bulb 2 will be
$\Rightarrow P{}_2 = {I^2}{R_2} = {\left( {\dfrac{{40}}{V}} \right)^2} \times \left( {\dfrac{{{V^2}}}{{100}}} \right)$
$\therefore P{}_2 = 16W$
Here it is clearly seen that
$64W>25W$
Therefore, Only bulb 1 or of 25 watt will get fused.
Hence option (B) is the correct answer.
Note: Any resistive electronic device consumes electrical power and it does not depend upon the direction of current.
In both the above cases, power is only consumed and this power consumed is given by
$P = \dfrac{{{V^2}}}{R} = {I^2}R = VI$
In any electrical circuit, law of conservation of energy is followed. i.e.
Net power supplied by all the batteries of the circuit =net power consumed by all the resistors in the circuit.<|endoftext|>
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Huma, Hubna and Seema received a total of ₹ 2,016
Question:
Huma, Hubna and Seema received a total of ₹ 2,016 as monthly allowance from their mother such that Seema gets ½ of what Huma gets and Hubna gets times Seema’s share. How much money do the three sisters get individually?
Solution:
From the question it is given that,
Total monthly allowance received by Huma, Hubna and Seem = ₹ 2,016
from their mother
Seema gets allowance = ½ of Huma’s share
Hubna gets allowance = of Seema’s share
= 5/3 of Seema’s share
= 5/3 of ½ of Huma’s share … [∵ given]
= 5/3 × ½ of Huma’s share
= 5/6 of Huma’s share
So,
Huma’s share + Hubna’s share + Seema’s share = ₹ 2,016
Let Huma’s share be 1,
1 + (5/6) Huma’s share + ½ Huma’s share = ₹ 2,016
(1 + (5/6) + ½) = ₹ 2,016
The LCM of the denominators 1, 6 and 2 is 6
(1/1) = [(1×6)/ (1×6)] = (6/6)
(5/6) = [(5×1)/ (6×1)] = (5/6)
(1/2) = [(1×3)/ (2×3)] = (3/6)
Then,
(6/6) + (5/6) + (3/6) = ₹ 2,016
(6 + 5 + 3)/ 6 = ₹ 2,016
(14/6) = ₹ 2,016
So, Huma’s share = ₹ 2,016 ÷ (14/6)
= 2,016 × (6/14)
= 144 × 6
∴Huma’s Share is = ₹ 864
Seema’s share = ½ Huma’s share
= ½ × 864
= ₹ 432
Hubna’s share = 5/6 of Huma’s share
= 5/6 × 864
= 5 × 144
= ₹ 720<|endoftext|>
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## P
To be sure, the right hand side of Eq. (3.24) is a kind of mathematical trick and most readers will not have seen in advance that this is the way to proceed. That is fine, part of learning how to use the tools is to apprentice with a skilled craft person and watch what he or she does and thus learn how to do it oneself. Note that some of the terms on the right hand side of Eq. (3.24) comprise the probability that K = k — 1. When we combine those terms and examine what remains, we see that
Equation (3.25) is an iterative relationship between the probability that K = k — 1 and the probability that K = k. From Eq. (3.23), we know explicitly the probability that K = 0. Starting with this probability, we can compute all of the other probabilities using Eq. (3.25). We will use this method in the numerical examples discussed below.
Although Eq. (3.24) seems to be based on a bit of a trick, here's an insight that is not: when we examine the outcome of Ntrials, something must happen. That is 0 Pr{K = k} = 1. We can use this observation to find the mean and variance of the random variable K. The expected value of K is
EfK} = £ kPrfK = k} = £ k i / (1 — p)N—k k=0 k=0 \ k /
There is nothing tricky about what we have done thus far, but another trick now comes into play. We know how to evaluate the binomial sum from k = 0, but not from k = 1. So, we will manipulate terms accordingly by first writing the binomial coefficient explicitly and then factoring out Np from the expression on the right hand side of Eq. (3.26)
= NpY (N — 1)! pk—1(1 — )N—k (k — 1)!(N — k)!P (1 P)
and we now set j = k — 1. When k = 1, j = 0 and when k=N, j = N — 1. The last expression in Eq. (3.27) becomes a recognizable summation:
0 0<|endoftext|>
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# Approximation error
(Redirected from Absolute error)
For a broader coverage related to this topic, see Approximation.
Graph of $f(x) = e^x$ (blue) with its linear approximation $P_1(x) = 1 + x$ (red) at a = 0. The approximation error is the gap between the curves, and it increases for x values further from 0.
The approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because
1. the measurement of the data is not precise due to the instruments. (e.g., the accurate reading of a piece of paper is 4.5 cm but since the ruler does not use decimals, you round it to 5 cm.) or
2. approximations are used instead of the real data (e.g., 3.14 instead of π).
In the mathematical field of numerical analysis, the numerical stability of an algorithm in numerical analysis indicates how the error is propagated by the algorithm.
## Formal Definition
One commonly distinguishes between the relative error and the absolute error.
Given some value v and its approximation vapprox, the absolute error is
$\epsilon = |v-v_\text{approx}|\ ,$
where the vertical bars denote the absolute value. If $v \ne 0,$ the relative error is
$\eta = \frac{\epsilon}{|v|} = \left| \frac{v-v_\text{approx}}{v} \right| = \left| 1 - \frac{v_\text{approx}}{v} \right|,$
and the percent error is
$\delta = 100\times\eta = 100\times\frac{\epsilon}{|v|} = 100\times\left| \frac{v-v_\text{approx}}{v} \right|.$
In words, the absolute error is the magnitude of the difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value. The percent error is the relative error expressed in terms of per 100.
### Generalizations
These definitions can be extended to the case when $v$ and $v_{\text{approx}}$ are n-dimensional vectors, by replacing the absolute value with an n-norm.[1]
## Examples
As an example, if the exact value is 50 and the approximation is 49.9, then the absolute error is 0.1 and the relative error is 0.1/50 = 0.002 = 0.2%. Another example would be if you measured a beaker and read 5mL. The correct reading would have been 6mL. This means that your percent error would be 16.67%.
## Uses of relative error
The relative error is often used to compare approximations of numbers of widely differing size; for example, approximating the number 1,000 with an absolute error of 3 is, in most applications, much worse than approximating the number 1,000,000 with an absolute error of 3; in the first case the relative error is 0.003 and in the second it is only 0.000003.
There are two features of relative error that should be kept in mind. Firstly, relative error is undefined when the true value is zero as it appears in the denominator (see below). Secondly, relative error only makes sense when measured on a ratio scale, (i.e. a scale which has a true meaningful zero), otherwise it would be sensitive to the measurement units . For example, when an absolute error in a temperature measurement given in Celsius is 1° and the true value is 2 °C, the relative error is 0.5 and the percent error is 50%. For this same case, when the temperature is given in Kelvin, the same 1° absolute error with the same true value of 275.15° K gives a relative error of 3.63e-3 and a percent error of only 0.363%. Celsius temperature is measured on an interval scale, whereas the Kelvin scale has a true zero and so is a ratio scale.<|endoftext|>
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# 7/21/2017 Abel's test - Wikipedia
Abel's test
In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The
test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test one is used with
series of real numbers, and the other is used with power series in complex analysis. Abel's uniform convergence test is a
criterion for the uniform convergence of a series of functions dependent on parameters.
Contents
1 Abel's test in real analysis
2 Abel's test in complex analysis
3 Abel's uniform convergence test
4 Notes
5 References
## Abel's test in real analysis
Suppose the following statements are true:
1. is a convergent series,
2. {bn} is a monotone sequence, and
3. {bn} is bounded.
## Then is also convergent.
It is important to understand that this test is mainly pertinent and useful in the context of non absolutely convergent series .
For absolutely convergent series, this theorem, albeit true, is almost self evident.
## Abel's test in complex analysis
A closely related convergence test, also known as Abel's test, can often be used to establish the convergence of a power series on
the boundary of its circle of convergence. Specifically, Abel's test states that if a sequence of positive real numbers is
decreasing monotonically (or at least that for all n greater than some natural number m, we have ) with
## then the power series
converges everywhere on the closed unit circle, except when z = 1. Abel's test cannot be applied when z = 1, so convergence at
that single point must be investigated separately. Notice that Abel's test implies in particular that the radius of convergence is at
least 1. It can also be applied to a power series with radius of convergence R 1 by a simple change of variables = z/R.[1]
Notice that Abel's test is a generalization of the Leibniz Criterion by taking z = 1.
Proof of Abel's test: Suppose that z is a point on the unit circle, z 1. For each , we define
https://en.wikipedia.org/wiki/Abel%27s_test 1/3
7/21/2017 Abel's test - Wikipedia
## By multiplying this function by (1 z), we obtain
The first summand is constant, the second converges uniformly to zero (since by assumption the sequence converges to
zero). It only remains to show that the series converges. We will show this by showing that it even converges absolutely:
## where the last sum is a converging telescoping sum. It should
be noted that the absolute value vanished because the sequence is decreasing by assumption.
Hence, the sequence converges (even uniformly) on the closed unit disc. If , we may divide by (1 z) and
obtain the result.
## Abel's uniform convergence test
Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions or an improper integration of
functions dependent on parameters. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the
proof relies on the same technique of summation by parts.
The test is as follows. Let {gn} be a uniformly bounded sequence of real-valued continuous functions on a set E such that
gn+1(x) gn(x) for all x E and positive integers n, and let {n} be a sequence of real-valued functions such that the series n(x)
converges uniformly on E. Then n(x)gn(x) converges uniformly on E.
Notes
1. (Moretti, 1964, p. 91)
References
Gino Moretti, Functions of a Complex Variable, Prentice-Hall, Inc., 1964
Apostol, Tom M. (1974), Mathematical analysis (2nd ed.), Addison-Wesley, ISBN 978-0-201-00288-1
Weisstein, Eric W. "Abel's uniform convergence test" (http://mathworld.wolfram.com/AbelsUniformConvergenceTest.htm
l). MathWorld.
Proof (for real series) at PlanetMath.org (https://web.archive.org/web/20070312202311/http://planetmath.org/encyclopedi
a/ProofOfAbelsTestForConvergence.html)<|endoftext|>
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Update all PDFs
# Operations on the number line
Alignments to Content Standards: 7.NS.A.1
A number line is shown below. The numbers $0$ and $1$ are marked on the line, as are two other numbers $a$ and $b$.
Which of the following numbers is negative? Choose all that apply. Explain your reasoning.
1. $a - 1$
2. $a - 2$
3. $-b$
4. $a + b$
5. $a - b$
6. $ab + 1$
## IM Commentary
There is a distinction in the Common Core State Standards between a fraction and a rational number. Fractions are always positive, and when thinking of the symbol $\frac{a}{b}$ as a fraction, it is possible to interpret it as $a$ equal-sized pieces where $b$ pieces make one whole. The rational numbers are the set of fractions taken together with their opposites: understanding rational numbers requires understanding both fractions and signed numbers. The standard 7.NS.1 signals a significant shift from working exclusively with positive numbers to working with signed numbers. The focus of this task is on the nature of signed numbers rather than the "part-whole" interpretation of fractions.
The purpose of this task is to help solidify students' understanding of signed numbers as points on a number line and to understand the geometric interpretation of adding and subtracting signed numbers. This task (like all tasks featured on the Illustrative Mathematics website) assumes that the number line is drawn to scale.
## Solution
1. $a$ is greater than $1$, so $a - 1$ is positive.
2. The distance between $a$ and $1$ appears to be less than the distance between $1$ and $0$, so it looks like $a$ is less than $2$. Thus $a-2$ is negative.
3. $b$ is negative, so $-b$ is positive.
4. The distance between $a$ and $0$ appears to be less than the distance between $b$ and $0$, so it looks like $|a|$ is less than $|b|$. Since $b$ is negative and a is positive, $a+b$ is negative.
5. $a-b$ = $a+ -b$. Since $b$ is negative, $-b$ is positive. $a$ is also positive. Thus, $a-b$ is positive.
6. Since $|a|$ and $|b|$ are both greater than $1$, $|ab|$ is also greater than 1 (this builds on the intuition students gained in fifth grade as in 5.NF.5). $ab$ is negative since $a$ is positive and $b$ is negative. Thus, $ab+1$ is negative.
#### Ashli says:
almost 5 years
I'm curious how revisiting a task like this a week later but with students marking off their own a, b, and 1 values would go. perhaps give blank number lines then trade and answer the same questions as the ones here. Perhaps as they develop more number sense turn the question around and have them place down a, b, 1 so that specific combinations are negative while others are positive. Brownie points to any teacher that makes this happen and reports back :)
#### smckj says:
I think this question should state that the comparative distances are accurate. Otherwise none of the solutions could be proven to be negative. If CCSS are supposed to help students think mathematically, they should reflect mathematical ideas including the concept that one can't make decisions(e.g., if an angle is 90 degrees) based on how something appears.
#### Emily Abshier says:
almost 5 years
Thanks for the excellent thoughts. Because if this, I was prepared for those conversations when I did this in class. I went ahead and told students they could assume distances were what they looked like, but also that any student who felt that he was advanced, and wanted a 4 (rubric-grading) needed to explain any such assumption he made. It allowed advanced students to show me so much more of what they knew while other students could still master the task. An advanced student might say, "negative as long as m is longer than n." Another student might just assume that fact and draw an arrow of length m attached to n.<|endoftext|>
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# RRB Group D Mock Test 2018: Free Online Practice Test – Attempt Now!
Sep 13, 2018 13:00 IST
RRB Group D 2018: Mock Test
RRB Group D Mock Test: Mock Test for Railway Group D Recruitment Exam 2018 is available here. In this article, you will get two Mock Tests for RRB Group D exam 2018. One Mock Test is provided by Railway Recruitment Board (RRB) and other by Jagran Josh. To access Railway Group D Mock Test by Railway Recruitment Board you need to click on the link given at the end of this article.
RRB Group D Mock Test by Jagranjosh.com is a collection of 100 most important questions which are expected to be asked in RRB Group D Recruitment Exam 2018. Detailed solutions of each and every question are also available here
## RRB Group D Mock Test Link (by Railway Recruitment Board)
Candidates preparing for this exam are advised to attempt both RRB Group D mock tests to prepare well for the exam. Candidates are also advised to learn latest syllabus and exam pattern of RRB Group D Recruitment Exam 2018-19.
RRB Group D Exam 2018: Important Dates, Syllabus, Exam Pattern & More
## Mock Test for RRB Group D Recruitment exam by Jagran Josh is given below
(Important: Candidates should note that in the exams questions from all the sections can be asked in any random manner and there is no fixed pattern. Here we have provided section-wise questions for the sake of simplicity)
### RRB Group D Mock Test 2018 (by Jagran Josh)
RRB Group D Mock Test 2018: Mathematics
Question:
Out of 11 men, 10 men spent Rs. 35 each for their meals. The 11th one spent Rs. 40 more than the average expenditure of all the nine. The total money spent by all of them was:
(a) Rs. 492.50
(b) Rs. 497.50
(c) Rs. 429
(d) Rs. 498.50
Explanation: Let the average expenditure be Rs. x Then,
11x = 10 × 35 + (x + 40) or 11x = x + 390 or 10x = 390 or x = 39
Total money spent = 11x = Rs. (11 × 39) = Rs 429
Question:
The profit percentage of X and Y is same on selling the articles at Rs. 3600 each but X calculates his profit on the selling price while Y calculates it correctly on the cost price which is equal to 20%. What is the difference in their profits?
(a) Rs.260
(b) Rs.160
(c) Rs.140
(d) Rs.120
Explanation: When profit calculate on S.P
Then Profit = 20% of 3600 = 720
When Profit calculate on CP = (x)
x + x/5 = 3600
x = 3000
Profit = 600
Required difference = Rs. 120
Question:
How many numbers are divisible by 275 between 1000 and 6000?
(a) 16
(b) 18
(c) 14
(d) 20
Explanation: Numbers (n) which are divisible by 275 between 1000 and 6000:
1100, 1375, _____, 5775
First Term, a = 1100
Common difference, d = 275
Last Term, Tn = 5775
Tn = a + (n ‒ 1) d
5775 = 1100 + (n ‒ 1) 275
4675 = (n ‒ 1) 275
n ‒ 1 = 17
Therefore, n = 18.
Question:
Evaluate cos 40o /sin 50o
(a) ‒1
(b) 1
(c) 2
(d) 0
Explanation:
Question:
A definite length of wire is used to make a square, an equilateral triangle and a circle such that their areas are equal. Which of the following will have a maximum wire leftover?
(a) Triangle
(b) Circle
(c) Square
(d) Equal for all
Explanation:
Question:
If 40% of a number is equal to two-third of another number, what is the ratio of first number to the second number?
(a) 4:7
(b) 5:3
(c) 1:2
(d) 8:9
Explanation:
Question:
The Simple Interest on Rs.5000 in 4 years @ y% p.a. equals the Simple Interest on Rs.8000 @ 10% p.a. in 2 years. Find the value of y.
(a) 4%
(b) 16%
(c) 32%
(d) 8%
Explanation:
Question:
A work is done by three people A, B and C. A alone takes 20 hours to complete a single product but B and C working together takes 8 hours, for the completion of the same product. If all of them worked together and completed 28 products, then how many hours have they worked?
(a) 160 hrs
(b) 128 hrs
(c) 120 hrs
(d) 154 hrs
Explanation: 1/A = 1/20 and 1/B + 1/C = 1/8 (Given)
1/A + 1/B + 1/C = 1/8 + 1/20 = 7/40
In 40 hours, working together they will complete 7 products.
Thus in 160 hours they will complete 28 products.
Question:
If x distance is covered at speed and half of this distance is covered in double the time. Then find the ratio of the two speeds.
(a) 2 : 1
(b) 4 : 1
(c) 3 : 1
(d) 1 : 1
Explanation:
Question:
The ages of Rahul and Sumit are in the ratio of 3: 7 respectively. After 5 years the ratio of their ages will be 1: 2. What is the difference in their ages?
(a) 2 years
(b) 8 years
(c) 20 years
(d) 12 years
Explanation: Let the ages of Rahul and Sumit be 3x and 7x respectively.
According to question,
Difference in their ages = 7x – 3x = 4x = 20 years.
Question:
A tap can fill an empty tank in 12 hours and another tap can empty half the tank in 10 hours. If both the taps are opened simultaneously, how long would it take for empty tank to be filled to its capacity?
(a) 30 hours
(b) 20 hours
(c) 15 hours
(d) 12 hours
Explanation: Post of the tank filled in 1 hour
Tank will be filled in 15 hrs.
Question:
402, 1115, 2541, 4680, ?
(a) 6666
(b) 7532
(c) 7436
(d) 7246
Explanation: 402 + 713 =1115
1115 + (2 × 713) = 2541
2541 + (3 × 713) = 4680
4680 + (4 × 713) = 7532
Question:
Find the value of b in the figure, if two straight lines AB and CD intersect each other O. (given ∠AOP = 75°)
(a) 22
(b) 21
(c) 35
(d) 24
Explanation: Since OC and OD are in the same line.
So, ∠AOC + ∠AOP + ∠POD = 180°
⇒ 4b° + 75° + b° = 180°
⇒ 5b° + 75° = 180°
⇒ 5b° = 105°
⇒ b = 21°
Question:
A certain sum yields Rs 10110 in 6 years and Rs 6740 in 3 years on compound interest. What is the sum?
a) Rs 4500
b) Rs 4493
c) Rs 4490
d) Rs 4498
Explanation:
Let P be the sum to be found.
In 3 years, P(1 + r/100)3= 6740 ------ (1)
And, In 6 years, P(1 + r/100)6= 10110------ (2)
Dividing both, we get,
(1 + r/100)3 = 10110/6740 = 3/2
Substituting this value in – (1)
P×(3/2) = 6740
P = 4493.33
Question:
In the figure LM || AB || NO. Find the value of ∠ANO?
(a) 15°
(b) 35°
(c) 25°
(d) 45°
Question:
The area of a circle having radius r is equal to that of a right triangle given that the base of the triangle is 4 times the radius. Find the ratio of altitude of triangle to the radius of circle.
(a) π∶2
(b) 1∶π
(c) 2∶ π
(d) None of these
Explanation:
Question:
The minimum value of 4sin2θ + 5cos2θ:
(a) 5
(b) 4
(c) -5
(d) 3
Explanation:
4sin2θ + 5cos2θ
4sin2θ + 4cos2θ + cos2θ
4(sin2θ + cos2θ) + cos2θ
4 + cos2θ
Minimum value of cosθ = ‒ 1
Required minimum value = 4 + 1 = 5
Question:
Two candidates participate in election. 10% votes did not cast their vote and 20% votes found invalid. Winner candidate got 75% of the valid votes and won by 10800 votes. Find the total number of votes?
(a) 27750
(b) 26500
(c) 30,000
(d) 29000
Explanation: Total no. of votes = 100%
Valid votes = 90/100 × 80 = 72%
So, difference = 36% votes = 10800
100% = 10800/36 × 100 = 30,000 votes.
Question:
Simplify the following expression:
Explanation:
Question:
If (x + 1/x) = 4, then find the value of x3 ‒ 1/x3
(a) 17√14
(b) 14√14
(c) √14
(d) 42√14
Explanation:
Question:
Find the value of x and y respectively from the figure given below where ∠c=2y.
(a) 120°, 40°
(b) 80°, 40°
(c) 40°, 120°
(d) 40°, 80°
Question:
81.3 × 40.6 × 160.2 = 2?
(a) 2.1
(b) 3.8
(c) 5.9
(d) 4.7
Explanation:
81.3 × 40.6 × 160.2
= 23(1.3) × 22(0.6) × 24(0.2)
= 2(3.9+1.2+0.8) = 25.9 = ? = 5.9
Question:
Find the perimeter of an isosceles triangle with ∠P = 90° and area 32 sq. cm.
(a) 16+8√2
(b) 16+32√2
(c) 8+2√2
(d) 4+√3
Question:
Find out the average of the remaining two numbers if the average of six numbers is 5. The average of two of them is 3.5, while the average of the other two is 3.8.?
(a) 5.5
(b) 4.6
(c) 7.7
(d) 6.8
Explanation: Sum of the remaining two numbers = (5 × 6) – [(3.5 × 2) + (3.8 × 2)]
= 30 – (7 + 7.6) = 30 – 14.6 = 15.4
So, required average = 15.4/2 = 7.7
Question:
If a + b + c = 6, a2 + b2 + c2 = 14 and a3 + b3 + c3 = 36, then value of abc is:
(a) 3
(b) 5
(c) 6
(d) 12
Explanation: a + b + c = 6
a 2 + b2 + c2 = 14
a3 + b3 + c3 = 36
Put value as
a =1, b = 2,c = 3
1+2+3 = 6
1+4+9 = 14
1+8+27 = 36
So, abc= 1×2×3 = 6
RRB Group D Mock Test 2018: General Intelligence & Reasoning
Question:
In a certain code language, ‘pen pencil’ is written as '\$%' and 'eraser sharpener' is written as '@#' and ‘pencil eraser’ is written as ‘\$@’. Then, what is the code for ‘pen’?
(a) #
(b) \$
(c) @
(d) %
Explanation: Here, in the first and third statements, the common code symbol is \$ and the common word is ‘pencil. So, \$ means pencil. Thus is the first statement % means pen.
Question:
Choose the correct mirror-image of the figure (X) from amongst the four alternatives (A), (B), (C) and (D) given along with it.
(a) a
(b) b
(c) c
(d) d
Explanation: The small triangle should be at right top and the diamond should be at right bottom.
Question:
Out of the four options given below, three are of a kind while one does not belong to the group. Choose the one which is unlike the others.
(a) Just
(b) Fair
(c) Favorable
(d) Equitable
Explanation:
Fair, just and equitable are all synonyms meaning impartial. Favorable means expressing approval.
Question:
In a certain code, WORKABLE is written as VOYZPILD, how will BLUNDERS be written in same code?
(a) CMVOEST
(b) TSEOVMC
(c) YOFMWVIH
(d) HIVWMFOY
Explanation: WORKABLE => DLIPZYOV (opposite letters in the table below) => VOYZPILD (reversing the order)
BLUNDERS => YOFMWVIH (opposite letters) => HIVWMFOY (reversing the order)
Question:
If P means 'add to', V means 'multiply by', M means 'subtract from' and L means 'divide by' then
30 L 2 P 3 V 6 M 5 = ?
(a) 18
(b) 28
(c) 31
(d) 103
Explanation: Using Correct Symbols, We have:
Given expression = 30 / 2 + 3 x 6 - 5 = 15 + 18 - 5 = 28
Question:
D, the son-in-law of B, is the brother-in-law of A who is the brother of C. How is A related to B?
(a) Brother
(b) Son
(c) Father
(d) None of these
Explanation: D is the brother-in-law of A, who is brother of C. C is wife of D. D is the son-in-law of B. B Can either father or mother of A.
Question:
'Fish’ is related to 'Pisciculture' in the same way as 'Bees' is related to:
(a) Horticulture
(b) Apiculture
(c) Sericulture
(d) Viticulture
Explanation: - Fish farming is called Pisciculture, Bee Farming is called Apiculture.
Question:
Choose the alternative which closely resembles the mirror image of the given figure
Explanation: Correct answer is option (b)
Question:
Arrange the following in a logical order:
1. Table
2. Tree
3. Wood
4. Seed
5. Plant
(a) 4, 5, 3, 2, 1
(b) 4, 5, 2, 3, 1
(c) 1, 3, 2, 4, 5
(d) 1, 2, 3, 4, 5
Explanation: All the given words are related to the making process of 'furniture', but the intensity increases in the order - Seed, Plant, Tree, Wood, and Table. Hence, the correct order is 4, 5, 2, 3, 1.
Question:
Given question have some statements and some conclusions. Choose the conclusion that logically follows:
Statements: Some red are green. Some green are yellow. No yellow is red.
Conclusions:
Some green that are yellow are red.
Some red that are green are yellow.
(a) If only conclusion I follows.
(b) If only conclusion II follows.
(c) If either conclusion I or conclusion II follows.
(d) If neither conclusion I nor conclusion II follows.
Explanation:
Question:
Choose the alternative which is closely resembles the mirror image of the given combination.
1965 INDOPAK
Explanation: Correct answer is option (d)
Question:
Out of the four options given below, four are of a kind while one does not belong to the group. Choose the one which is unlike the others.
(a) APBQ
(b) CRDT
(c) EUFV
(d) GWHX
Explanation: In all other, the first and the third as well as the second and the fourth are consecutive letters in the English alphabet.
Question:
Manoj went to the movie nine days ago. He goes to the movies only Thursday. What day of the week today?
(a) Tuesday
(b) Thursday
(c) Saturday
(d) Sunday
Explanation: Clearly, nine days ago, it was Thursday. So, today is Saturday.
Question:
What should replace ‘?’ in the figure
(a) 53 and Z
(b) 51 and Y
(c) 50 and Y
(d) 52 and U
Explanation: Numerical series:
In fig 1, 9 x 9 – 5 x 3 = 66, similarly figure 2 and 3 to give 51.
Alphabet series:
All the alphabets beginning from figure 1 are evenly placed after 5 positions of the preceding alphabet.
Like,
A (+5) → F (+5) → K ---------------------------T (+5) → Y
Question:
The following question has one statement followed by two conclusions. Read and understand them carefully and then select your answer option as given below.
Statement: Bureaucrats marry only intelligent girls. Gudiya is very intelligent.
Conclusions:
Gudiya will marry a bureaucrat.
Gudiya will not marry a bureaucrat.
(a) Only conclusion I follows.
(b) Only conclusion II follows.
(c) Neither conclusion I nor II follow.
(d) Either conclusion I or II follows.
Explanation: The data does not mention whether all intelligent girls are married to bureaucrats. So, either I or II may follow.
Question:
If ‘+’ stands for ‘-‘ , ‘-‘ stands for ‘x’, ’x’ stands for ‘÷’and ‘÷’stands for ‘+’ .Then, what is the value of 56x7÷13-11+15-8÷2-7?
(a) 30
(b) 60
(c) 95
(d) 45
Explanation: Changing the symbols as given in the problem the above expression is
= 56 ÷ 7 + 13 x 11 – 15 x 8 + 2 x 7
We get, 8 + 143 – 120 + 14 = 165 – 120 = 45
Question:
In a certain code language ‘LIBERAL’ is coded as ‘MJCFSBM’, then how is the word ‘REDUCTION’ coded in that language?
(a) SFEVDUJPO
(b) EDCTBSHNM
(c) SFDUCTJPO
(d) SFEVCTJPO
Explanation:
Word: L I B E R A L
Pattern: +1+1+1+1+1+1+1
Code: M J C F S B M
Similarly, the code for REDUCTION is SFEVDUJPO.
Question:
PQUT: VWAZ:: LMQP: __? __
(a) RSUT
(b) RSYX
(c) RSWV
(d) RSVW
Explanation: The letters in the second part are moved six times to their right as explained below:
P → q, r, s, t, u, V
Q → r, s, t, u, v, W
U → v, w, x, y, z, A
T → u, v, w, x, y, Z
Similarly,
L → m, n, o, p, q, R
M → n, o, p, q, r, S
Q → r, s, t, u, v, W
P → q, r, s, t, u, V
Question:
'Teacher' is related to 'Teaching' in the same way as 'Doctor' is related to:
(a) Medicine
(b) Prescription
(c) Examination
(d) Healing
Explanation: - A doctor “examines” a patient, whereas a teacher “teaches” students.
Question:
In a certain code, ‘commit also make policy’ is written as ‘%e4 !y6 #t6 @o4’; ‘policy craze anger mobile’ is written as ‘!y6 @r5 %e6 #e5’; ‘allow mild course prize’ is written as ‘!e5 %d4 #e6 @w5’, and ‘craze manner pump artist’ is written as ‘%r6 #e5 !p4 @t6’. Then, what does ‘#e6 #e5 @04’ stand for?
(a) artist mild craze
(b) also make course
(c) craze also course
(d) commit course mobile
Explanation:
#e6 – course
#e5 – craze
@04 – also
Question:
Among the five given figure, four figures follow the same pattern and one figure is different from these four. Choose the figure which is different from the rest.
Explanation: In all other figures, the three squares have the same halves shaded.
Question:
Two positions of a cube are as follows.
Point out which number will be on the top if number 3 is at the bottom.
(a) 1
(b) 5
(c) 6
(d) 2
Explanation: The faces adjacent to 3 are 1, 4, 2 and 5. So, 6 is on the top when 3 is at the bottom.
Question:
What will come in place of the question mark in the following number series?
114 115 107 134 70 ?
(a) 140
(b) 35
(c) 195
(d) 150
Explanation: 114 115 107 134 70 ?
+13 –23 +33 –43 +53
Hence, question mark (?) will be replaced by 195.
Question:
Which answer figure completes the question figure?
Explanation:
Question:
One morning at 6 O’ clock, Vineet started walking with his back towards the Sun. Then, he turned towards left, walked straight and then turned towards right and walked straight. Then, he again turned towards left. Now, in which direction is he facing?
(a) North
(b) South
(c) West
(d) East
Explanation: According to the question, the direction diagram will be as follows:
So, Vineet is facing towards South.
Question:
If U + V means U is the brother of V, W – X means W is the father of S, X ÷ Y means X is the sister of Y, Y × Z means Z is the mother of Y. Which of the following means that N is the mother of O?
a) L + M ÷ N × O
b) L – M × O ÷ P
c) M + L ÷ O × N
d) N ÷ M × L ÷ O
Explanation: M + L ÷ O × N means M is the brother of L and L is the sister of O and N is the mother of O.
Question:
Arrange the given words in alphabetical order and tick the one that comes in the last?
(a) Efflorescent
(b) Entry
(c) Entreat
(d) Ensure
Explanation: Efflorescent, Ensure, Entreat, Entry.
Question:
Statements: All purses are mobile. All mobile are phones. Some mobile are rings. No ring is purse.
Which Conclusion is right?
(a) Some purse is ring.
(b) No mobile is ring.
(c) Some phones are purse.
(d) All rings are mobile.
Explanation:
Question:
A man starts walking towards Northeast diagonally from the left corner of a square plot and turns right after reaching the center of the plot. After some distance he again turns right and continues walking. Which direction is he facing now?
(a) East
(b) West
(c) South
(d) North
Explanation:
Question:
Select the missing number from the given responses.
(a) 45
(b) 60
(c) 70
(d) None of these
Explanation: In the first box, (2 × 4 × 1 = 8) similarly in the second and third box. The number in place of question mark will be (6 × -2 × -5 = 60)
General Science
Question:
By which process can sea water be purified?
(a) Evaporation
(b) Fractional Distillation
(c) Filtration
(d) Distillation
Explanation: Distillation is a procedure by which two liquids with different boiling points can be separated or it is a process in which the components of a substance or liquid mixture are separated by heating it to a certain temperature and condensing the resulting vapours.
Question:
Boyle's law relates to which state of matter?
(a) Solid state
(b) Liquid stage
(c) Gaseous state
(d) None of the above
Explanation: Boyle's law, the pressure of certain quantities of gas on stable heat is inversely proportional to its volume.
Question:
What type of artificial propagation method is used in sugarcane, banana and cactus trees?
(a) Layering
(b) Grafting
(c) Cutting
(d) Regeneration
A small part of a plant which is removed by making a cut with a sharp knife is called cutting. And above mentioned plants are grown by means of cutting.
Question:
Fungi are plants that lack:
(a) Oxygen
(b) Carbon dioxide
(c) Chlorophyll
(d) None of these
We know plants prepare their own food and so are known as autotrophs. With the help of photosynthesis they make food in which they produce glucose from carbon dioxide and sunlight. Also, oxygen is released by plants which are further used by humans and other animals. But Fungi lack chlorophyll and do not engage in photosynthesis.
Question:
Due to which phenomena the stick if immersed in water appears to be bent?
(a) Reflection
(b) Dispersion
(c) Refraction
(d) Scattering
Question:
In projectors which lenses are used?
(a) Convex lens
(b) Concave lens
(c) Bipolar lens
(d) Both (a) and (b)
Question:
Which gas is safe and an effective extinguisher for all confined fires?
(a) Nitrogen dioxide
(b) Carbon dioxide
(c) Sulphur dioxide
(d) Nitrous Oxide
Question:
The cooking gas (LPG) mainly consists of:
(a) Butane
(b) Ethene
(c) Ethyne
(d) Propene
Question:
What will be the litmus test if the solution is basic?
(a) Red litmus will turn to blue
(b) Blue litmus will turn to red
(c) No change in colour
(d) It will change into orange pink.
Question:
Name an enzyme that digests fat?
(a) Lipase
(b) Sucrase
(c) Maltase
(d) Fructose
Explanation: Fats are lipids and one of the three major food groups needed for proper nutrition. The digestive enzyme lipase is required to digest fat. It hydrolyzes lipids, the ester bonds in triglycerides to form fatty acids and glycerol.
Question:
In what way iodine can be separated from a mixture of potassium chloride and iodine?
(a) Filtration
(b) Sedimentation
(c) Distillation
(d) Sublimation
Explanation: Sublimation is a chemical process where solid is converted into a gas without going through a liquid stage. At standard, atmospheric pressure, a few solids which will sublime are iodine, carbon dioxide, naphthalene and arsenic.
Question:
Name an inert diatomic gas which is neither combustible nor helps in combustion?
(a) Carbon Dioxide
(b) Hydrogen
(c) Nitrogen
(d) None of the above
Question:
The non-metal which is liquid at room temperature is:
(a) Mercury
(b) Bromine
(c) Carbon
(d) Helium
Question:
The magnetic field lines outside a bar magnet:
(a) Originate from the South pole and end at its North Pole
(b) Originate from the North pole and end at its East Pole
(c) Originate from the North Pole and end at its South Pole
(d) Originate from the South pole and end at its West Pole
Question:
A good fuel is one possess:
(a) High calorific value and low ignition temperature
(b) Low calorific value and low ignition temperature
(c) High calorific value and moderate ignition temperature
(d) Low calorific value and moderate ignition temperature
Question:
In a hydroelectric power plant more electrical power can be generated if water falls from a greater height because:
(a) Its temperature increases.
(b) Larger amount of potential energy is converted into kinetic energy.
(c) The electricity content water increases with height.
(d) More water molecules dissociate into ions.
Question:
The radiation present in the sunlight that gives us the feeling of hotness is
(b) Infra-red
(c) Red
(d) Ultra-violet
Question:
Rainbow is formed due to the combination of
(a) refraction and absorption
(b) dispersion and focusing
(c) dispersion and total internal reflection
(d) refraction and scattering
A rainbow is formed due to a combination of dispersion and total internal reflection.
Question:
The oxidant which is used as an antiseptic is:
(a) KBrO3
(b) KMnO4
(c) CrO3
(d) KNO3
Explanation: Potassium permanganate (KMnO4) is an oxidising agent with disinfectant and astringent properties due to which it is used as an antiseptic.
Question:
The rise of sea-water during high tide is caused by the gravitational pull of the:
(a) Sun
(b) Earth
(c) Moon
(d) Mars
Question:
The gas emancipating through paddy field is :
(a) Ethane
(b) Methane
(c) Nitrogen
(d) All of these
Question:
What is the speed of sound in air?
(a) 330 m/s
(b) 332 m/s
(c) 334 m/s
(d) 336 m/s
The speed of Sound in Air (0o C) is 332 m/s and in Air (20o C) is 343 m/s.
Question:
Which of the following is an air-borne disease?
(a) Measles
(b) Typhoid
(c) Pink eye
(d) None of the above
Airborne diseases are the infections spread by airborne transmissions including Chickenpox, Anthrax, Influenza, Measles, Smallpox, Cryptococcosis and Tuberculosis.
Question:
In poorly ventilated buildings which one of the following inert gases can be accumulated?
(a) Helium
(b) Neon
(c) Argon
Question:
Which technique is used to separate the substances from a mixture?
(a) Chromatography
(b) Racing Forging
(c) Assembling
(d) None of the above
Explanation: Chromatography is one of the most important technique for the separation of a mixture by passing it in solution or suspension through a medium in which the components move at different rates. Mixtures that are suitable for separation by chromatography include inks, dyes and colouring agents in food.
RRB Group D Mock Test 2018: General awareness & Current Affairs
Question:
HRD Ministry launches Atal rankings for which institutions?
(a) Secondary schools
(b) Primary Schools
(c) Higher Education Institutions
(d) Nursery
The Union Human Resource Development Minister Prakash Javadekar has launched the Innovation Cell and Atal Ranking of Institutions on Innovation Achievements (ARIIA) in New Delhi. The development aims to promote a culture of innovation and research in higher education.
Question:
Which city is hosting Organization of the Petroleum Exporting Countries (OPEC) Forum 2018?
(a) Tehran
(d) Vienna
The Organization of the Petroleum Exporting Countries (OPEC) Forum 2018 began in Vienna, Austria on June 20, 2018. India’s Petroleum Minister Dharmendra Pradhan attended the forum.
Question:
President Ram Nath Kovind recently approved the amendments to the Minimum Wages Act of which state?
(a) Bihar
(b) Delhi
The Minimum Wages (Delhi) Amendment Act, 2017 was officially notified on May 4, 2018 following the approval of the Bill by President Ram Nath Kovind on April 26, 2018. This new legislation amends the Minimum Wages Act, 1948 and was passed by the Delhi Assembly on August 10, 2017. The bill was first tabled in 2015.
Question:
When was International Labour Day observed?
(a) April 30
(b) May 2
(c) May 1
(d) April 29
The International Labour Day was observed on May 1, 2018. On the occasion, several programms were organised across India to acknowledge the contribution of the labourers in development of the society. The Union Ministry of Labor and Employment also awarded various organizations that are working for the welfare of workers and labourers.
Question:
When is the World Liver Day observed every year?
(a) April 16
(b) April 17
(c) April 18
(d) April 19
The World Liver Day (WLD) is observed every year on April 19 to build awareness and understanding how important the liver is for our body and how liver ailments can be treated. Liver is a unique organ in human body and has a special capacity of regeneration. It has been seen that even 75% of liver can be safely removed without any untoward consequences because of this capacity of regeneration.
Question:
Which state government will be launching an app-based taxi service?
(a) Maharashtra
(b) Kerala
(c) Delhi
(d) Goa
The state government of Goa has decided to launch its own app-based taxi service to some key tourist destinations with a hope to solve the state’s problematic tourist taxi issue that has been plaguing its tourism sector for years.
Question:
Who has been appointed as the CEO of Ayushman Bharat mission?
(a) Bharat Mittal
(b) Indu Bhushan
(d) Vineet Jain
Indu Bhushan has been appointed as the Chief Executive Officer of the Union Government’s ambitious Ayushman Bharat National Health Protection Mission. Bhushan is currently the Director General of East Asia Department, Asian Development Bank in Manila, Philippines.
Question: ISRO launched which communication satellite on March 29?
(a) GSAT-6A
(b) GSAT-6 2B
(c) PSAT-4 E
(d) X-PAT 1
The Indian Space Research Organisation on March 29, 2018 launched communication satellite GSAT-6A from Sriharikota, around 105 km away from Chennai. The satellite will provide a platform for developing technologies such as demonstration of the 6-metre S-Band Unfurlable Antenna, hand-held ground terminals and network management techniques that could be useful in satellite-based mobile communication applications.
Question:
The Union Government has launched the first Indian Sign Language Dictionary comprising how many words?
a) 2000
b) 5000
c) 3000
d) 1500
The Union Minister for Social Justice and Empowerment, Thaawarchand Gehlot on March 23, 2 018 launched the first Indian Sign Language (ISL) dictionary containing 3000 words at a function in Delhi. The main aim of developing the dictionary is to remove the communication barriers between the deaf and hearing communities, as it is focused on providing more information in Indian sign language.
Question:
Who became the first Indian gymnast to win an individual medal at the Gymnastics World Cup that was held in Melbourne recently?
(a) Deepa Kumari
(b) Tejasvini Sharma
(c) Kripa Rani
(d) Aruna Buddha Reddy
Aruna Buddha Reddy scripted history in February 2018 by winning a bronze medal at the Gymnastics World Cup that was held in Melbourne, Australia. Reddy became the first Indian gymnast to bag an individual medal at the tournament.
Question: Who won the best actress award at the Oscar awards 2018?
(a) Frances McDormand
(b) Meryl Streep
(c) Jennifer Lawrence
(d) Sally Hawkins
Frances McDormand won the best actress award at the Oscar awards 2018 for the movie ‘Three Billboards Outside Ebbing, Missouri’. Other actresses who won nomination in the category included Sally Hawkins, Margot Robbie, Saoirse Ronan and Meryl Streep.
Question:
The Reserve Bank of India (RBI) recently launched the Ombudsman Scheme for which organisations?
(a) Non-Banking Financial Companies
(b) Banking Sector
(c) Mutual Funds
(d) Insurance Companies
The Reserve Bank of India (RBI) has launched Ombudsman Scheme for redressal of complaints against the Non-Banking Financial Companies (NBFCs). The scheme will provide a cost-free and expeditious complaint redressal mechanism relating to deficiency in the services by NBFCs covered under the scheme.
Question:
What was the theme of World Environment Day 2018?
(a) Beat Plastic Pollution
(b) No to Plastics
(c) Plastic: An Illness
(d) Plastic: Damaging World
The theme of 2018 World Environment Day was Beat Plastic Pollution.
Question:
Which nation is the latest to withdraw from the UN refugee programme?
(a) Tanzania
(b) Zambia
(c) Kenya
(d) Ghana
Tanzania has announced its withdrawal of United Nation’s “Comprehensive Refugee Response Framework” citing security reasons and lack of funds. Tanzania has long been considered as safe haven for refugees, particularly from conflict hit Democratic Republic of Congo (DRC) and Burundi.
Question:
What is the salary of President as per Union Budget 2018?
(a) Rs 5 lakh
(b) Rs 4 lakh
(c) Rs 3 lakh
(d) Rs 2 lakh
Emoluments will revised for the President to Rs 5 lakh, Rs 4 lakh for Vice-President and Rs 3.5 lakh per month for Governors. Emoluments to MPs will be refixed with effect from April 1, 2018.
Question:
Which Indian female boxer won gold at the India Open Boxing tournament in the 48kg weight category?
(a) Mary Kom
(b) Pinki Rani
(c) Lovlina Borgohain
(d) L Sarita Devi
The five-time World Champion and Olympic bronze medallist, Mary Kom on February 2, 2018 claimed the gold medal in the India Open International Boxing tournament at Thyagraj Indoor Stadium in New Delhi. Kom defeated Josie Gabuco from the Philippines in a split 4-1 decision to grab the gold in the women's 48kg weight category.
Question:
The World Economic Forum has ranked India at which position on the Inclusive Development Index 2018?
(a) 52
(b) 62
(c) 60
(d) 78
India has been ranked at the 62nd place among emerging economies on an Inclusive Development Index, released by World Economic Forum (WEF). India's position is much below its neighbours, China and Pakistan, which are ranked at 26th and 47th positions respectively.
Question:
What is the name of India’s first and fastest supercomputer?
(a) Ayush
(b) Digvijay
(c) Pragati
(d) Pratyush
Union Minister of Earth Sciences Harsh Vardhan dedicated to India, the nation’s first and fastest multi-petaflops supercomputer at the Indian Institute of Tropical Meteorology (IITM) in Pune on 8 January 2017. Named ‘Pratyush’, which means the Sun, the supercomputer will be a national facility for improving weather and climate forecasts and services under the umbrella of the Ministry of Earth Sciences (MoES).
Question:
India has risen to which rank in the latest ICC T20 rankings?
(a) Third
(b) Second
(c) First
(d) Fourth
Question:
Which scheme will be launched this year as per Budget 2018 to improve the Health Sector of the country?
(a) Rashtriya Swasthya Bima Yojana
(b) National Health Policy 2018
(c) Swasth Bharat
(d) National Health Protection Scheme
National Health Protection Scheme will be launched to cover 10 crore poor and vulnerable families. Under this, up to Rs 5 lakh will be provided to each family per year in secondary and tertiary care institutions. This scheme will have 50 crore beneficiaries.
### RRB Group D Mock Test Link 2018 (by Railway Recruitment Board)
Mock Test for Railways Group D post is also available on official websites of RRBs. The Mock Test is available in English, Marathi, Gujarati, Tamil, Malayalam, Telugu, Bengali, Punjabi, Assamese, Urdu, Kannada, Konkani, Manipuri, Hindi, Odia languages.
In order to attempt the RRB Group D Mock Test Online, candidates need to visit any of the official websites of RRBs or alternatively, they can click on the direct link given below
RRB Group D Mock Test Link 2018 (by Railway Recruitment Board)<|endoftext|>
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MathsClass
A blog about teaching and learning in a maths classroom.
@mathslinks — New post on MathsClass: Excel for Mathematics General mths.co/4242 29 Aug ago
Teaching Surface Area
Thursday, 25 November 2010 | 1 Comment
I like teaching surface area, I think it’s an interesting topic. Yet, I find kids struggle with the concept. Not understanding the basics of area and then getting over the prior knowledge of solids meaning volume are two aspects that cause some difficulty.
This is how I break down surface area for my low ability Year 8 class.
Lesson 1 – Introducing Surface Area
1. Investigate Surface Area using Centicubes (comparing with Volume).
Concepts: counting the faces of the centicubes rather than counting the cubes, reinforced by creating multiple solids with different surface areas using the same number of cubes, understand what is meant by a 3 × 2 × 2 rectangular prism. Layed out in “Making Solids: Investigation” (Page 1 of download below). This is based on an activity in the book Access to Stage 5.1 Maths.
1. From a net, make a solid and find the surface area.
Concepts: calculating area by counting squares, see that a rectangular prism is made up of six rectangles (3 pairs of congruent rectangles). Make net (Page 3 of download below). Print to card, and cut in half (no tabs on these nets, I use sticky tape to hold together).
1. Find the surface area by drawing a net and counting squares.
Similar to 2., but this time students draw their own net. Concepts: the drawing of a net from an image of a solid, understand how the edges of the rectangles join. Draw nets – Page 4 of download below.
1. Find the surface area from nets where the area of each face is shown.
Lesson 2 – Rectangular Prisms
1. Find the surface area of solids where the area of some faces shown (e.g. show the front, top and side of a rectangular prism).
2. Concepts: understand that the faces of a rectangular prism come in pairs.
3. Find the surface area of a cube given the length of a single edge.
1. Find the surface area of a rectangular prism given the length of 3 edges.
Concepts: determine which lengths are used for each face and that faces have a matching face so the area of a face can be doubled.
Lesson 3 – Triangular Prisms
1. Review area.
1. Students draw a 3cm square and find it’s area. Then, draw a diagonal and find the area of the resulting triangle(s).
2. Students draw a 4 by 5cm rectangle and find it’s area. Then, draw a diagonal and find the area of the resulting triangle(s).
1. Determine a means for finding the area of a triangle.
1. Make two triangular prisms.
Concepts: understand how a triangular prism is constructed from two triangles and 3 rectangles, understand how the side lengths of the triangles relate to the side lengths of the rectangles, see the common side length of all the rectangles. Use: Paper Model of a Triangular Prism
1. Show my triangular prism t-shirt box for which I’ve made peel off sides, peel off each side and form the net.
1. Complete some exercises given diagrams of triangular prisms with side lengths.
Lesson 4 – Smarties
1. Determine the surface area of a Smarties box
1. Design a Toblerone box
This work is licensed under a Creative Commons License (?).
Posted in • Lesson IdeaSurface AreaPrintableWorksheet | Short URL: http://mths.co/2047
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Comments
Nordin Zuber on 29 November 10 at 09:43 PM #
Got to love any activities that uses Smarties or Toblerone I’m doing one tomorrow to build a frequency histogram of the colour of the Smarties in the box.
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About
Simon Job — ninth year of teaching maths in a public high school in Western Sydney, Australia.
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Researchers at University West in Sweden are using nanoparticles in the heat-insulating surface layer that protects aircraft engines from heat. In tests, this increased the service life of the coating by 300%. The hope is that motors with the new layers will be in production within two years. The surface layer is sprayed on top of the metal components. Thanks to this extra layer, the engine is shielded from heat. The temperature can also be raised, which leads to increased efficiency, reduced emissions, and decreased fuel consumption.
The ceramic layer is subjected to great stress when the enormous changes in temperature make the material alternately expand and contract. Making the layer elastic is therefore important. The new layer has been tested thousands of times in thermal shock tests to simulate the temperature changes in an aircraft engine. The new coating layer lasts at least three times as long as a conventional layer, while it has low heat conduction abilities.
One of the most important issues for the researchers to solve is how they can monitor what happens to the structure of the coating over time, and to understand how the microstructure in the layer works.<|endoftext|>
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A group of scientists have been able to create a mouse using only male DNA. No female was involved in the creation of this animal apart from them being used as incubators for the all-male created offspring. The scientists used stem cells from two adult male mice and created embryos which were later placed into surrogate mothers.
Since the female genes are missing, the embryos had to be genetically modified in order to grow enough to be born. Sadly, these pups created only by using male DNA died a short few days after birth. On the other hand, some mice that used only their mothers DNA were able to reach adulthood and some even had offspring of their own.
Scientists performed these tests in order to determine why mammals only reproduce sexually, and when two parents of the opposite sex are present. Other vertebrates though are sometimes able to reproduce even when only one parent is present. This group so far contains sharks, turkeys, and snakes. In females of those species, there is a chance that an unfertilized egg produces fully functional offspring.
Until now, researchers were never able to achieve androgenesis, male-only reproduction in mammals. There has been a case where researchers were able to make a zebrafish with only a father’s DNA sample, but, until now, no such experiment worked on mammals.
In the most recent experiment, where only a mother’s DNA was used, the mouse pups were somewhat smaller than usual. This happens due to the fact that some genes that are inherited from the mother are imprinted, and imprinting causes some genes to be more or less active than usual.
To combat this, scientists used CRISPR to remove three imprinted regions near genes important when it comes to the production of an embryo. The pups that came from these genes grew up normally and achieved normal fertility.
For the all-male pups, the results are much less promising as only one percent of the 1,023 embryos produced pups. In these cases, the scientists had to cut out six regions of DNA in order to create genes that were able to produce a functioning embryo. But the pups were born too big and died soon after birth. Cutting an additional region created smaller pups, but they lived only 48 hours at best.
This research offers great insight into how the imprinted regions affect normal gene development. If scientists are able to better understand how this works they may even be able to save some endangered species. The last northern white rhinoceros died earlier this year and if we were able to produce offspring from only male specimens that might not have happened.
The process is expensive but well worth looking into.<|endoftext|>
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Preparing the cannon for firing started with placing a cartridge of black powder, which is gunpowder, made from paper, flannel or wool. How much powder was used varied based on the distance to the target, the size of the gun and what type of projectile is used. In order to get the cartridge to the bottom of the tube a pushed into the tube, breach, with a rammer (long stick).
A friction primer is used to ignite the black powder that is in the tube. The friction primer consists of a hollow tube that fits into the vent hole. At its top there is an opening through which a serrated wire can be inserted. The wire has a loop on it through which a lanyard, or rope, is attached.
When it is time to fire the cannon the rope on the friction primer is pulled. The serrated wire creates enough heat from friction to ignite the black powder that is in the primer tube. This fire ignites the black powder in the breech. The explosion that results from the breech powder being ignited propels the projectile out the end of the tube.
Sciencing Video Vault
Aiming the cannon is done by pointing the piece, the cannon, with a breech sight for the up and down elevation and the cannon was moved left or right for lateral positioning.<|endoftext|>
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Similarity
In geometry, two objects are similar (~) if the ratio of their corresponding sides are equal.
In the figures on the left:
Similar objects' corresponding angles are also equal. Figures could also be rotated, reflected, or translated, making similarities sometimes harder to identify.
Example 1. Find the values of the missing side lengths in the figure below.
We know that <ABC = <DEF and <ACB = <DFE, so <BAC = <EDF. Corresponding angles of both triangles are equal, so
Since corresponding side ratios are the same, we can set up equations and solve for the unknown lengths x and y.
A quadrilateral is a 2D figure with four edges and vertices. The sum of the interior angles is 360 degrees.
Example 2. The perimeter of the rectangle is 12. Find the length of the longer side.
Since the sum of all four sides is 12, we can set up an algebraic equation and solve for x.
The longer side is 2x, so its length is 2*2 = 4.
Example 3. Quadrilaterals ABCD and WXYZ are similar. Find the measure of angle X.
Since ABCD and WXYZ are similar, <X = <B. We can find angle B by subtracting angles A, B, and C from 360 degrees.
Therefore,
It is important to remember that the remaining three angles in WXYZ are also equal to their corresponding angles in ABCD.<|endoftext|>
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Factual Description (nonfiction), 16 words, Spanish Level aa (Grade K), Lexile BR50L
In the book Spring, students learn about some of the animals, plants, activities, and types of weather associated with spring. The simple repetitive text, supportive pictures, and use of the high-frequency word the support beginning readers.
Use the reading strategy of visualizing to understand text
Identify main idea and details
Discriminate initial /b/ sound
Identify initial consonant Bb
Grammar and Mechanics
Recognize and understand that nouns are naming words
Recognize and write the high-frequency word the
You may unsubscribe at any time.<|endoftext|>
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I thought this was going to be an easy one, and it would just be an extension of nouns. However, it seems pronouns are just as complicated.
Pronouns are essentially words that take the place of nouns. And from my understanding, their only purpose is to make the text more interesting.
Instead of writing ‘Sam wants to be a lawyer, therefore, Sam needs to go to law school’.
Sam wants to be a lawyer, therefore, he needs to go to law school’.
There are many different categories of pronoun, there are: personal, relative, subject and object, demonstrative, indefinite, reflexive, intensive, possessive, reciprocal and lastly interrogative.
So… let us start with the personal pronouns.
The personal pronouns are: I, me, you, he, she, her, him, it, we, us, they and them.
Despite the term ‘personal’ they do not have to refer to a person – what is it?
They are essentially the pronouns that are associated with ‘person’ in writing i.e. 1st, 2nd, 3rd.
These are used to connect relative clauses – which can be restrictive (provides essential information about the noun) and non-restrictive (can be left out without affecting the meaning of the clause) – to independent clauses. Relative clauses are used to identify the noun that came before them.
The relative pronouns are: which, that, whom, whose, who, when, what.
My writing, which is relatively poor, is improving.
Subject and object pronouns
Who and whom. When referring to a subject use the ‘who’ pronoun, and when referring to an object use whom. I will look at subject and object in another post. But in short, the object is acted upon by the subject.
To whom, should I send this letter?
Who will receive the letter?
A demonstrative shows distance as I spoke about in this post.
The demonstrative pronouns are: that, this, these, those.
This is used for singular items that are nearby, whereas, these not those (over there) is used for many items that are close.
Are used when you need to refer to something unspecific, one, none, other, some, anybody, everybody and no one.
Reflexive and intensive pronouns
reflexive pronouns are used when both the subject and object of a verb refer to the same person or thing. They have self or selves on the end. Himself, themselves etc
The writer set himself the task of writing about pronouns.
Intensive pronouns are more unnecessary as a category in my opinion; they are similar to reflexive pronouns but they do a different thing, they add emphasis.
I wrote the blog post myself. The ‘myself’ is unnecessary but it adds emphasis.
Not technically a pronoun but it is important. The antecedent is the noun that the pronoun refers to.
My girlfriend (antecedent) bakes me cakes, I love her (pronoun) for that.
its, his, her, our, their, My, your and whose
Seems obvious, and it is. Basically, they show possession
Absolute pronouns (mine, yours, his, hers, ours, theirs) can be substituted for the thing that belongs to the antecedent.
BloggerX is working on his blog post.
They are absolute because they stand alone and do not modify nouns.
Reciprocal pronouns are each other and one another
each other refers to two things, one another refers to multiple things.
These are used when two or more things are acting in the same way towards the other.
The blogger and commenters are talking to one another.
These, as the name suggest are used in questioning: whose, who, what, which
who wants to leave a like and follow?<|endoftext|>
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Ron Nichols, USDA NRCS North Carolina A farmer inspects the health of his soil through look, feel, and smell in Greensboro, North Carolina.
We have all heard the expression “cheaper than dirt.” But many experts disagree. Soil is a vital resource that the UN’s Food and Agriculture Organization (FAO) estimates contributes about USD $16.5 trillion in ecosystem services annually.1 In fact, FAO named 2015 the International Year of the Soils in order to highlight the importance of soils in our food system.
Unfortunately, arable soil is depleting very rapidly due to erosion, by around 24 billion tons each year.2 This rate of erosion is 10 to 100 times greater than the rate at which soil is being replenished.3 The major contributing factors are urban development, desertification, and industrial agriculture. The use of chemicals, intensive machinery, and monoculture are increasing productivity in the short term but leading to fallow soil and desertification over the long term. The most widely discussed solutions around these issues include polyculture, reforestation, and climate-smart agricultural practices. But, what if the reason we do not see soil being replenished is because we are not properly valuing it? I believe soil can provide a way to increase food access in urban food deserts, increase healthy diets among low-income communities, and shield communities from increasingly volatile global markets. To do this, we can look to the world of economics for a solution.
Some practitioners, artists, and scholars are exploring the idea of soil as a currency. Economists, agronomists, and ecologists have already agreed and estimated the economic benefits we receive from soil ecosystem services. Because we can create certain types of topsoil and because we know how valuable it is, we can create an economic system that is based on the value of soil.
Imagine a world where households provide their organic waste to urban gardeners and farmers in exchange for currency or points. After a prescribed amount of time, that household will collect enough points to “buy” produce from that farmer. Both parties benefit in this arrangement. The farmer needs inputs to create valuable compost for his/her operations, and the household has free access to healthy local produce. In addition, nearby restaurants and cafes can get involved, providing input materials for farmers, buying ingredients for their menus, and selling salads and sandwiches to participating households, all with an alternative currency based on soil.
The points accrued could be based not only on the quantity of waste, but also on the type. For instance, to create quality compost, you need both materials high in nitrogen (greens) and materials high in carbon (browns). The volumes required for each of these differ as well, contributing to the waste-food calculation. To be clear, the waste in question is primarily actual waste that cannot or will not be consumed as food again. Examples include egg shells, coffee grounds, used paper towels, and kitchen scraps. This system does not incentivize throwing out edible food. Participants will be trained on what constitutes “waste” and would be encouraged not to throw out food simply to acquire soil currency. Furthermore, there are components to this system that will help to check this moral hazard. Soil currency can only buy more fresh produce, so the participant does not gain anything by throwing out edible food. Different growing seasons for different crops will limit participants’ ability to game the system. Lastly, a part of this system could be to provide participants with waste bins designed for the right amount of browns and greens, further encouraging the right kind of participation.
Rebecca Murphey. Household compost.
There is a lot of research on how much food is wasted in both the industrialized and developing world, and where in the supply chain this happens, before or after the consumer.4 Households and industry are wasting perfectly good food at the same time that millions of people are experiencing hunger and malnutrition. Food waste is becoming more and more accepted as a serious issue to be addressed and soil currency should not undermine this progress. The goal of soil currency is to incentivize people to compost who ordinarily would not, and, to eat more fresh fruits and vegetables.
This might sound like pie in the sky, and in many ways it is still just an idea at this point. There are, however, groups who are trying to make this a reality, like Hello Compost in the Bronx, NYC. Hello Compost is a “home composting service” that collects waste from residents in specially designed pouches and delivers that waste to a nearby urban farm called Project EATS.5,6 Participating residents receive “credits” for the waste they produce via a mobile app, where they can track their progress and redeem their credits for food grown at a Project EATS site. The urban farm can use the compost it has made or sell it to make income. The Hello Compost operation is still in the early stages and has yet to include nearby businesses that serve food, as described earlier. However, after one year in operation, this project has contributed significantly to making soil currency a reality for working class residents of the Bronx.
Soil currency provides answers to several problems. First, soil currency raises awareness of the importance of soil by placing a monetized value on it. This economic incentive will not only produce more soil, but should increase the amount of fresh produce a household consumes, because the resulting kitchen scraps now have value as the inputs to make compost. If households are buying more produce because it equals “free food,” we might be able to assume these households are eating healthier, whole fruits and vegetables. This is an area that needs further study, but if true, even in some cases, could be a groundbreaking solution that would encourage healthy eating habits. However, the issue of moral hazard still needs to be addressed in such a study and through the implementation of the project.
Second, this economic system will work best in low-income and marginalized communities, where the need is greatest. Often, people in low-income communities do not have access to healthy food either because it is too far away (in communities located in so-called “food deserts”) or it is too expensive. Soil currency could help subsidize access to healthy food, without spending any legal tender to get it.
Third, soil currency will decrease the potency of greenhouse gases (GHG), and perhaps, the amount that is emitted into the atmosphere. When organic waste is sent to a landfill, it decomposes anaerobically (without oxygen), which produces methane. If that same waste is put toward a compost pile that is being turned, the waste decomposes aerobically (with oxygen) and therefore produces carbon dioxide, a GHG 21 times less potent than methane. Landfills and methane are significant contributors to emissions and climate change in the United States.7 If enough soil is produced, we may see an offset of the CO2 created because the soil (with the help of plants) will sequester carbon from the atmosphere.
Lastly, it directly addresses the issue of declining arable soil by incentivizing ordinary citizens to make more of it, albeit in much smaller quantities than is needed. Although raising awareness is a central objective of soil currency, it does so in the practical application of creating compost. It should be noted that compost is but one of many types of soil, most of which take millennia to form, so this concept will not replenish all soil types. However, finished compost is a very good one for growing food and is considered one of the best ways to replenish the Earth with sustainable agriculture. Furthermore, when using space-efficient vertical gardens and composters like those designed and built by the non-profit Can YA Love,8 one maximizes the limited amount of space for food and compost production as well. This is critical in an urban setting where there is not much space for composting or agriculture.
The example of Hello Compost shows that soil currency can have an impact. But, some critical questions need to be answered in order to scale this up and across different communities. Mainly, what is the correct monetary value to assign soil?
Tiffany Woods / Oregon State University. Finished compost soil at Oregon State University’s student-run organic farm in Corvallis, Oregon. Compost soil is ideal for growing food.
This is difficult to answer from a technical perspective, as well as a practical one. Soil provides many services for our planet, and is, therefore, hard to valuate. The main service soil provides in a waste-to-food system, such as this one, is acting as the medium to grow food. Therefore, to place a monetary value on that soil, you would need to know the cost of various crops in your area, the amount of compost needed to grow said crops, and the amount of waste needed for the correct amount of compost. Once this exchange rate is calculated for a particular area, a farmer can then plan how much he/she would need to grow for the participating households throughout the season, as well as how much inputs for compost he/she would receive.
However, the question remains: will the value of soil be enough for people to change their behavior? This key question has not been answered because soil currency has not existed nor been implemented (until very recently). Perhaps the price of a particular crop is not enough to persuade people to make compost out of their waste? However, adding the value of carbon storage, water retention, and of other ecosystem services soil provides, could push the value high enough to make this idea worthwhile. If there is adequate demand for the compost produced, could it be sold in other markets, thus adding more value and more incentive? Other studies have already shown that improvements in soil health (increased humus content) directly correlate to increased economic value for farmers.9 This information is being used to persuade farmers to go organic. This same beneficial relationship can be used to persuade households and urban gardeners to compost.
If we can place an accurate and high enough value on soil, it may change people’s behavior to eat more healthy foods, create quality topsoil, increase community resilience, and mitigate food insecurity. Soil currency will not be a panacea, but it could be another tool in our arsenal to help us plan for a more food secure future.
- Food and Agricultural Organization of the United Nations (FAO). The State of the World’s Land and Water Resources for Food and Agriculture: Managing Systems at Risk [online] (2011). http://www.fao.org/docrep/017/i1688e/i1688e.pdf.
- United Nations Convention to Combat Desertification (UNCCD). Land Degradation Neutrality: Resilience at Local, National and Regional Level [online]. http://www.unccd.int/Lists/SiteDocumentLibrary/Publications/V2_201309-UN….
- Grantham Centre for Sustainable Futures. A Sustainable Model for Intensive Agriculture [online] (2015). http://grantham.sheffield.ac.uk/wp-content/uploads/2015/12/A4-sustainabl….
- Food and Agricultural Organization of the United Nations (FAO). Global Food Losses and Food Waste [online] (2011). http://www.fao.org/docrep/014/mb060e/mb060e00.pdf.
- Hello Compost [online] http://hellocompost.com/.
- Project EATS [online]. http://projecteats.org/about-us/.
- United States Environmental Protection Agency. EPA Overview of Greenhouse Gases Methane Emissions [online]. http://www3.epa.gov/climatechange/ghgemissions/gases/ch4.html.
- Can YA Love [online]. www.canyalove.org.
- Sait, G. Humus Saves the World. Nutrition Matters [online] (April 2013). http://blog.nutri-tech.com.au/humus-saves- the-world/.<|endoftext|>
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The autonomous community of Catalonia – one of the 17 autonomous communities that form the country of Spain – voted on October 1, 2017 in a referendum for independence from Spain. Voting day was marred by violent clashes between Catalan voters on one side, and Spanish police and the Guardia Civil on the other side. It was also marred by the seizure of some ballot boxes by the Spanish authorities. One of the main reasons for these clashes was that the Constitutional Court of Spain had already suspended the referendum (on September 8) on constitutional grounds. Was the Catalan referendum – whose results state that 92.01% of participating voters backed independence in a 43.03% voting turnout (not accounting for the missing ballot boxes) – unconstitutional? This article will look at just this question without discussing any political, economic or social implications.
The following are links to the Spanish Constitution – in English and in Spanish. The Spanish constitution defines ideals to be honored by Spain such as democracy, justice, liberty, security and wellbeing (preamble), the political and cultural structure of Spain (preliminary part), fundamental rights and duties of its citizens (Part I), the powers and responsibilities of the monarch (Part II), the powers and responsibilities of the legislature which is also referred to as Cortes Generales (Part III), the structure of the central government (Part IV), checks and balances between the executive and the legislative branch (Part V), the powers and responsibilities of the judiciary (Part VI), the role of the government in the economy and finances (Part VII), territorial organization of Spain (Part VIII), the powers and responsibilities of the Constitutional Court of Spain, and the process to add amendments to the constitution (Part X).
Regarding the matter of this article, arguably the most important part of the constitution is Part VIII (Territorial Organization of the State), especially Chapter 3 which discusses the autonomous communities. However, looking deeper into the supreme law of Spain, other parts of the Spanish constitution merit further examination too, especially Section 2 of the Preliminary Part which states that the Spanish homeland is indivisible and based on the solidarity among all Spaniards:
“The Constitution is based on the indissoluble unity of the Spanish Nation, the common and indivisible homeland of all Spaniards; it recognizes and guarantees the right to self-government of the nationalities and regions of which it is composed and the solidarity among them all.”
The language in Section 2 in and of itself suggests that the referendum in Catalonia is unconstitutional.
Looking at Part IX of the constitution, Section 161, paragraph 2 gives the Constitutional Court of Spain the power to suspend provisions and resolutions adopted by the bodies of the autonomous communities that have been brought to the Constitutional Court of Spain by the federal government as an appeal, and “ratify or lift the suspension” within five months. This means that the September 8 suspension should be decided on by the Constitutional Court of Spain – particularly to be ratified or lifted – by February 8, 2018. The suspension document from September 8 also mentions Section 30 of the constitution as part of the decision to suspend the referendum. However, Section 30 merely states that “Citizens have the right and the duty to defend Spain” (in paragraph 1) and that “The duties of citizens in the event of serious risk, catastrophe or public calamity may be regulated by law” (in paragraph 4). If defending Spain includes defending its territorial integrity regardless of whether it has been attacked militarily or has had one of its 17 autonomous communities, or either of the cities of Ceuta and Melilla, declare war on it, then the referendum is truly unconstitutional. The Constitutional Court of Spain’s September 8 decision potentially sets a precedent in Spanish case law that outlaws any autonomous community’s aspirations to independence.
In Part VIII, Section 138 defines the principle of solidarity throughout Spain, and that includes all the 17 autonomous communities:
“The State guarantees the effective implementation of the principle of solidarity proclaimed in section 2 of the Constitution, by endeavoring to establish a fair and adequate economic balance between the different areas of the Spanish territory and taking into special consideration the circumstances pertaining to those which are islands.”
Therefore, if Catalans demand independence from Spain as a result of a referendum under the pretext that too much of its resources is being redistributed from their autonomous community to poorer autonomous communities within Spain, Section 138 makes their anger at the central government unjustified unless the principle of solidarity has been defined in a legal precedent or a bill was passed by the Cortes Generales and signed by the King into law (Section 91), and having that law’s solidarity threshold passed.
In fact, Section 158 makes any anger by Catalans’ part for economic reasons unjustified on constitutional grounds. Paragraph 2 states that “With the aim of redressing interterritorial economic imbalances and implementing the principle of solidarity, a compensation fund shall be set up for investment expenditure, the resources of which shall be distributed by the Cortes Generales among the Self-governing Communities and provinces, as the case may be.” Therefore, Spain’s legislative body – the Cortes Generales – with the King’s approval – is the decision maker on how the compensation fund should be spent. That decision, as paragraph 1 suggests, should happen on the basis of “proportion to the amount of State services and activities for which [the autonomous communities] have assumed responsibility and to guarantee a minimum level of basic public services throughout Spanish territory.” Paragraph 1 doesn’t necessarily have the potential to incentivize growing sizes of autonomous governments within the autonomous communities because if addressing imbalances is based on the proportion of the amount of state services and activities for which the autonomous communities have assumed responsibility, then poorer autonomous communities might be incentivized to grow their governments on one hand, but on the other hand the guarantee is to a minimum level of basic public services – meaning that the central government may allocate anywhere between the minimum level and above.
The October 1 referendum appears to be unconstitutional considering the language of Section 92 as well. It states that “Political decisions of special importance may be submitted to all citizens in a consultative referendum (paragraph 1); The referendum shall be called by the King on the President of the Government’s proposal after previous authorization by the Congress (paragraph 2); An organic act shall lay down the terms and procedures for the different kinds of referendum provided for in this Constitution (paragraph 3).” The October 1 referendum was not consultative (seeking advice), it was binding. Therefore, it appears to be unconstitutional in two aspects of Section 92: it does not ask the rest of the Spaniards as to whether they approve of an independent Catalonia and it is not advisory but instead unilateral in decision.
In Part VIII, Section 145 also prevents autonomous communities from uniting into a federation (paragraph 1) under any circumstances. It is interesting to note that Section 145 represents a contrast to the Iraqi constitution, as discussed in a different article, which provides “one or more governorates [with] the right to organize into a region based on a request to be voted on in a referendum.”
In Part VIII, Section 153 gives another reason to conclude that the October 1 referendum in Catalonia was unconstitutional. Section 153 states that control over the bodies of the autonomous communities shall be exercised by the Constitutional Court in matters pertaining to the constitutionality of their regulatory provisions having the force of law. This gives the Spanish government the right to appeal the Catalan referendum to the Constitutional Court of Spain which on September 8 declared the referendum unconstitutional.
Section 155 appears to further justify the Spanish government’s reaction to the Catalan referendum. It provides the Spanish government with flexibility on reaction to any action that is “seriously prejudicial to the general interest of Spain” – and it could be argued that the Catalan referendum is counter to the principle of solidarity in Section 2 discussed above – and reaction to any action that has not provided a satisfactory response to inquiries by the Spanish government.
The October 1 referendum is unconstitutional based on the following:
- Section 2 of the Preliminary Part which states that the Spanish homeland is indivisible and based on the solidarity among all Spaniards.
- Section 161, paragraph 2 which gives the Constitutional Court of Spain the power to suspend provisions and resolutions adopted by the bodies of the autonomous communities that have been brought to the Constitutional Court of Spain by the federal government as an appeal, and “ratify or lift the suspension” within five months.
- Section 158 provides the Spanish government with the power to redistribute wealth as part of “redressing interterritorial economic imbalances.”
- The referendum is not consultative as per Section 92, Paragraph 1.
The constitution also gives the Spanish government the flexibility to act however it sees fit to protect “the general interest of Spain,” and Spanish Prime Minister Mariano Rajoy already stated that his government could resort to Section 155.<|endoftext|>
| 3.765625 |
815 |
United States of AmericaPA
High School Core Standards - Geometry Assessment Anchors
# 10.03 Chords of a circle
Lesson
A chord is a line segment with endpoints on the circumference of a circle. The diameter is the longest chord of a circle, and it divides the circle into two semi-circles. All other chords divide the circle into a major arc and minor arc. A minor arc has a corresponding central angle of less than $180^\circ$180°, while the major arc has corresponding central angle of greater than $180^\circ$180°.
If two chords of a circle are congruent, what can we conclude about the arcs of those chords?
#### Exploration
1. Using the applet below, move points $C$C and $D$D to change the lengths of the chords. Move point $E$E to change the location of $\overline{EF}$EF around the circle. Move point $B$B to change the size of the circle.
2. Is it always true that the arcs $EF$EF and $CD$CD are the same length? Explain your reasoning (hint: click the checkbox to show the circle radii).
The applet above demonstrates following theorem relating the chords and arcs of a circle.
Theorem
In the same circle or in congruent circles, two arcs are congruent if and only if their corresponding chords are congruent.
As the applet alluded to, we can prove this using congruent triangles. Since we know that congruent arcs are always formed by congruent central angles, we can deduce the following corollary to our theorem.
Corollary
In the same circle or in congruent circles, chords are congruent if and only if their corresponding central angles are congruent.
### Bisecting arcs and chords
If a line, line segment, or ray divides an arc into two congruent arcs, then we say it bisects the arc.
#### Exploration
A special phenomenon happens when the radius of a circle is perpendicular to a chord. Use the applet below to test create and test a conjecture. The radius $\overline{AG}$AG is perpendicular to the chord $\overline{CD}$CD, so what does it do to the chord $\overline{CD}$CD and the minor arc of $CD$CD?
If you hypothesized that the radius bisects the arc and the chord, you were correct. We can prove our conjecture using what we know of congruent triangles (hint: use the checkbox in the applet to show the radii of the circle). Since we can prove our conjecture, it's a theorem!
Theorem
If a diameter or radius of a circle is perpendicular to a chord, then it bisects the chord and its arc.
The converse of the theorem can also be proven using congruent triangles.
Converse
The perpendicular bisector of a chord is a diameter (or radius) of the circle.
### Congruent chords are equidistant
We can also see another phenomenon relating two chords and their distance from the center of the circle. The following theorem can also be proven using congruent triangles.
Theorem
In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
#### Practice questions
##### Question 1
$C$C is the center of the circle. Calculate $x$x.
##### Question 2
Find the length of $\overline{AB}$AB in circle $O$O.
##### Question 3
What is the length of $x$x?
### Outcomes
#### CC.2.3.HS.A.8
Apply geometric theorems to verify properties of circles.
#### G.1.1.1.3
Use chords, tangents, and secants to find missing arc measures or missing segment measures.<|endoftext|>
| 4.71875 |
896 |
, 01.02.2020cleik
# The length of the rectangle is 5cm more than its width when its area is 150cm²
Width = 5
Lengtg = 10
Step-by-step explanation:
REPRESEBTATION :
Let x be the width
Let x+5 be the length
Area = 50
Mathematical Sentence:
Since Area = Length x Width
Area = x(x+5)
50 = x(x+5)
x(x+5)=50
x²+5x-50=0
Solution:
x²+5x-50=0 ➡️ (x-5)(x+10)=0 ➡️ x-5=0; x+10=0
x = 5 or x = -10
(since there's no such thing as a negative width or length, kaya we will use x = 5)
Width = x ➡️ x=5
Length = x+5 ➡️ 5+5=10
The width of the rectangle is 5, while the length is 10
The length and the width of perimeter is 25
Representation:
?m
| |
| A= 50cm² | ?m
| |
L, W= A÷?=?
P= ?m+?m+?m+?m
Equation:
A= L=(?), and W=(L-5)
P= L+L+W+W
Solution:
A= 50cm²
L, W= A÷?=?
L, W= 50÷10= 5
L= 10
W= (10-5)
A= L= 10 W= (10-5)
L= 10m
W= 5m
P= L+L+W+W
P= 10+10+5+5
P= 30m
L= 10m
W= 5m
P= 30m
that's I know
length(15) width(10)
Explanation:
rectangle=L x W
15x10=150
Step-by-step explanation:
L=x+5
W=x
A=50
A=lw
50=x(x+5)
x²+5x-50=0
A
D
C
A
B
Step-by-step explanation:
basta, tama 'yan magtiwala ka lang.
I don't think if I'm correct or incorrect...?
1. (n) (n+2)=195
n^2 + 2n - 195 = 0
2. (x) (x+5)=50
x^2 + 5x - 50 = 0
3. incomplete problem
4. (h) (h+3)=75
h^2 + 3h - 75 = 0
The length is 15 and the width is 10
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| 4.46875 |
1,047 |
# Find Maximum and Minimum Values of X3 +3xy2 -15x2-15y2+72x. - Applied Mathematics 1
Sum
Find maximum and minimum values of x3 +3xy2 -15x2-15y2+72x.
#### Solution 1
Given f(x) = x3 +3xy2 -15x2-15y2+72x… (1)
STEP 1] for maxima, minima,(delf)/(delx)=0;(delf)/(dely)=0
3x2+3y2-30x+72=0 and 6xy – 30y=0
∴ y (6x-30) =0
y=0, x= 5
For x=5; From Equation 3x2+3y2-30x+72=0, we get y2-1=0
Y=±1
Hence (4,0) , (6,0) , (5,1) ,(5,-1) are the stationary points.
STEP 2] Now,r=(del^2f)/(delx^2)=6x-30;
"S"=(del^2f)/(delxdely)=6y;
t=(del^2f)/(dely^2)=6x-30
STEP 3] for (x, y) ≡ (4, 0), r = -6, s = 0, t = -6;
rt –s2=(-6)(-6)-0=36>0 and r< 0.
This shows that the function is maximum at (4, 0)
∴ From Equation (1)
Fmax=f (4, 0) =43+0-15(42) +0+72(4) =64 – 240 + 288
Fmax=112
STEP 4] For (x,y)≡(6,0)
r=6, s=0, t=6
rt-s2=36 but r=6>0
This shows that function is minimum at (6, 0).
From Equation (1),
Fmin=f(6,0)=63+0-15(6)2+0+72(6)=108.
STEP 5] For(x, y) ≡ (5, 1)
r=0, s=6, t=0;
(rt-s2)<0
This shows that at (5, 1) and (5,-1) function is neither maxima nor minima.
#### Solution 2
Given f(x) = x3 +3xy2 -15x2-15y2+72x… (1)
STEP 1] for maxima, minima,(delf)/(delx)=0;(delf)/(dely)=0
3x2+3y2-30x+72=0 and 6xy – 30y=0
∴ y (6x-30) =0
y=0, x= 5
For x=5; From Equation 3x2+3y2-30x+72=0, we get y2-1=0
Y=±1
Hence (4,0) , (6,0) , (5,1) ,(5,-1) are the stationary points.
STEP 2] Now,r=(del^2f)/(delx^2)=6x-30;
"S"=(del^2f)/(delxdely)=6y;
t=(del^2f)/(dely^2)=6x-30
STEP 3] for (x, y) ≡ (4, 0), r = -6, s = 0, t = -6;
rt –s2=(-6)(-6)-0=36>0 and r< 0.
This shows that the function is maximum at (4, 0)
∴ From Equation (1)
Fmax=f (4, 0) =43+0-15(42) +0+72(4) =64 – 240 + 288
Fmax=112
STEP 4] For (x,y)≡(6,0)
r=6, s=0, t=6
rt-s2=36 but r=6>0
This shows that function is minimum at (6, 0).
From Equation (1),
Fmin=f(6,0)=63+0-15(6)2+0+72(6)=108.
STEP 5] For(x, y) ≡ (5, 1)
r=0, s=6, t=0;
(rt-s2)<0
This shows that at (5, 1) and (5,-1) function is neither maxima nor minima.
Concept: Maxima and Minima of a Function of Two Independent Variables
Is there an error in this question or solution?<|endoftext|>
| 4.5 |
933 |
# 【数论】乘法逆元
## Definition
$x \times y \equiv 1 \pmod p$
## Algorithm
#### Proof
$x \times x^{-1} \equiv 1 \pmod p~~~~~~(1)$
$\gcd(x,~p) = d \neq 1~~~~~~(2)$
$x \times x^{-1} = k \times p + 1~~~~~~(3)$
$(3)$ 的等号两侧同时除以 $(2)$ 中的 $d$
$\frac{x \times x^{-1}}{d}~=~\frac{k \times p + 1}{d}~~~~~~(4)$
$\frac{x}{d} \times x^{-1}~=~\frac{p}{d} \times k + \frac{1}{d}~~~~~~(5)$
$\frac{x}{d}$$\frac{p}{d}$ 都是整数，进而 $\frac{p}{d} \times k$ 是整数，$\frac{x}{d} \times x^{-1}$ 是整数。
#### 求单个数字的逆元
##### Algorithm 1
$x \times x^{-1}~\equiv 1 \pmod p$
$x \times x^-1 = 1 + kp$
$y = -k$，移项得到
$x \times x^{-1} + y \times p = 1$
$ax + by = 1$
##### Algorithm 2
$x^{\phi(p)} \equiv 1 \pmod p$
$x^{\phi(p) - 1} \equiv x^{-1} \pmod p$
#### 求 $n$ 以内所有正整数模 $p$ 的逆元
$inv_i \equiv \left\lfloor\frac{p}{i}\right\rfloor \times inv_{p \bmod i} \pmod p$
$inv_1 = 1$
##### Proof
$p = ki + r$
$0 \equiv ki + r \pmod p$
$r \equiv -ki \pmod p$
$i^{-1} \equiv -kr^{-1} \pmod p$
## Code
### Ex_Gcd
#include <iostream>
typedef long long int ll;
ll x, p;
void Ex_gcd(const ll a, const ll b, ll &X, ll &Y);
int main() {
std::cin >> x >> p;
ll a, b;
Ex_gcd(x, p, a, b);
std::cout << (a % p + p) % p << std::endl;
return 0;
}
void Ex_gcd(const ll a, const ll b, ll &X, ll &Y) {
if (b == 0) {
X = 1; Y = 0;
} else {
Ex_gcd(b, a % b, Y, X);
Y -= a / b * X;
}
}
### 欧拉定理
#include <iostream>
typedef long long int ll;
ll X, p;
ll mpow(ll x, ll y);
int main() {
std::cin >> X >> p;
std::cout << mpow(X, p - 2) << std::endl;
return 0;
}
ll mpow(ll x, ll y) {
ll _ret = 1;
while (y) {
if (y & 1) (_ret *= x) %= p;
y >>= 1;
(x *= x) %= p;
}
return _ret;
}
### 线性求逆元
#include <cstdio>
const int maxn = 3000005;
int n, p;
int inv[maxn], factinv[maxn];
int main() {
scanf("%d%d", &n, &p);
factinv[1] = inv[1] = 1;
printf("%d\n", 1);
for (int i = 2; i <= n; ++i) {
inv[i] = 1ll * (p - p / i) * inv[p % i] % p;
printf("%d\n", inv[i]);
factinv[i] = 1ll * factinv[i - 1] * inv[i] % p;
}
return 0;
}
posted @ 2020-01-06 20:29 一扶苏一 阅读(470) 评论(0编辑 收藏 举报<|endoftext|>
| 4.40625 |
1,441 |
Eczema is not a single disease but a reaction pattern of the skin produced by many different conditions.
There are at least eleven different types of skin conditions that produce eczema, such as:
- Pompholyx (dyshidrotic eczema);
- Allergic contact dermatitis;
- Atopic dermatitis;
- Nummular eczema;
- Stasis dermatitis;
- Seborrheic eczema;
- Fungal infections;
- Xerotic (dry skin) eczema;
- Lichen simplex chronicus.
Eczema affects about 10 percent to 20 percent of infants and commonly starts in infancy, with 65% of patients developing symptoms in the first year of life and 90% develop symptoms before age 5.
An estimated 31.8 million people in the United States are estimated to have symptoms of eczema. In the United Kingdom, it has been estimated that up to 16 million people could be living with eczema.
Eczema is usually itchy. For many individuals, the itch is commonly only mild, or moderate.
But in some cases, it can become much worse and you might develop an extremely inflamed skin.
Some individuals develop red bumps or clear fluid-filled bumps that look “bubbly” and, when scratched, add wetness to the overall appearance.
Additionally, in adults, chronic rubbing produces thickened plaques of skin.
In babies and children, it is seen in varying severity and different places on the body depending on age. Face, head, and/or feet are the more usual areas for flares in babies and young children than adults.
Patients with eczema are prone to have food allergies causing either more subtle reactions or, in some cases, anaphylaxis.
There is also a risk of the skin becoming infected with pathogenic bacteria. Signs of a bacterial infection can include:
- generally feeling unwell;
- fluid oozing from the skin;
- a high temperature (fever);
- a yellow crust on the skin surface;
- the skin becoming sore and swollen;
- small yellowish-white spots appearing on the skin.
Moreover, eye complications may occur, with symptoms including:
- infection (conjunctivitis);
- inflammation of the eyelid (blepharitis);
- itching around the eyelids.
Viral infections, especially herpes simplex virus, are more common in people with eczema.
The exact cause of this disease is unknown, but it’s thought to be connected to an overactive response by the body’s immune system to an irritant.
It is this response that causes the symptoms of eczema.
Moreover, it can also be triggered by environmental factors such as pollen and smoke. Mineral and vitamin deficiency can also play a big role in eczema.
Vitamins A, C, E, B are particularly important, due to the fact that some individuals become deficient in trace minerals, which are much harder to absorb with our soils being so depleted in modern times.
Foods can also trigger the disease, especially chemical food additives (such as colorings and preservatives), dairy products, meat, eggs, or nuts.
Common risk factors include:
- excessive sweating – the sodium found in perspiration can lead to irritation as it dehydrates the skin
- mother’s age at child’s birth – kids who are born to mothers who are later in their childbearing years are more likely to develop eczema;
- occupation – jobs which put you in contact with certain solvents, metals, or cleaning supplies substantially increase your chance of this skin condition;
- low humidity – the lack of moisture in low humidity areas may trigger the skin to react in a negative way;
- allergies – people who have allergies have a higher risk of developing eczema. Some allergens include – certain foods, molds, household dust mites, animal dander, and plant pollen.
Your doctor will commonly be able to diagnose eczema by assessing your skin and asking some questions about the condition, like:
- whether it has affected the typical areas, like skin creases, in the past;
- whether you have any other conditions which may be related to your eczema, like – asthma or allergies;
- whether there is a history of atopic eczema in your family;
- whether you have flare-ups of severe symptoms;
- when the symptoms first began;
- whether the rash is itchy.
Commonly recommended treatment for eczema include:
- topical steroids – low potency topical corticosteroids are available over the counter for short-term treatment;
- antihistamines – these medications control the itch;
- bandages – they allow the body to heal underneath;
- cytokine therapies – they are usually used in severe cases and target the immune system response;
- emollients (moisturizers) – they are used every day to stop the skin from becoming dry
- get a humidifier since dry air can be stressful for your skin;
- light therapy – this treatment is used for people who either who rapidly flare again after treatment or don’t get better with topical treatments.
Avoid common irritants such as:
- animal fur or dander;
- wool or synthetic fibers;
- soaps and detergents;
- cigarette smoke;
- some cosmetics and perfumes;
- dust mites;
- dust or sand;
- substances like mineral oil, chlorine, or solvents.
Take Shorter Showers Baths
Limit your showers and baths to maximum of 10 minutes. Also, bath oil may be helpful. In addition, use warm, rather than hot, water.
Some foods, such as cows’ milk and eggs, can trigger eczema symptoms.
Spiritual Causes of Eczema
Spiritual causes may include:
- Trying to keep the world at a distance;
- Feeling inadequate or unworthy;
- Isolating and locking yourself inside because of fear in coming out;
- Someone or something getting under your skin or feeling overexposed;
- Extreme sensitivity to circumstances and emotions around you.
Psychological stresses can also provoke or aggravate it, presumably by suppressing normal immune mechanisms.
”I release all fear I am holding in my skin.”
”I am comfortable and confident.”
”Peace and harmony, love and joy surround me and indwell me.”
”I clear all the ways I am afraid to let go of the old pattern thinkings to allow new ways to emerge.”
”I am safe and secure.”
”I release any feelings of being unsettled that I am holding in my skin.”<|endoftext|>
| 3.703125 |
1,509 |
# Factors of 8
The factors of 8 are the numbers that divide 8 exactly without leaving any remainder. There are four factors of 8, they are 1, 2, 4, and 8. Hence, the highest factor is 8 that divides itself. If we add all the factors, then the sum will be equal to 15.
The pair factors of 8 are (1,8) and (2,4). These pair factors when multiplied together results in the original number. 8 is also a perfect cube, thus its prime factor will be a single digit. We can determine the factors of 8 by the division method and prime factorisation method.
## What are the Factors of 8?
A number or an integer that divides 8 exactly without leaving a remainder, then the number is a factor of 8. As the number 8 is an even composite number, it has more than two factors. Thus, the factors of 8 are 1, 2, 4 and 8.
## Pair Factors of 8
A pair of numbers, which are multiplied together resulting in an original number 8 is called the pair factors of 8. As discussed above, the pair factors of 8 can be positive or negative. The positive and negative pair factors of 8 is given below:
Positive Pair Factors of 8:
Positive Factors of 8 Positive Pair Factors of 8 1 × 8 (1, 8) 2 × 4 (2, 4)
Negative Pair Factors of 8:
Negative Factors of 8 Negative Pair Factors of 8 -1 × -8 (-1, -8) -2 × -4 (-2, -4)
## How to Find Factors of 8?
8 is a whole number that is also an even number. Since it is a single-digit number, therefore, it is much easier to find the factors of 8.
There are two methods to find the factors of 8:
• Division method
• Prime factorisation method
## Factors of 8 by Division Method
The factors of 8 can be found using the division method. In the division method, we need to find the integers which divide 8 exactly without leaving a remainder, and those integers are considered as the factors of 8. Now, let us find the factors of 8 using the division method.
• 8/1 = 8 (Factor is 1 and Remainder is 0)
• 8/2 = 4 (Factor is 2 and Remainder is 0)
• 8/4 = 2 (Factor is 4 and Remainder is 0)
• 8/8 = 1 (Factor is 8 and Remainder is 0)
If we divide 8 by any numbers other than 1, 2, 8 and 8, it leaves a remainder. Hence, the factors of 8 are 1, 2, 4 and 8.
## Prime Factorization of 8
The prime factorization of 8 is the process of writing the number as the product of its prime factors. Now, let us discuss the process of finding the prime factors of 8.
• Divide 8 by the smallest prime number, i.e. 2. 8/2 = 4
• Divide 4 by the smallest possible prime number, 4/2 = 2
• 2 is itself a prime number and is divisible by 2
• Therefore, the prime factorization of 8 is 2 × 2 × 2 or 23.
Another method to find the prime factors of 8 is the factor tree method.
### Factor Tree
Hence, the number of prime factors of 8 is one. The prime factorisation of the whole number 8 is 23. The exponent in the prime factorisation is 3. When you add the number 1 with the exponent, i.e.,3 +1 = 4. Therefore, the number 8 has 4 factors.
## Facts – Factors of 8
• Factors of 8 – 1, 2, 4 and 8
• Prime factorisation – 2 x 2 x 2
• Prime factor of 8 – 2
• Pair factors – (1,8) and (2,4)
• Sum of factors of 8 – 15
### Related Articles
Links Related to Factors Factors of 15 Factor of 36 Factors of 48 Factors of 18 Factors of 24 Factors of 25 Factors of 42 Factors of 60 Factors of 35 Factors of 81 Factors of 75 Factors of 56
## Solved Examples on Factors of 8
Example 1:
Find the common factors of 8 and 5.
Solution:
The factors of 8 are 1, 2, 4 and 8.
The factors of 7 are 1 and 5.
As 5 is a prime number, the common factor of 8 and 5 is 1.
Example 2:
Find the common factors of 8 and 9.
Solution:
Factors of 8 = 1, 2, 4 and 8.
Factors of 9 = 1, 3, and 9.
Hence, the common factor of 8 and 9 is 1.
Example 3:
Find the common factors of 8 and 4.
Solution:
The factors of 8 are 1, 2, 4 and 8.
The factors of 4 are 1, 2 and 4.
Hence, the common factors of 8 and 4 are 1, 2, and 4.
### Practise Questions on Factors of 8
1. What is the lowest factor of 8?
2. Is there any odd number that is a factor of 8?
3. How many even numbers are the factors of 8?
Stay tuned with BYJU’S to know about factor 8 and the factors and prime factors of other numbers. Download BYJU’S – The Learning App to better experience and clarification.
## Frequently Asked Questions on Factors of 8
### What are the factors of 8?
The factors of 8 are 1, 2, 4 and 8.
### What is the prime factorization of 8?
The prime factorization of 8 is 2 × 2 × 2 or 23.
### What is the sum of all factors of 8?
The sum of all factors of 8 is 15.
### What is the greatest common factor of 8 and 7?
The greatest common factor of 8 and 7 is 1.
### What are the positive pair factors of 8?
The positive pair factors of 8 are (1, 8) and (2, 4).
### Write down the negative pair factors of 8.
The negative pair factors of 8 are (-1, -8) and (-2, -4).
### Is 2 a factor of 8?
Yes, 2 is a factor of 8. If 8 is divisible by 2, it leaves a quotient of 4 and remainder 0, and hence 2 is a factor of 8.<|endoftext|>
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When was oil first used as fuel?
Crude oil is called petroleum. The rocks in which petroleum is found lie deep underground. The oil is reached by drilling below the earth's surface.
In some places, petroleum seeps to the surface of the ground through cracks. These seepages, or oil springs, were easy for men to locate. And this crude oil from surface seepages was known to most ancient peoples. Some oil was burned in lamps and torches.
The real history of oil began in the 19th century. The Industrial Revolution brought a need for better lamp fuels to light the new factories. In the United States, oil lay close to the surface in many regions and it was often used as medicine.
The first man who thought of drilling for oil was a New York lawyer named George Bissell. He sent a sample of Pennsylvania crude oil to a scientist at Yale University, Benjamin Silliman. Silliman reported that petroleum yielded many useful products: lamp oils, lubricating oils, illuminating gas, paraffin wax for candles, and others. Silliman's report convinced businessmen that there was money to be made in oil.
Bissell hired a man named Edwin Drake to drill for oil near Titusville, Pennsylvania. On August 27, 1859, they struck oil. The news spread quickly. Men rushed to buy or lease land where oil might be found, and the oil rush was on. Oil fever spread to other parts of the United States, to Canada, and to Europe. New uses for petroleum products were found, including its use as fuel, and the demand for oil increased. Today, the search for new oil fields is still going on all over the world.<|endoftext|>
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# Expressions / Equations
Expressions are groups of mathematical symbols that can be written and manipulated according to definite rules for simplifying or for identifying relationships. For example, (a + 2b)(a – 2b) is an expression. Equations consist of expressions on the two sides of an equals sign, which can be manipulated in order to find the value of one or more variables or “unknowns.” Word problems are usually solved through the formulation and solution of one or more equations.
• 3 Expression problems – General
• 2 Expression problems with radicals
• 3 Equation problems – General
• 2 Simultaneous Linear Equation problems
Expressions Example 1)
3m + n = x and 2m + 4n = 2x, so what is m/n?
A. 1/3
B. 2
C. 1/2
D. 1/4
E. 1/8
Explanation: Notice that both of these expressions in m and n are equal to some multiple of x. So if we make them equal to the same multiple of x, then we can set them equal to each other and eliminate x. Notice also that we don’t need to find the value of m or n, but just the value of m/n.
3m + n = x and 2m + 4n = 2x
So, 6m + 2n = 2x
and
6m + 2n = 2m + 4n
6m – 2m = 4n – 2n
4m = 2n
m/n = 2/4 = 1/2
So C is the correct answer.
Expressions Example 2)
If x is an integer, for what value of x is |3x – 28| minimized?
A. -10
B. -9
C. 0
D. 9
E. 10
Explanation: Since the absolute value of any expression is always positive or 0, |3x – 28| is always positive or 0. Since x is an integer 3x is never equal to 28, so the minimum value will come when x = 9 or x = 10. If x = 9, then 3x – 28 = -1, and the absolute value of -1 is 1. If x = 10, then 3x – 28 = 2, and the absolute value of 2 is 2. So |3x – 28| has its minimal value when x = 9.
Therefore, D is the correct answer.
Expressions Example 3)
If uv = mn and vw = nk, which one of the following equalities is valid for all non-zero numbers?
A. uw = nk/v
B. m/u = w/n
C. m/u = k/w
D. w/n = k/m
E. mu = vn
Explanation:
This is not a straightforward problem, so in order to solve it, let’s look at each of the possible answers in turn. Bear in mind that none of these variables is ever equal to 0, but they can be equal to any other number, and they’re not necessarily equal to each other:
A. uw = nk/v
Since we know that vw = nk, then it follows that w = nk/v. Substituting w into answer A for nk/v, we get
uw = w
which is false, for example, if u is 2. So answer A is invalid.
B. m/u = w/n
We know uv = mn, so m/u = v/n. But m/u = w/n implies w = v, which is not necessarily true. So answer B is invalid, also.
C. m/u = k/w
uv = mn means v/n = m/u
vw = nk means v/n = k/w
So, m/u = k/w (Answer C) is valid.
D. w/n = k/m
From vw = nk, we get w/n = k/v. So w/n = k/m implies m = v, which is not necessarily true. So answer D is invalid, also.
E. mu = vn
This implies m = vn/u. Substituting into uv = mn, we get uv = vn2/u, which means u2 = n2, which is not necessarily true. So answer E is also invalid.
Therefore, C is the correct answer.
Notice that we eliminated each incorrect answer by manipulating the answer’s equation and one or more of the two equations we were given originally. We then compared results. We did the same to determine that C was correct.
\begin{align}\sqrt {75} + \sqrt {108} =\end{align}
A. \begin{align}11\sqrt{3}\end{align}
B. \begin{align}5\sqrt{3}\end{align}
C. \begin{align}3\sqrt{5}\end{align}
D. \begin{align}5\sqrt{5}\end{align}
E. \begin{align}11\sqrt{7}\end{align}
Explanation: Here we have the sum of square roots of two seemingly different numbers. But if we take the prime factorizations of the two and move the factors around a little
\begin{align}75 = 3 \cdot 5^2\end{align}
\begin{align}108 = 3^3 \cdot 2^2 = 3 \cdot 3^2 \cdot 2^2\end{align}
we can see that each of these numbers contains a square times the number 3. So if we take their square roots, we get
\begin{align}5\sqrt{3} + 6\sqrt{3} = 11\sqrt{3}\end{align}
So A is the correct answer.
Note that the important thing to do here is to take the prime factorization of the two numbers. Then we could see how to proceed further.
\begin{align}\frac{1}{\sqrt{2}}\, – \sqrt{8} + \frac{\sqrt{72}}{4} =\end{align}
A. \begin{align}2\end{align}
B. \begin{align}18\sqrt{2}\end{align}
C. \begin{align}3\sqrt{5}\end{align}
D. \begin{align}9\end{align}
E. \begin{align}0\end{align}
Explanation: This problem requires some prime factorization, and other manipulation of radicals for its solution:
\begin{align}8 = 2^2 \cdot 2\end{align}
\begin{align}72 = 3^2 \cdot 2^2 \cdot 2\end{align}
When we take the square roots of these two numbers, we get
\begin{align}\sqrt{8} = 2\sqrt{2}\end{align}
\begin{align}\sqrt{72} = 6\sqrt{2}\end{align}
We also need to “rationalize” the denominator for the first term. This is done by multiplying this fraction by 1 in the form of \begin{align}\frac{\sqrt{2}}{\sqrt{2}} :\end{align}
\begin{align}(\frac{1}{\sqrt{2}}) \cdot (\frac{\sqrt{2}}{\sqrt{2}}) = \frac{\sqrt{2}}{2}\end{align}
Now we are ready to solve the problem:
\begin{align}\frac{1}{\sqrt{2}}\, – \sqrt{8} + \frac{\sqrt{72}}{4} =\end{align}
\begin{align}\frac{\sqrt{2}}{2}\, – {2}\sqrt{2} + \frac{{6}\sqrt{2}}{4} =\end{align}
\begin{align}\frac{\sqrt{2}}{2}\, – {2}\sqrt{2} + \frac{{3}\sqrt{2}}{2}\end{align}
Now we rearrange terms, and we get
\begin{align}\frac{\sqrt{2}}{2}\, – {2}\sqrt{2} + \frac{{3}\sqrt{2}}{2} =\end{align}
\begin{align}\frac{{3}\sqrt{2}}{2} + \frac{\sqrt{2}}{2}\, – {2}\sqrt{2} = \end{align}
\begin{align}\frac{{4}\sqrt{2}}{2}\, – {2}\sqrt{2} = \end{align}
\begin{align}{2}\sqrt{2}\; – {2}\sqrt{2} = 0\end{align}
So E is the correct answer.
Equations Example 1)
Today Ralph and his 3 triplet sisters celebrate their birthday. He is 11, and they are 5. In how many years will the combined age of the triplets be twice Ralph’s age?
A. 5
B. 7
C. 8
D. 9
E. 10
Explanation: In word problems we normally use a variable to represent the quantity we are seeking. So we let x equal the number of years until the combined ages of the triplets will be twice Ralph’s age. That being the case, we can use the information we’re given in the problem to formulate an equation:
2(11 + x) = 3(5 + x)
In x years, Ralph will be 11 + x years old, and twice his age at that time will be 2(11 + x). The triplets will be 5 + x years old, and since we’re combining the triplets’ ages, we multiply 5 + x by 3. We then set these two quantities equal to each other, and that gives us the formula for when their combined age will be twice Ralph’s.
Now we solve the equation for x:
22 + 2x = 15 + 3x
22 – 15 = 3x – 2x
7 = x
In 7 years Ralph will be 18 and the triplets will be 12. Since 3 · 12 =36, this is twice Ralph’s age 7 years from now.
Equations Example 2)
21 coins, each worth 5, 10, or 25 cents, add up to $2.15. If there are 2 more dimes (10 cent coins) than nickels (5 cent coins), how many quarters (25 cent coins) are there? A. 2 B. 3 C. 5 D. 6 E. 8 Explanation: This problem involves 3 unknowns – the numbers of quarters (q), dimes (d), and nickels (n) – and such problems tend to be messy. We’ll find it easier to solve this by picking various possibilities for the number of quarters, and then seeing how each one works. Since we’ve only got 5 possible answers, we’ll try them until we find the right one. Let’s start with answer A – 2 quarters. If q = 2, then since there are 21 coins altogether d + n = 19 But d = n + 2, so n +2 + n = 19 2n = 17 n = 8 1/2 which is absurd, since we can’t have half a nickel. So answer A is out. In fact, the same thing will happen whenever q is an even number, so we’ve already eliminated all the answers except B and C. Let’s try B. If q = 3, then d + n = 18 n + 2 + n = 18 2n = 16 n = 8 d = 10 Then, since q = 3 represents the quarters, d = 10 the dimes, and n = 8 the nickels, by adding the values of these coins together, we get 3∙25 + 10∙10 + 8∙5 = 75 + 100 + 40 = 215 which is the amount they’re supposed to add up to, so the correct answer is B. What would happen if we chose C, in which q = 5? Then n + d = 16 2n + 2 = 16 n = 7 d = 9 In this case then, 5∙25 + 9∙10 + 7∙5 = 125 + 90 + 35 = 250 so the coins add up to too large an amount. Therefore, answer C is incorrect. Equations Example 3) MyRetail estimated revenue of$768 on sale of its inventory of a certain product. But when 8 more were found in stock, the store decided to lower its price by $8 each, and in selling all the product, they still received$768, as planned. What was the price they originally planned to charge?
A. $32 B.$24
C. $36 D.$28
E. $30 Explanation: This problem can be solved with equations, but in this case it is easier and faster to simply plug in values from the list of answers, and see what works. Notice that the values of answers C, D, and E do not divide 768 evenly, and so these answers must be incorrect. So let’s try answer A: 768 / 32 = 24 which means that the original quantity was 24. But they found 8 more, so now the quantity is 32, and since they reduced the price by$8, the new price is $24. Thus, the original price was$32, and answer A is correct.
Simultaneous Linear Equations Example 1)
Seamus loves professional sports. In the past 3 years he’s attended 180 baseball, basketball, and soccer games. He’s attended 4 times as many basketball games as soccer games, and 90 more baseball games than basketball games. How many baseball games has he gone to?
A. 50
B. 30
C. 62
D. 45
E. 130
Explanation: Let b be the number of baseball games, k the number of basketball games, and s the number of soccer games Seamus has gone to in the past 3 years. Then we have the following relationships:
b + k + s = 180
k = 4s
b = k + 90
So,
b = 4s + 90
b + k + s = (4s + 90) + 4s + s = 180
9s = 180 – 90 = 90
s = 10
k = 40
b = 130
Simultaneous Linear Equations Example 2)
200 children at a school carnival were each given 1 blue, 1 red, and 1 green chip for use, respectively, at the fishing booth, the merry-go-round, and the throwing gallery. Afterward it was determined that 80% of the blue, 75% of the red, and 40% of the green chips had been used. If none of the children had three chips left over, and 20% of the children had one chip left over, how many had two chips left over?
A. 80
B. 60
C. 85
D. 75
E. 55
Explanation: One difficulty with this problem is caused by the description that’s given. When it says a child had x (0, 1, 2, or 3) chips left over, it really means that the child used 3 – x of the chips. So, let n be the number of children who used 1 chip, or in other words, had two chips left over. Let w be the number that used 2 chips, and let r be the number that used all three chips. Then we need to find the value of n. We have the following relationships:
n + w + r = 200 (since every child used at least one chip)
w = .2∙200 = 40 (since 20% of the children used two chips)
So,
n + 40 + r = 200
n + r = 160
The number of chips used (or spent) was
80% of the blue
75% of the red
40% of the green
and since there were 200 of each color chip, then the total number of chips spent was
(.8 + .75 + .4) x 200 = 390
Now we need a formula showing how the chips were spent. Since 390 chips were spent in all, then that total number can be calculated by adding the number of chips spent by n children, w children, and r children:
n + 2w + 3r = 390
This formula follows from the fact that the w children who spent 2 chips contribute 2w to the total of 390 chips spent, and the r children contribute 3r to the number spent. Since w = 40, then
n + 80 + 3r = 390
n + 3r = 310
We know from above that
n + r = 160
so subtracting this equation from the one above it, we get
2r = 150
r = 75
So finally of the 200 children:
40 children used two chips
75 used three chips
85 used one chip
That is, n = 85.
So C is the correct answer, as the children using one chip (85) still had two left over.
\begin{align}(p + 2) (p\, – 2) = 2p\, – 1\end{align}and\begin{align}(q + 2) (q\, – 2) = 2q\, – 1\end{align}, also\begin{align}p > q.\end{align}What is the value of\begin{align}q\end{align}?
A. \begin{align}2\end{align}
B. \begin{align}-1\end{align}
C. \begin{align}4\end{align}
D. \begin{align}-2\end{align}
E. \begin{align}1\end{align}
Explanation: These equations are identical, except for the variable in each of them. Additionally, the left side of both is the factoring of the difference of two squares. Therefore, each of these is the same quadratic equation, which will have two solutions. We will find the two solutions, and set one of them equal to p and the other equal to q. We start by expanding each equation so we get:
\begin{align}p^2 – 4 = 2p\, – 1\end{align}
\begin{align}q^2 – 4 = 2q\, – 1\end{align}
Now, let’s put everything on the left side and make the right side equal 0, so that these are in the normal form for quadratic equations. We will then factor them so that we can see their solution:
\begin{align}p^2 – 2p\, – 3 = 0\end{align}
\begin{align}(p\, – 3) (p + 1) = 0\end{align}
\begin{align}p = 3,\end{align}or \begin{align}p = -1\end{align}
\begin{align}q^2 – 2q\, -3 = 0\end{align}
\begin{align}(q\, -3) (q + 1) = 0\end{align}
\begin{align}q = 3,\end{align}or \begin{align}q = -1\end{align}
Since \begin{align}p > q,\end{align} then \begin{align}p = 3\end{align} and \begin{align}q = -1.\end{align}
So B is the correct answer.
If \begin{align}a^2b^4 -5 = 4ab^2,\end{align}then what is one value of\begin{align}a\end{align}in terms of\begin{align}b\end{align}?
A. \begin{align}\frac{-1}{5b^2}\end{align}
B. \begin{align}4b^2\end{align}
C. \begin{align}5b^2\end{align}
D. \begin{align}\frac{5}{b^2}\end{align}
E. \begin{align}-b^2\end{align}
Explanation: We can think of this problem as a quadratic equation in\begin{align}a,\end{align}where the various powers of\begin{align}b\end{align}are coefficients. So:
\begin{align}a^2b^4 – 5 = 4ab^2\end{align}
\begin{align}a^2b^4 – 5 -4ab^2 = 0\end{align}
\begin{align}b^4a^2 – 4b^2a\, – 5 = 0\end{align}
The expression on the left side of the equals sign can be factored into
\begin{align}(b^2a\, – 5) (b^2a + 1) = 0\end{align}
So,
\begin{align}b^2a\, – 5 = 0,\end{align}
\begin{align}a = \frac{5}{b^2}\end{align}
and
\begin{align}b^2a + 1 = 0\end{align}
\begin{align}a = \frac{-1}{b^2}\end{align}
are the two possible solutions for\begin{align}a\end{align}in terms of\begin{align}b.\end{align}The only one of these that shows up in the answers is\begin{align}\frac{5}{b^2},\end{align}which is answer D.<|endoftext|>
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We have already published a few posts about dinosaurs, the largest creatures that have ever roamed on Earth but in fact, in the prehistoric era, almost all animals as well as plants were much bigger than their modern counterparts. From an elephant-size sloth to a car-size armadillo, check out these 25 giant prehistoric ancestors that would make their contemporary relatives look like dwarfs.
These days, the polar bear and the Kodiak bear are considered the largest species of bear, both of them weighing as much as 1500 pounds (680 kg), which is an impressive size but compared to the short-faced bear, these are still lightweight. This extinct bear inhabited North America during the Pleistocene epoch until 11,000 years ago and was – when standing on its hind legs – up to 12 feet (3.7 m) tall. Moreover its estimated to have weighed over 3,000 pounds (1360 kg).
While today, the largest eagle species of eagles such as the Philippine, harpy or wedge-tailed eagle usually weigh up to 15 pounds (6.7 kg) and have wingspans of up to 7 feet (about 213 cm), the Haast´s eagle was considerably larger. Once native to New Zealand where it became extinct some 600 years ago, this raptor could weigh more than 35 pounds (16 kg), and though it had a relatively short wingspan for their size, it was still much longer than the one of the modern eagles.
We currently know 5 different species of tapir that range in size from 330 to 700 pounds (150-300 kg) but from the Middle Pleistocene to about 4,000 years ago, there was a creature called megatapirus that was bigger, considerably heavier and more massive. Living in today’s China and Vietnam, the extinct prehistoric tapir may have weighed up to 1,100 pounds (500 kg).
Endemic to South America, Megatherium was a genus of elephant-sized ground sloths. Living from the late Pliocene to the end of the Pleistocene, it was one of the largest land mammals known, weighing up to 4 tons and measuring up to 20 feet (6 m) in length. Scientists suggest that Megatherium was capable of bipedal locomotion and had a long tongue used for putting leaves into its mouth.
There are three known species of wombats today, ranging in weight from 44 to 77 pounds (20 – 35 kg) and usually reaching a length of less than 40 inches (1 m). It’s hard to believe that these cute marsupials native to Australia evolved from a creature the size of a rhinoceros. Diprotodon, a member of a group of unusual species collectively called the “Australian megafauna” and the largest marsupial ever, was up to 10 feet (over 3 m) long and weighed over 6,100 pounds (almost 2800 kg).<|endoftext|>
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# Annuities for Loans
## Calculating equal periodic payments for a loan using present values.
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Annuities for Loans
Many people buy houses they cannot afford. This causes major problems for both the banks and the people who have their homes taken. In order to make wise choices when you buy a house, it is important to know how much you can afford to pay each period and calculate a maximum loan amount.
Joanna knows she can afford to pay 12,000 a year for a house loan. Interest rates are 4.2% annually and most house loans go for 30 years. What is the maximum loan she can afford? What will she end up paying after 30 years? ### Annuities for Loans The present value can be found from the future value using the regular compound growth formula: \begin{align*}PV(1+i)^n &= FV\\ PV &= \frac{FV}{(1+i)^n}\end{align*} You also know the future value of an annuity: \begin{align*}FV=R \cdot \frac{(1+i)^n-1}{i}\end{align*} So by substitution, the formula for the present value of an annuity is: \begin{align*}PV=R \cdot \frac{(1+i)^n-1}{i} \cdot \frac{1}{(1+i)^n}=R \cdot \frac{(1+i)^n-1}{i(1+i)^n}=R \cdot \frac{1-(1+i)^{-n}}{i}\end{align*} The present value of a series of equal payments \begin{align*}R\end{align*} with interest rate \begin{align*}i\end{align*} per period for \begin{align*}n\end{align*} periods is: \begin{align*}PV=R \cdot \frac{1-(1+i)^{-n}}{i}\end{align*} This formula can also be used to find out other information such as how much a regular payment should be and how long it will take to pay off a loan. Take a1,000,000 house loan over 30 years with a nominal interest rate of 6% compounded monthly. You are not given the monthly payments, \begin{align*}R\end{align*}. To find \begin{align*}R\end{align*}, solve for \begin{align*}R\end{align*} in the formula given above.
\begin{align*}PV=\ 1,000,000, \ R=?, \ i=0.005, \ n=360\end{align*}
\begin{align*}PV &= R \cdot \frac{1-(1+i)^{-n}}{i}\\ 1,000,000 &= R \cdot \frac{1-(1+0.005)^{-360}}{0.005}\\ R &= \frac{1,000,000 \cdot 0.005}{1-(1+0.005)^{-360}} \approx 5995.51\end{align*}
It is remarkable that in order to pay off a $1,000,000 loan you will have to pay$5,995.51 a month, every month, for thirty years. After 30 years, you will have made 360 payments of $5995.51, and therefore will have paid the bank more than$2.1 million, more than twice the original loan amount. It is no wonder that people can get into trouble taking on more debt than they can afford.
### Examples
#### Example 1
Earlier, you were asked about how much Joanna can afford to take out in a loan. Joanna knows she can afford to pay 12,000 a year to pay for a house loan. Interest rates are 4.2% annually and most house loans go for 30 years. What is the maximum loan she can afford? What does she end up paying after 30 years? You can use the present value formula to calculate the maximum loan: \begin{align*}PV=12,000 \cdot \frac{1-(1+0.042)^{-30}}{0.042} \approx \202,556.98\end{align*} For 30 years she will pay12,000 a year. At the end of the 30 years she will have paid \begin{align*}\12,000 \cdot 30=\360,000\end{align*} total
How long will it take to pay off a $20,000 car loan with a 6% annual interest rate compounded monthly if you pay it off in monthly installments of$500? What about if you tried to pay it off in monthly installments of 100? \begin{align*}PV = \20,000, \ R=\500, \ i=\frac{0.06}{12}=0.005, \ n=?\end{align*} \begin{align*}PV &= R \cdot \frac{1-(1+i)^{-n}}{i}\\ 20,000 &= 500 \cdot \frac{1-(1+0.005)^{-n}}{0.005}\\ 0.2 &= 1-(1+0.005)^{-n}\\ (1+0.005)^{-n} &= 0.8\\ n &= -\frac{\ln 0.8}{\ln 1.005} \approx 44.74 \ months\end{align*} For the100 case, if you try to set up an equation and solve, there will be an error. This is because the interest on $20,000 is exactly$100 and so every month the payment will go to only paying off the interest. If someone tries to pay off less than $100, then the debt will grow. #### Example 3 It saves money to pay off debt faster in order to save money on interest. As shown earlier, interest can more than double the cost of a 30 year mortgage. This example shows how much money can be saved by paying off more than the minimum. Suppose a$300,000 loan has 6% interest convertible monthly with monthly payments over 30 years. What are the monthly payments? How much time and money would be saved if the monthly payments were larger by a fraction of \begin{align*}\frac{13}{12}\end{align*}? This is like making 13 payments a year instead of just 12. First you will calculate the monthly payments if 12 payments a year are made.
\begin{align*}PV &= R \cdot \frac{1-(1+i)^{-n}}{i}\\ 300,000 &= R \cdot \frac{1-(1+0.005)^{-360}}{0.005}\\ R &= \1,798.65\end{align*}
After 30 years, you will have paid 647,514.57, more than twice the original loan amount. If instead the monthly payment was \begin{align*}\frac{13}{12} \cdot 1798.65=1948.54\end{align*}, you would pay off the loan faster. In order to find out how much faster, you will make your unknown. \begin{align*}PV &= R \cdot \frac{1-(1+i)^{-n}}{i}\\ 300,000 &= 1948.54 \cdot \frac{1-(1+0.005)^{-n}}{0.005}\\ 0.7698 &= 1-(1+0.005)^{-n}\\ (1+0.005)^{-n} &= 0.23019\\ n &= -\frac{\ln 0.23019}{\ln 1.005} \approx 294.5 \ months\end{align*} 294.5 months is about 24.5 years. Paying fractionally more each month saved more than 5 years of payments. \begin{align*}294.5 \ months \cdot \1,948.54=\573,847.99\end{align*} The loan ends up costing573,847.99, which saves you more than $73,000 over the total cost if you had paid over 30 years. #### Example 4 Mackenzie obtains a 15 year student loan for$160,000 with 6.8% interest. What will her yearly payments be?
\begin{align*}PV=\160,000, \ R=?, \ n=15, \ i=0.068\end{align*}
\begin{align*}160,000 &= R \cdot \frac{1-(1+0.068)^{-15}}{0.068}\\ R & \approx \17,345.88\end{align*}
#### Example 5
How long will it take Francisco to pay off a $16,000 credit card bill with 19.9% APR if he pays$800 per month? Note: APR in this case means nominal rate convertible monthly.
\begin{align*}PV=\16,000, \ R=\600, \ n=?, \ i=\frac{0.199}{12}\end{align*}
\begin{align*}16,000 &= 600 \cdot \frac{1-\left(1+\frac{0.199}{12}\right)^{-n}}{\frac{0.199}{12}}\\ n &= 24.50 \ months\end{align*}
### Review
For problems 1-10, find the missing value in each row using the present value for annuities formula.
Problem Number \begin{align*}PV\end{align*} \begin{align*}R\end{align*} \begin{align*}n\end{align*} (years) \begin{align*}i\end{align*} (annual) Periods per year 1. $4,000 7 1.5% 1 2.$15,575 5 5% 4 3. $4,500$300 3% 12 4. $1,000 12 2% 1 5.$16,670 10 10% 4 6. $400 4 2% 12 7.$315,000 $1,800 5% 12 8.$500 30 8% 12 9. $1,000 40 6% 4 10.$10,000 6 7% 12
11. Charese obtains a 15 year student loan for $200,000 with 6.8% interest. What will her yearly payments be? 12. How long will it take Tyler to pay off a$5,000 credit card bill with 21.9% APR if he pays $300 per month? Note: APR in this case means nominal rate convertible monthly. 13. What will the monthly payments be on a credit card debt of$5,000 with 24.99% APR if it is paid off over 3 years?
14. What is the monthly payment of a $300,000 house loan over 30 years with a nominal interest rate of 2% convertible monthly? 15. What is the monthly payment of a$270,000 house loan over 30 years with a nominal interest rate of 3% convertible monthly?
To see the Review answers, open this PDF file and look for section 13.7.
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
TermDefinition
annuity An annuity is a series of equal payments that occur periodically.
future value In the context of earning interest, future value stands for the amount in the account at some future time $t$.
loan A loan is borrowed money. Loans are commonly repaid with interest.
mortgage A loan is money borrowed that has to be paid back with interest. A mortgage is a loan specifically for a house.
present value In the context of earning interest, present value stands for the amount in the account at time 0.<|endoftext|>
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Coral reefs are formed by the hard limestone skeletons of millions of tiny polyps as they die. Layers build up to form coral rock structures that become a community for a wide variety of sea creatures who form remarkable and colourful relationships. Growing in shallow warm water, coral reefs can't survive below 20 degrees Celsius. The most famous coral reef in the world, the Great Barrier Reef, stretches over 2000 km off the northern Queensland coast of Australia.
Belonging to the phylum Coelenterata, coral is made up of tiny polyps which secrete calcium carbonate creating a limestone skeleton which protects the soft base, stomach, mouth opening and tentacles. The polyps remain retracted within the skeleton but expand at night to catch microscopic plankton with their tentacles.
There are approximately 9500 species of phylum Coelenterata including jellyfish and sea anemones which share similar characteristics to coral. Coral polyps have an amazing symbiotic relationship with zooxanthellae algae providing a suitable environment for the algae to survive. In return, the algae use sunlight to create sugars giving the polyp energy in a process known as photosynthesis. The algae also contribute to the stunning colours of the coral by combining its chlorophyll with fluorescent colour pigments.
Coral reproduction has a number of forms thanks to each polyp possessing both male and female sexual organs. As such, coral reproduces with both sexual and asexual reproduction. Some corals fertilise and incubate eggs internally however many corals use a process called 'spawning' where all the coral in a certain region will release eggs and sperm on the same night. This event is controlled by the phases of the moon and is an amazing display for those lucky enough to catch it.
The larva that are created through this process are known as 'planula' and drift for a few days on the ocean's surface before settling to the seabed and attaching to a solid. The young then build structure by secreting calcium carbonate which hardens into the skeleton within which the polyps live.
The biggest threat to coral is the Crown of Thorns starfish which surrounds a structure and eats it alive by dissolving it with the digestive enzymes of its stomach. However, coral can also be affected by unfavourable temperatures, variations in sea levels and overexposure to the sun resulting in radiation. Water run-off from the mainland containing harmful chemicals and fertilizers can also damage the reef ecosystem.
The Great Barrier Reef is renowned for the amazing variety of marine life that exists within its waters including 1500 types of fish and 4000 species of shell fish. You'll also find manta rays, sea urchins, sponges, worms, sea anemones, sea cucumbers, sea stars, green, hawksbill and loggerhead turtles.<|endoftext|>
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Fourteenth Amendment to the United States Constitution
|This article is part of a series on the|
|Constitution of the|
United States of America
Preamble and Articles|
of the Constitution
|Amendments to the Constitution|
|Full text of the Constitution and Amendments|
The Fourteenth Amendment (Amendment XIV) to the United States Constitution was adopted on July 9, 1868. It was one of the Reconstruction Amendments. The amendment discusses citizenship rights and equal protection of the laws. It was proposed in response to issues related to former slaves following the American Civil War. This amendment was bitterly contested. Southern states were forced to ratify it in order to regain representation in Congress. The Fourteenth Amendment is one of the most litigated parts of the Constitution. It forms the basis for landmark decisions such as Roe v. Wade (1972), and Bush v. Gore (2000). It remains the most important Constitutional amendment since the Bill of Rights was passed in 1791.
Summary[change | change source]
At the end of the Civil War, Abraham Lincoln freed the slaves. The problem was, he did not ask Congress. Congress had not passed a law to free slaves. Meanwhile, some states still had slavery. The Thirteenth Amendment freed the slaves. It became law in late 1865. Three years later the Fourteenth Amendment provided civil rights. Republicans controlled Congress during this period. They wanted to give full citizenship to freed slaves. But they also realized that giving civil rights to blacks, it opened the door for women's suffrage. It would lead to giving women the right to vote, which Congress did not want to do. If only section one was included in the amendment, the wording "all persons born or naturalized in the United States" would include women. For this reason, the word "male" was inserted in section two so the amendment would be approved by Congress.
Section one - citizenship[change | change source]
The first section of the Fourteenth Amendment gave citizenship to “all persons born or naturalized in the United States”, and "subject to the jurisdiction thereof". To be subject to the jurisdiction thereof, the parents of a infant born in the United States must already be naturalized or citizens, as they are the only people subject to the jurisdiction of the United States. This included freed slaves, but does not include foreigners or aliens. The second clause, commonly called the Privileges and Immunities Clause, states that "the citizens of each state shall be entitled to all privileges and immunities of citizens in the several states." This gave all Americans the protection of civil rights under the law. It forbids states from denying citizens their life, their liberty or their property without due process. States could not deny persons "equal protection of the laws." It meant that for the first time all people would have the same protection no matter what their color. The fact that states were mentioned makes them responsible for these protections the same as the federal government. The Fourteenth Amendment is cited more often in law suits than any other amendment.
Section two - apportionment[change | change source]
The second section changed a part of the original Constitution which counted slaves as three-fifths of a person. This was for the purpose of determining how many U.S. congressmen a state could have (apportionment). The second section established that every citizen would be counted as one person.
Sections three, four and five[change | change source]
The third section was intended to be strict with members of the Confederacy who fought against the United States. It required a two-thirds vote of Congress to allow leaders of the Confederacy to regain their citizenship or hold office. To be allowed to hold a federal office, former confederates had to swear an oath to uphold the constitution. Section four said the federal government would not repay Confederate debts. Section five means what it says, Congress will enforce the provisions of the 14th amendment.
Text[change | change source]
Section 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
Section 2. Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed. But when the right to vote at any election for the choice of electors for President and Vice President of the United States, Representatives in Congress, the Executive and Judicial officers of a State, or the members of the Legislature thereof, is denied to any of the male inhabitants of such State, being twenty-one years of age, and citizens of the United States, or in any way abridged, except for participation in rebellion, or other crime, the basis of representation therein shall be reduced in the proportion which the number of such male citizens shall bear to the whole number of male citizens twenty-one years of age in such State.
Section 3. No person shall be a Senator or Representative in Congress, or elector of President and Vice President, or hold any office, civil or military, under the United States, or under any State, who, having previously taken an oath, as a member of Congress, or as an officer of the United States, or as a member of any State legislature, or as an executive or judicial officer of any State, to support the Constitution of the United States, shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof. But Congress may, by a vote of two-thirds of each House, remove such disability.
Section 4. The validity of the public debt of the United States, authorized by law, including debts incurred for payment of pensions and bounties for services in suppressing insurrection or rebellion, shall not be questioned. But neither the United States nor any State shall assume or pay any debt or obligation incurred in aid of insurrection or rebellion against the United States, or any claim for the loss or emancipation of any slave; but all such debts, obligations and claims shall be held illegal and void.
Section 5. The Congress shall have power to enforce, by appropriate legislation, the provisions of this article.
Related pages[change | change source]
- Dred Scott v. Sandford
- American Civil War
- Reconstruction of the United States
- Andrew Johnson
- Jim Crow laws
- Civil Rights Act of 1964
- Voting Rights Act of 1965
References[change | change source]
|Wikimedia Commons has media related to Fourteenth Amendment to the United States Constitution.|
- Nick Malik (7 February 2014). "Explaining the 14th Amendment in Non-Technical English". Nick Malik. Retrieved 11 April 2015.
- "Voting Rights and the 14th Amendment". George Mason University. Retrieved 12 April 2015.
- "Privileges and Immunities Clause". Legal Information Institute, Cornell University Law School. Retrieved 31 March 2016.
- "Primary Documents in American History; 14th Amendment to the U.S. Constitution". The Library of Congress. Retrieved 11 April 2015.
- Martin Kelly. "14th Amendment Summary". About.com. Retrieved 11 April 2015.
- "14th Amendment". SHMOOP UNIVERSITY. Retrieved 11 April 2015.
- "14TH AMENDMENT". Laws.com. Retrieved 11 April 2015.
- "Constitution of the United States: Amendments 11–27". National Archives and Records Administration. Archived from the original on June 11, 2013. Retrieved June 11, 2013.<|endoftext|>
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# Distribution of integration constant (c) in separable differential equation
This equation describes leaking water form a conical tank. We are interested in finding $t$ when $h(t) = 0$ (time it takes to empty the tank).
$$\frac{dh}{dt} = - \frac{5}{6h^{3/2}}, h(0) = 20$$
Since this is separable, I separate the equation and solve:
$$6h^{3/2}dh = −5dt$$
$$\frac{12}{5}h^{5/2} = −5t + c$$
Using $h(0) = 20$ I find that $c = 1920\sqrt{5}$. Then, solving $h(t) = 0$, I find that $t= 384\sqrt{5}$.
Why is that the way I distribute the 6/5 have an impact on the solution? All of thesee equations results in different values for $t$:
$$-6h^{3/2}dh = 5dt$$ $$\frac{6}{5}h^{3/2}dh = -dt$$ $$-\frac{6}{5}h^{3/2}dh = dt$$ $$etc.$$
Why? I understand distributing 6/5 affects $c$, but since it's an arbitrary constant, why does it matter?
Physically, it makes no sense (there can't be an infinite number of times it takes to empty a tank of water).
Thoughts? I must be getting this wrong.
-
Are you "solving for $h(t)$" before trying to figure out $h(t)=0$? You can't just plug in $h(t)=0$ and then solve for $t$ in the resulting equation, because $h$ is not independent of $t$. So you must first rewrite $\frac{12}{5}h^{5/2} = -5t + 1920\sqrt{5}$ as $$h(t) = \left(-\frac{25}{12}t + 800\sqrt{5}\right)^{2/5},$$ and then solve $h(t)=0$ for $t$.
Likewise with the other ways of solving the differential equation: before you can solve $h(t)=0$ for $t$, you need to have an explicit expression for $h$ in terms of $t$, rather than an implicit one. Well, you don't have to go all the way to the expression above, it is enough to get to $$(h(t))^{5/2} = -\frac{25}{12}t + 800\sqrt{5}$$
I think you will find that if you clear that constant factor from $h$ first, you will always get the same $t$ for $h(t)=0$.<|endoftext|>
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1. ## Recurrence relation ( Advanced counting Techniques)
A model for the number of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years.
(a) Find a recurrence relation for {Pn}, where Pn is the number of lobsters caught in year n, under the assumption for this model.
(b) Find Pn if 100000 lobsters were caught in year 1 and 300000 were caught in year 2.
2. $\displaystyle P_{n+2}=\frac{P_n + P_{n+1}}{2}$
3. Hello, bhuvan!
I know some techniques for this problem.
I hope they're appropriate for your course.
A model for the number of lobsters caught per year is based on the assumption
that the number of lobsters caught in a year is the average of the number
caught in the two previous years.
(a) Find a recurrence relation for $\displaystyle P(n)$, the number of lobsters caught in year $\displaystyle n.$
. . $\displaystyle P(n) \;=\;\frac{P(n-1) + P(n-2)}{2}$
(b) Find $\displaystyle P(n)$ if $\displaystyle P(1) = 100,\!000\text{ and }P(2) = 300,\!000.$
From (a), we have: .$\displaystyle 2P(n) \;=\;P(n-1) + P(n-2)$ .[1]
We conjecture that $\displaystyle P(n)$ is an exponential function: .$\displaystyle P(n) \,=\,X^n$
Then [1] becomes: .$\displaystyle 2X^n \:=\:X^{n-1} + X^{n-2} \quad\Rightarrow\quad 2X^n -X^{n-1} - X^{n-2} \:=\:0$
. . Divide by $\displaystyle X^{n-2}\!:\quad 2X^2 - X - 1 \:=\:0 \quad\Rightarrow\quad (X - 1)(2X + 1) \:=\:0$
. . And we have two roots: .$\displaystyle X \;=\;1,\;\text{-}\tfrac{1}{2}$
Then: .$\displaystyle f(n) = 1^n\:\text{ or }\:f(n) = \left(\text{-}\tfrac{1}{2}\right)^n$
Form a linear combination of these roots:
. . $\displaystyle f(n) \;=\;A\left(1^n\right) + B\left(\text{-}\tfrac{1}{2}\right)^n \quad\Rightarrow\quad f(n) \;=\;A + B\left(\text{-}\tfrac{1}{2}\right)n$
We know the first two values of this sequence:
. . $\displaystyle \begin{array}{ccccc} f(1) = 100,\!000\!: & A - \frac{1}{2}B &=& 100,\!000 & {\color{blue}[2]} \\ \\[-3mm] f(2) = 300,\!000\!: & A + \frac{1}{4}B &=& 300,\!000 & {\color{blue}[3]} \end{array}$
Subtract [2] from [3]: .$\displaystyle \tfrac{3}{4}B \:=\:200,\!000 \quad\Rightarrow\quad B \:=\:\frac{800,\!000}{3}$
Substitute into [3]: .$\displaystyle A + \tfrac{200,\!000}{3} \:=\:300,000 \quad\Rightarrow\quad A \:=\:\frac{700,\!000}{3}$
Therefore: .$\displaystyle f(n) \;=\;\frac{700,\!000}{3} + \frac{800,\!000}{3}\left(\text{-}\frac{1}{2}\right)^n \quad\Rightarrow\quad f(n) \;=\;\frac{100,\!000}{3}\bigg[7 + \frac{8}{(\text{-}2)^n} \bigg]$
4. Thanks you very much !!
Do you know how to solve below problem i am trying to solve this but i am confuse..
Consider the non homogeneous linear recurrence relation an=2an-1+2^n
(a) show that an=n2^n is a solution of this relation.
i tried to solve this by putting an value in
2an-1+2^n=2(n2^n)+2^n)=2^n(2n+1) but i am not getting right answer as far as i think.
5. It works !
$\displaystyle 2a_{n-1}+2^n=2(n-1)2^{n-1}+2^n=2n2^{n-1}-2.2^{n-1}+2^n=n2^n=a_n$
6. can you please explain it ? i am confuse how you got 2(n-1)2^n-1
in 2(n-1)2^n-1+2^n since we have an=n2^n ??
7. Never mind i got it.. Thank You for your effort.<|endoftext|>
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Education and Awareness Water Quality
Water quality in streams can directly affect water quality in the aquifer because of rapid recharge through karst features, such as fractures and sinkholes in streambeds. The reverse is also true where springs contribute to river flows. The health of the creek is highly dependent on maintaining adequate spring flows, making recharge and groundwater management in the larger region critical to maintaining a healthy system in Cypress Creek.
Find out how to analyze a sample for an unknown contaminant.
Articles and Studies
EPA has developed human health benchmarks for approximately 394 pesticides to help states, tribes and water systems better understand whether pesticides they may detect in drinking water or sources of drinking water may present a public health risk.
EPA Releases Guidance for Sampling and Field Testing During Water Contamination Incidents
"Water is powerful. It can be beautiful and brutal, cleansing and contaminates, lethal and life-giving. Water is essential to life, yet 748 million people in the world still struggle to find it.
It's time to take notice"
"While water covers 71 % of the earth’s surface, less than 1 % of that is accessible fresh water. We must protect our limited fresh water supplies from harmful contaminants. Non-point source pollution is the biggest threat to water quality today, meaning that many small actions build into a big problem."
"Almost a billion people live without clean drinking water. We call this the water crisis. It's a crisis because it only starts with water- but water effects everything in life."
"Water is a precious, yet finite resource essential for live, with no adequate substitutes. Supplying and allocating water of adequate quality and in sufficient quantity is one of the major challenges facing society today."<|endoftext|>
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## Wednesday, March 08, 2006
### Mathematical Limits
One of the most important ideas in calculus is the concept of the mathematical limit. Limits relate to continuous functions and the basic idea is that if as a value the argument x approaches the value a, the difference between the limit L and f(x) can be arbitrarily small.
Definition 1: Mathematical Limit:
A function f(x) has a limit of L at point a if given any number ε, there exists a positive number δ such that:
if x-a lies between and , then f(x) - L lies between and ε
This definition is very similar to the definition of a continuous function (see here) and it is not surprising that the two concepts are very closely related.
In today's blog, I will need two definitions in order to prove the Squeeze Law relating to mathematical limits.
Definition 2: Open Interval :
x is an element of an open interval (α, β) if x is greater than α and x is less than β
Definition 3: Deleted Neighborhood
A deleted neighborhood is a set of points that result from deleting a single point in an open interval.
Lemma 1: Constant Law for Limits
if f(x) = C, then lim (x → a) f(x) = C
Proof:
(1) Let δ = 1
(2) if x - a lies between and , we know that f(x) = C.
(3) So, we know that f(x) - C = C - C = 0 which is less than any positive value ε
QED
Lemma 2: Product Law
if lim (x → a) f(x) = L and lim(x → a)g(x) = M, then
lim(x → a)[f(x)*g(x)] = L * M
Proof:
(1) Let ε be any nonzero value. We will prove that f(x)g(x) - LM lies between and
(2) Since the limit of f(x) = L, we know that there exists δ1 such that:
if x - a is between 1 and 1, then f(x) - L is between and
Since by definition, if x - a is between 1 and 1, then f(x) is between -L and +L.
(3) We also know that there exists δ2 such that:
if x - a is between 2 and 2, then f(x) - L is between -ε/(2M) and ε/(2M).
The definition for limits is that for any given positive value (ε), we can find a positive value (δ) to get the result (see above if review is needed).
(4) And there exists δ3 such that:
if x - a is between 3 and 3, then g(x) - M is between -ε/(2L) and ε/(2L)
(5) Let δ = min(δ123)
(6) Now, if x - a is between and , then:
(a) f(x) - L is between -ε/(2M) and +ε/(2M)
(b) g(x) - M is between - ε/(2L) and + ε/(2L)
(c) M[f(x) - L] is between (M)[-ε/(2M)] and (M)[+ε/(2M)] which is between -ε/2 and ε/2.
(d) f(x)[g(x) - M] is between (L)[-ε/(2L)] and (L)[+ε/(2L)] which is between -ε/2 and ε/2.
(e) If we add (c) + (d), we get:
f(x)M - LM + f(x)g(x) - f(x)M = f(x)g(x) - LM
(f) So, f(x)g(x) - LM is between (-ε/2 + -ε/2) and (+ε/2 + +ε/2) which means that it is between and
(7) So LM is the limit for f(x)g(x).
QED
Lemma 3: Squeeze Law
Suppose f(x), g(x), h(x) are functions such that
(a) f(x) ≤ g(x) ≤ h(x) for a deleted neighborhood (α, β) where point a is removed.
(b) lim (x→ a) f(x) = L = lim(x→a)h(x).
Then:
lim (x→ a) g(x) = L
Proof:
(1) Let ε be an arbitary number.
(2) Using the definition of limits, we know that there exists δ1 and δ2 such that:
if x-a lies between 1 and 1, then f(x)-L lies between and
if x-a lies between 2 and 2, then h(x)-L lies between and
(3) Let δ = min(δ12)
(4) We know that δ is greater than 0. [By the definition of mathematical limit]
(5) If x-a in between and , we know that f(x) and h(x) are both points of the open interval (L-ε, L+ε) [Again, from the definition of mathematical limit]
(6) So L-ε is less than f(x) ≤ g(x) ≤ h(x) which is less than L + ε
(7) Combining (#5) and (#6), this gives us that for any given ε, there exists a δ such that:
if x-a is in between and , then g(x)-L is between and
(8) From (#7), L is also a limit for g(x) as x approaches a.
QED
Lemma 4: Substitution Law
If lim (x → a) g(x) = L and lim (x → L) f(x) = f(L), then lim (x → a) f(g(x)) = f(L)
Proof:
(1) Let ε be any positive real value.
(2) Because lim (y → L) f(y) = f(L), we also know that there exists a value δ1 such that:
if (y - L) is between 1 and 1, then f(y) - f(L) is between and
(3) Because lim (x → a) g(x) = L, we know that there exists a value δ2 such that:
if (x - a) is between - δ2 and 2, then g(x) - L is between 1 and 1
(4) But this means if y = g(x), then:
if (x -a ) is between 2 and 2, then y - L is between 1 and 1 and f(g(x)) - f(L) is between and
(5) This then proves that:
lim (x → a) f(g(x)) = f(L)
QED
Lemma 5: lim (x → a) (1/x) = 1/a if a ≠ 0
Proof:
(1) Let ε be any positive real number.
(2) Assume that a is greater than 0.
(3) abs(1/x - 1/a) = abs([a - x]/ax) = abs([x -a]/ax) = (1/a)abs(x-a)/abs(x)
(4) Let us assume that abs(a-x) is less than a/2.
We can do this since abs(x-a) approaches 0 as x moves toward a.
(5) Then x -a is between -a/2 and +a/2 which means that x is between a/2 and 3a/2.
(6) This gives us that abs(x) is greater than a/2 and 1/abs(x) is less than 2/a.
(7) So that abs(1/x - 1/a) = abs(x-a)*(1/a)*abs(1/x) which is less than abs(x-a)*(1/a)(2/a) = 2/a2 * abs(x-a)
(8) Let δ be the minimum of a/2 and a2ε/2
(9) Then if x - a is between and , then:
abs(1/x - 1/a) is less than (2/a2)(a2ε/2) = ε
This then proves that lim (x → a) (1/x) = 1/a for when a is greater than 0.
(10) Assume that a is less than 0
(11) Then
abs(1/x - 1/a) = abs(x-a)/(-a)*1/abs(x)
(12) If we assume that abs(x-a) is less than -a/2, then:
x - a is between -a/2 and +a/2, then x is between -3a/2 and -a/2.
(13) So abs(x) is greater than -a/2.
(14) So 1/abs(x) is less than -2/a.
(15) In this case, then:
abs(1/x - 1/a) = abs(x-a)/(-a)*1/abs(x) which is less than 1/(-a)*(-2/a)*abs(x-a) = 2/a2*abs(x-a)
(16) Let δ be the minimum of -a/2 and a2ε/2
(17) Then if x - a is between and , then:
abs(1/x - 1/a) is less than (2/a2)(a2ε/2) = ε
This then proves that lim (x → a) (1/x) = 1/a for when a is less than 0.
QED
Lemma 6: Reciprocal Law
if lim(x → a) g(x) = L and L ≠ 0, then lim (x → a) 1/g(x) = 1/L
Proof:
(1) Let f(x) = 1/x
(2) lim (x → a) f(x) = lim (x → a) (1/x)
(3) Using Lemma 5 above, we have:
lim (x → a) f(x) = 1/L = f(L)
(4) Applying the Substitution Law (Lemma 4 above) gives us:
lim (x → a) 1/g(x) = lim (x → a) f(g(x)) = f(L) = 1/L
QED
Lemma 7: Quotient Law
if lim (x → a) f(x) = L and lim (x → a) g(x) = M ≠ 0, then:
lim (x → a) f(x)/g(x) = L/M
Proof:
(1) Using the Product Law above, we have:
lim (x → a) f(x)/g(x) = lim (x → a)f(x) * lim(x → a)1/g(x)
(2) Using the Reciprocal Law above:
lim (x → a) 1/g(x) = 1/M
(3) Combining step #1 and step #2 gives us:
lim (x → a) f(x)/g(x) = L*(1/M) = L/M
QED
Lemma 8: abs(a + b - c - d)) ≤ abs(a - c) + abs(b - d)
Proof:
(1) If (a-c),(b-d) are the same sign, then abs(a + b - c -d) = abs(a -c) + abs(b - d)
(2) If (a-c),(b-d) are not the same sign, then abs(a - c + b - d) is less than abs(a -c) + abs(b-d).
QED
Corollary 8.1: Addition Law
if lim (x → a) f(x) = L and lim(x → a)g(x) = M, then
lim(x → a)[f(x)+g(x)] = L + M
Proof:
(1) Let ε be any nonzero value.
(2) Since the limit of f(x) = L, we know that there exists δ1 such that:
if x - a is between 1 and 1, then f(x) - L is between -ε/2 and +ε/2
Since by definition, if x - a is between 1 and 1, then f(x) is between -L and +L.
(3) Since the limit of g(x) = L, we know that there exists δ2 such that:
if x - a is between 2 and 2, then g(x) - M is between -ε/2 and +ε/2
Since by definition, if x - a is between 2 and 2, then g(x) is between -M and +M.
(4) Let δ = min(δ12)
(5) Now, if x - a is between and , then:
(a) f(x) - L is between -ε/2 and +ε/2
(b) g(x) - M is between - ε/2 and + ε/2
(c) By Lemma 8 above, abs([f(x) + g(x)] - (L + M)) ≤ abs(f(x) - L) + abs(g(x) - M)
(d) abs(f(x) - L) + abs(g(x) - M) ≤ η/2 + η/2 = η
(7) So L+M is the limit for f(x)+g(x).
QED
References
#### 7 comments :
Anonymous said...
You must be smart, but I feel like you just wrote all that in chinese. Didn't help me understand much.
Larry Freeman said...
It takes some time getting used to formal language if you are interested.
If it's not for you, I suggest checking out other web sites such as Wikipedia.org, cut-the-knot.org, and betterexplained.com.
-Larry
kiberu emma said...
thanx this really helped.
kiberu emma said...
and I'm not so convinced about your quotient and reciprocal law.
Larry Freeman said...
Hi Kiberu,
Which step are you unclear about?
-Larry
TechOn top said...
I am really thankful to you for such nice information.
Max Dee said...
You really explained it well. Hats off to you. Anybody who didn't understand it can also use online limit solver to solve their mathematics problems.<|endoftext|>
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# What Is A Rule For The Sequence?
## What is the general rule for the sequence?
The Rule.
Because all arithmetic sequences follow a similar pattern, you can use a general formula to find the formula for the sequence.
The formula is this: an = a1 + d ( n – 1 ).
## What is a sequence of numbers called?
A sequence is a list of numbers in a certain order. Each number in a sequence is called a term . Each term in a sequence has a position (first, second, third and so on). For example, consider the sequence {5,15,25,35,…} In the sequence, each number is called a term.
## What are the two types of sequence?
A sequence is a set of numbers, called terms, arranged in some particular order.An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is called the common difference.A geometric sequence is a sequence with the ratio between two consecutive terms constant.
## What are the four rules of maths?
The four basic mathematical operations–addition, subtraction, multiplication, and division–have application even in the most advanced mathematical theories. Thus, mastering them is one of the keys to progressing in an understanding of math and, specifically, of algebra.
## What is a pattern problem?
Looking for a pattern is another strategy that you can use to solve problems. The goal is to look for items or numbers that are repeated or a series of events that repeat. The following problem can be solved by finding a pattern.
## What is a rule in math terms?
An algebraic rule is a mathematical expression that relates two variables and is written in the form of an equation. There are many constant algebraic rules, such as area = length x width. You can also create your own rule when given a set of variables.
## What is the formula of sequence?
An arithmetic sequence can be defined by an explicit formula in which an = d (n – 1) + c, where d is the common difference between consecutive terms, and c = a1.
## What are the 4 types of sequences?
What are Some of the Common Types of Sequences?Arithmetic Sequences.Geometric Sequences.Harmonic Sequences.Fibonacci Numbers.
## What does sequence mean?
noun. the following of one thing after another; succession. order of succession: a list of books in alphabetical sequence. a continuous or connected series: a sonnet sequence. something that follows; a subsequent event; result; consequence.
## What is sequence and example?
A sequence is an ordered list of numbers . The three dots mean to continue forward in the pattern established. Each number in the sequence is called a term. In the sequence 1, 3, 5, 7, 9, …, 1 is the first term, 3 is the second term, 5 is the third term, and so on.
## What is sequence and time?
From Wikipedia, the free encyclopedia. A sequential time is one in which the numbers form a normal sequence, such as 1:02:03 4/5/06 (two minutes and three seconds past 1 am on 4 May 2006 (or April 5, 2006 in the United States) or the same time and date in the “06” year of any other century).
## Why is sequencing important?
Sequencing is a very important concept for preschool children to develop since it allows children to recognize patterns that make the world more understandable and predictable. … Helping children sequence can help them learn routines and develop key academic skills like reading comprehension and scientific inquiry.<|endoftext|>
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Question Video: Comparing the Volumes of a Cylinder and a Cube given Their Dimensions | Nagwa Question Video: Comparing the Volumes of a Cylinder and a Cube given Their Dimensions | Nagwa
# Question Video: Comparing the Volumes of a Cylinder and a Cube given Their Dimensions Mathematics • Second Year of Preparatory School
## Join Nagwa Classes
Which has the greater volume, a cube whose edges are 4 cm long or a cylinder with a radius of 3 cm and a height of 8 cm?
02:31
### Video Transcript
Which has the greater volume, a cube whose edges are four centimetres long or a cylinder with a radius of three centimetres and a height of eight centimetres?
To answer this question, we need to remember how to calculate the volume of both a cube and a cylinder. Letβs begin with the cube. A cube is a special type of rectangular prism or cuboid, in which all three of the cubeβs dimensions are the same. We can refer to them as π.
The volume of a cube is therefore π multiplied by π multiplied by π, which we can write as π cubed. We take the side length of the cube and then cube it. So the volume of the cube in this question, which has a side length or edge length of four centimetres, is four cubed. You may know what four cubed is equal to. But if not, we could first work out four squared, which is 16, and then multiply this by four. 16 multiplied by four is 64. So the volume of the cube is 64 centimetres cubed.
Next, letβs consider the cylinder. A cylinder is described by two measurements, its height β and its base radius π. Its volume is given by ππ squared β. Here, ππ squared gives the area of the circular base. And then, we multiply by eight, which is the depth of the prism. Our cylinder has a radius of three centimetres and a height of eight centimetres. So its volume is given by π multiplied by three squared multiplied by eight. Three squared is equal to nine. And nine multiplied by eight is equal to 72. So the volume of the cylinder as a multiple of π is 72π.
Now, if we have a calculator, we can evaluate 72π. And it gives 226.194 continuing. So we can see that the volume of the cylinder will be larger than the volume of the cube. But we can also see this if we donβt have a calculator. The number π is approximately equal to 3.14. So 72π means 72 multiplied by something a little bit bigger than three. We can see that this is going to be bigger than 64 because 72 is already bigger than 64. And then, weβre multiplying it by three.
So we can conclude that the solid which has the larger volume is the cylinder.
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# Median
Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]
## Overview
In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, the median is not unique, so one often takes the mean of the two middle values.
At most half the population have values less than the median and at most half have values greater than the median. If both groups contain less than half the population, then some of the population is exactly equal to the median.
## Popular explanation
The big difference between the median and mean is illustrated in a simple example.
Suppose 19 paupers and 1 billionaire are in a room. Everyone removes all money from their pockets and puts it on a table. Each pauper puts £5 on the table; the billionaire puts £1 billion (i.e.£109) there. The total is then £1,000,000,095. If that money is divided equally among the 20 people, each gets £50,000,004.75. That amount is the mean amount of money that the 20 people brought into the room. But the median amount is £5, since one may divide the group into two groups of 10 people each, and say that everyone in the first group brought in no more than £5, and each person in the second group brought in no less than £5. In a sense, the median is the amount that the typical person brought in. By contrast, the mean is not at all typical, since nobody in the room brought in an amount approximating £50,000,004.75.
## Non-uniqueness
There may be more than one median: for example if there are an even number of cases, and the two middle values are different, then there is no unique middle value. Notice, however, that at least half the numbers in the list are less than or equal to either of the two middle values, and at least half are greater than or equal to either of the two values, and the same is true of any number between the two middle values. Thus either of the two middle values and all numbers between them are medians in that case.
## Measures of statistical dispersion
When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the range, the interquartile range, the mean absolute deviation, and the median absolute deviation. Since the median is the same as the second quartile, its calculation is illustrated in the article on quartiles.
Working with computers, a population of integers should have an integer median. Thus, for an integer population with an even number of elements, there are two medians known as lower median and upper median. For floating point population, the median lies somewhere between the two middle elements, depending on the distribution.So if there is not a middle number and there is two numbers left that is an example
## Medians of probability distributions
For any probability distribution on the real line with cumulative distribution function F, regardless of whether it is any kind of continuous probability distribution, in particular an absolutely continuous distribution (and therefore has a probability density function), or a discrete probability distribution, a median m satisfies the inequalities
$\operatorname{P}(X\leq m) \geq \frac{1}{2} \quad\and\quad \operatorname{P}(X\geq m) \geq \frac{1}{2}\,\!$
or
$\int_{-\infty}^m \mathrm{d}F(x) \geq \frac{1}{2} \quad\and\quad \int_m^{\infty} \mathrm{d}F(x) \geq \frac{1}{2}\,\!$
in which a Riemann-Stieltjes integral is used. For an absolutely continuous probability distribution with probability density function f, we have
$\operatorname{P}(X\leq m) = \operatorname{P}(X\geq m)=\int_{-\infty}^m f(x)\, \mathrm{d}x=0.5.\,\!$
Medians of particular distributions: The medians of certain types of distributions can be easily estimated from their parameters: The median of a normal distribution with mean μ and variance σ2 is μ. In fact, for a normal distribution, mean = median = mode.The median of a uniform distribution in the interval [a, b] is (a + b) / 2, which is also the mean.The median of a Cauchy distribution with location parameter x0 and scale parameter y is x0, the location parameter.The median of an exponential distribution with parameter $\lambda$ is the natural log of 2 divided by the scale parameter: $\frac{\ln 2}{\lambda}$The median of a Weibull distribution with shape parameter k and scale parameter $\lambda$ is $\frac{(\ln 2)^{1/k}}{\lambda}$
## Medians in descriptive statistics
The median is primarily used for skewed distributions, which it represents differently than the arithmetic mean. Consider the multiset { 1, 2, 2, 2, 3, 9 }. The median is 2 in this case, as is the mode, and it might be seen as a better indication of central tendency than the arithmetic mean of 3.166….
Calculation of medians is a popular technique in summary statistics and summarizing statistical data, since it is simple to understand and easy to calculate, while also giving a measure that is more robust in the presence of outlier values than is the mean.
## Theoretical properties
### An optimality property
The median is also the central point which minimizes the average of the absolute deviations; in the example above this would be (1 + 0 + 0 + 0 + 1 + 7) / 6 = 1.5 using the median, while it would be 1.944 using the mean. In the language of probability theory, the value of c that minimizes
$E(\left|X-c\right|)\,$
is the median of the probability distribution of the random variable X. Note, however, that c is not always unique, and therefore not well defined in general.
### An inequality relating means and medians
For continuous probability distributions, the difference between the median and the mean is less than or equal to one standard deviation. See an inequality on location and scale parameters.
## Efficient computation
Even though sorting n items takes in general O(n log n) operations, by using a "divide and conquer" algorithm the median of n items can be computed with only O(n) operations (in fact, you can always find the k-th element of a list of values with this method; this is called the selection problem).<|endoftext|>
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# In a regular parallelepiped, the base area is 144 cm2, and the height is 14 cm
In a regular parallelepiped, the base area is 144 cm2, and the height is 14 cm, determine the diagonal of the parallelepiped and the area of the flat surface of the parallelepiped.
Since the parallelepiped is regular, there is a square at its base. Let us determine the length of the side of the base of the parallelepiped if the area of the base is 144 cm2. AB = BC = DC = AD = √144 = 12 cm.
Consider a right-angled triangle ABD, whose legs are 12 cm.Let’s find the hypotenuse BD.
BD ^ 2 = AD ^ 2 + AB ^ 2 = 144 + 144 = 288.
ВD = 12 * √2.
Consider a right-angled triangle BDD1, whose leg DD1 is equal to the height of the parallelepiped DD1 = 14 cm, leg BD = 12 * √2, and the hypotenuse D1B is the parallelepiped’s diagonal.
D1B ^ 2 = DD1 ^ 2 + DB ^ 2 = 142 + (12 * √2) 2 = 196 + 288 = 484.
D1B = 22 cm.
Determine the surface area of the parallelepiped.
S = 2 * AB ^ 2 + 4 * AB * AA1 = 2 * 144 + 4 * 12 * 14 = 288 + 672 = 960 cm2.
Answer: D1B = 22 cm. S = 960 cm2.
One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.<|endoftext|>
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Black Hawk (1767-1838). The American Indian chief of the Sauk tribe, Black Hawk was the leader of the last war against white settlers in the Northwest Territory. He had a band of about 1,000 followers, many of whom were women, old men, and children.
Black Hawk was born in a Sauk village near the mouth of the Rock River in Illinois. In the War of 1812 he was recruited by the British to fight against the United States government. The Indians’ grievances increased after the war as settlers continued to take over their fields and homes.
In 1804 several members of the Sauk and Fox tribes had signed a treaty ceding all their lands east of the Mississippi River to the United States. Under Chief Keokuk, some of the Indians moved across the river to Iowa, but Black Hawk claimed the treaty was not valid. Forced to move in 1831, Black Hawk led his warriors and their families back into Illinois the following spring. American troops, aided by the Illinois militia, set out after them, and fighting soon broke out. Other tribes failed to help Black Hawk’s band, and it was crushed in August.
Black Hawk was imprisoned for a time and then was taken East, where he met President Andrew Jackson. Later he was allowed to return to Iowa. His autobiography, dictated to a government interpreter, is an American classic. He died in Iowa in 1838.
Black Hawk War
The Black Hawk War (1832) was the last major Indian-white conflict east of the Mississippi River. In 1804 representatives of the Sauk and Fox tribes signed a treaty abandoning all claims to land in Illinois. Although expected to remove to Iowa, they were permitted to remain east of the Mississippi until their former lands were sold. The Sauk leader, Black Hawk (1767-1838), opposed the treaty and rose to prominence when he fought for the British during the War of 1812.
When the Indians were finally ordered into Iowa in 1828, Black Hawk sought in vain to create an anti-American alliance with the Winnebago, Potawatomi, and Kickapoo. In 1829, 1830, and 1831, Black Hawk’s band returned across the Mississippi for spring planting, frightening the whites. When the Indians returned in 1832, a military force was organized to repulse them.
For 15 weeks Black Hawk was pursued into Wisconsin and then westward toward the Mississippi. He received no substantial support from other tribes, some of which even aided in his pursuit. On Aug. 3, 1832, the remnants of his band were attacked as they attempted to flee across the river and were virtually annihilated. Black Hawk escaped but soon surrendered. Imprisoned for a short time, he later settled in a Sauk village on the Des Moines River.
- Black Hawk, Life of Black Hawk (1833; repr. 1994);
- Nichols, Roger, Black Hawk and the Warrior’s Path (1992);
- Stevens, Frank, The Black Hawk War (1993);
- Whitney, Ellen, ed., The Black Hawk War, 2 vols. (1970-78).<|endoftext|>
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# The last digit of the number ${7^{886}}$is$a.{\text{ 9}} \\ b.{\text{ 7}} \\ c.{\text{ 3}} \\ d.{\text{ 1}} \\$
Last updated date: 17th Sep 2024
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Hint- last digit means we need to check the unit digit in ${n^x}$.For finding the unit digit number we must know the concept of cyclicity of power to a number.
For finding the last digit of the number, we will have to check unit digit in ${n^x}$
So, check the unit digit in ${7^{886}}$
${7^1} = 7$
So, the unit digit is 7.
${7^2} = 49$
So, the unit digit is 9.
${7^3} = 343$
So, the unit digit is 3.
${7^4} = 2401$
So, the unit digit is 1.
${7^5} = 16807$
So, the unit digit is 7.
${7^6} = 117649$
So, the unit digit is 9.
Now we can see that after 4 steps the unit digits are repeating.
Now in the given number the power of 7 is 886.
So divide 886 by 4
$\Rightarrow \frac{{886}}{4} = 221\frac{2}{4}...............\left( 1 \right)$
Now we are generalizing the number of form and the corresponding unit digit can be written as
$4n = 1 \\ 4n + 1 = 7 \\ 4n + 2 = 9 \\ 4n + 3 = 3 \\$
Now as we see from equation (1) that the power of 7 which is 886 is of form
$886 = 221\left( 4 \right) + 2 \\ {\text{i}}{\text{.e}}{\text{. }}4n + 2 \\$
Hence the unit digit would be 9.
So, the correct answer is option (a).
Note- In such types of questions if we asked to find out the last digit of some ${n^x}$number, first we have to check the unit digit and find after how many steps it gets repeated, then divide the power of number with the number of steps after which the unit digit is repeating then generalize the number as above we will get the required last digit of the given number.<|endoftext|>
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# Law of Sines… How? When? (NancyPi)
81,811
Published on May 5, 2020 by
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If you need to "solve the triangle" in a trigonometry problem, it just means to find all the missing angles and sides of the triangle.
WHAT is the LAW of SINES? Say that you're given an oblique triangle, meaning one that is not a right angle triangle, and you need to solve it. If you know two angles and at least one of the sides, you can solve with the Law of Sines trigonometry formula. The Law of Sines says that the sine of one angle over the side opposite that angle, equals the sine of the next angle over its opposite side, which equals sine of the last angle over its opposite length. These three ratios equal each other, and you can use these proportions to solve for unknowns.
Say that you're given that angle B is 34 degrees, angle C is 110 degrees, and side b is 14. We know two angles and one side. What's missing, or what we're looking for, is side a, side c, and the measure of angle A. That missing third angle, angle A, we can find very quickly without using any special new law or trig identity, since the sum of the degrees in a triangle is always 180 degrees. Subtracting the two known angles (34 and 110 degrees) from the total 180 degrees gives that the measure of angle A is 36 degrees.
HOW to use the Law of Sines: to find the missing two side lengths, we do need the Law of Sines theorem. First write all the ratios from the Law of Sines: the sine of each angle, over the length opposite. We write the sine of 36 degrees over the opposite side, side a. This equals sine of the next angle, 34 degrees, over its opposite side 14. And finally, that equals sine of the last angle, 110 degrees, over the opposite length, the unknown c.
If we look at just the leftmost two ratios, we see that we can use that equation to solve for a. So we separate that part out to solve: sin(36)/a = sin(34)/14. An easy way to solve a proportion like this is to cross-multiply. This gives us 14sin(36) = a*sin(34). Now it's more clear how to solve for side a, by just dividing out sin(34) from both sides. So we get that a is equal to 14*sin(36)/sin(34). To get a number value that's practical as the length of a triangle side, you can use your calculator to get a decimal number. CAUTION: since we were given angles in degrees, make sure your calculator is in degree mode, not radian mode, so that you don't get the wrong answer. The side length a is then approximately 14.72.
Now there's just one more unknown to find, the side c. To find the remaining missing side of a triangle, you can use a different pair of ratios to solve, the equation with the rightmost two ratios: sin(34)/14 = sin(110)/c. In general, to solve for one of the unknowns, use the ratio that has what you want to find in it, as the only unknown, and set it equal to a ratio where you know everything already, and you will get the answer. We cross-multiply to get c*sin(34) = 14sin(110). When we get c alone and use a calculator, we find that side length c is approx. 23.53. Now we've completely solved the triangle for all the missing sides and angles.
How do you know WHEN to use the LAW of SINES? If you're given two angles and a side (AAS or ASA cases), or two sides and an angle opposite one of those sides (SSA), you can use the Law of Sines property. If you have two angles and one side (AAS/ASA), you can use the Law of Sines just as we did to find the missing sides. WARNING: for the SSA case, when you have two sides and an angle that's opposite one of them, you can also use the Law of Sines to solve, but instead of having one solution, there may be no solution or two solutions. The SSA case is also called the "ambiguous case".
For more triangle solver math help as well as videos with trig identities, trigonometry problems, geometry, algebra, algebra 2, and calculus, check out: http://nancypi.com
Editor: Miriam Nielsen of zentouro @zentouro
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• You should set up your website to accept BAT Tokens if you have not already ๐
• Honestly i learn more math from here that in school keep up nancy
• I was thinking that she quit but here she is back… We missed you hope you will make some more vids.
• Thank you teacher excellent Lesson Math _ Sines in trigonometry – RAOUF
• triangles are nothing but deflated squares
• ูุงููู ูุงูููุง ุฌู
ุงู
• Iโm not here for math. Iโm here for her
• YOURE BACK!! this just made my entire week
• alison thorpe great
• u made my week ๐
• Hey!
Excuse me
• My thought exactly! Trig time!
• I Love you Nancy! Thank you for being an INCREDIBLE math teacher. I’ve been watching your video’s for years to refresh math concepts
• Thank you
• OMG THANK YOU SO MUCH!!! MY horrible trig teacher tried to teach this to the class but nobody could understand a single thing. I knew youโd be there to help. Thank you very much NancyPi!!! ๐
• You are a lifesaver during quarantine
• Why teachers don’t explain like this at the schools? Why? Thanks Nanci!
• They did the same thing. But we were confused, not really focused in it. And he took an entire hour for that. ๐๐คฃ Thanks Nancy
• I bet I know what’s coming next! Clearly it will be a video on the law of tangents.
• I knew this stuff because after all, I am a professor of mathematics. Despite having watched many of her videos, I am still attempting to figure out how she writes on a seemingly transparent board that’s facing the camera and she’s behind it. Does anyone have theories as to how this is possible?
• Nancy youre back! where have you been my calc grades have been slipping
• Nancy the cutie ‘pi’ is back again ๐
• Thank you for sharing this lessons Nancy you are awesome ๐๐ ๐ฎand cute ๐ณ๐โ๏ธ
• Safe return ๐ฌ
• 2016, what do you effing want?
• I may have seen this somewhere before. But now I understand it.
• Lmao how does anyone actually learn any math lol
• @Greg Bell I was just saying it can be distracting because of her looks ๐
• Mam plz make a series on numerical methods………… plz plz plz
• Your Channel is awesome. Wish I had professors that explained stuff like this when I was taking Calculus.
• Nice calm explanations. I suggest do not speed up the video when start writing, it’ a little confusing
• Heyy!!!
• Welcome back
• Hola Nancy, es grandioso que hayas subido un nuevo vรญdeo, tus explicaciones son muy claras pero tu sonrisa me distrae. Hi Nancy, it is great that you have uploaded a new video, your explanations are very clear but your smile distracts me.
• First video in six months, did you make parole?
• You welcome back!
• Hey nanci love you. I’ve been watching your videos from 2 years. A lot of lova from your biggest fan. Perfect example of beauty with brain
• Sorry its love
• So glad to see you back!
• Hey can you make a video on principal equations
• I love you sister . Please accept Islam . If you have any questions I will answer them .
• back?
• Nancy is back ๐ฅบ
• Glad to have you back!
• I love math. I really, really, reeeeeally love math.
• I found you wanting to review Trig I have not used in 30 years. Then I stroked 4years ago but remembered how much I loved Trig thru engineering. I still remember 90% of it just with out CAD (use of). I want to say how nice it is learning from you…….
• ๐ต๏ธโโ๏ธ
Multiply by
. ๐ธ
โคโค both sides and be reciprocal
• Do a livestream session! Would like the Queen of Maths to be my temporary teacher during quarantine๐คฉ๐
• Nancy where were you?? You launched a video after so many days
• Long long time……..
• Wow nice to glad to see you after long time….. Thank you ๐๐๐๐๐
• Beauty and brains! You are
• Sheโs back!!! Thank goodness!!
• Welcome back Nancy
• love u mam
• We need teacher like you in India.
Shaanti se padhai hoygi.. just calmmm relax and…
• Hello Nancy (& Maths Learners, Lovers & Experts). I thank you for this piece of Maths, and glad to see you still use your great MathTeaching Style. KUTGW !
• You came back, Nancy! ๐
• after long time. happy
• 5:20 “Seems like a garbage idea to me” best part hahahaha
• Mi loco no se Inglรฉs pero quede encantado con ella
• Mate, where have you been? Anyways, great to see you back
• Thank you Nancy pi
• Beautiful ice queen.
• Hi’ Nancy your back! I like this vedio, it’s remind me when I was studying during my college,.love this math subject,.thanks for sharing,.
• I could write.. like this, but then, I might sound… like, you ๐
• I’m not gonna lie to you Nancy but it’s a little bit late for this now ๐๐
• So what did people do before calculators?
• NancyPi is like the = sign in our equations, she helps us solve our problems in life. Life/<3=math(NancyPi)<--watch her videos to solve this equation.
• Wow it’s Nancy! Glad you’re alive. Nice video keep it up!
• good to see you again,
• Ah, okay, she’s back. All good then.
• Was scrolling through recommendations, saw the thumbnail and it brought me back to freshman year in College trig. Just wanted to say thank you for all the help your videos have given me throughout my college classes! Iโm not sure I wouldโve made it through all the math classes without it.
• Omg this is hot to watch
• Would it help to find the sin. of each angle so then you can deal with numbers in the equation and then divide things.
• cool
• Hay there !! Its a pleasure to see you after long time. Just to see you is also pushing adrenaline in body , you are incarnation of titanese the Goddess of wisdom
• This could be useful when installing air-conditioning. You can easily calculate the angle of the fandangle for optimum efficiency. OR as Burton Cummings once said, “baby, if you got the curves, I got the angles.
• Great video and clear as ever! Thanks for sharing, cheers!
RJ
• Verry good. Thank you teacher lovely ๐๐๐
• WOAHHH WAIT THIS IS THIS WEEK’S HOMEWORK!!!!! AND NANCY YOU’RE BACK!!! ๐๐๐๐
• Nancy can you please put Cosine rule on your list for videos please? It would be very much appreciated โค๏ธโค๏ธโค๏ธ๐
• Can you do a vid on trig identities double angle and half angle identities as well?
• Hi Nancy, We missed you so so so….. much
• Yayyy our faourite teacher is back
• If beauty with brain had a face
• Welcome back๐๐๐
• I think you would not make a video again.
• Stay at home and give us more videos , thank you
• Nancy Pi where have been… it has been a long time
• Thank you Nancy, we are actually doing this THIS WEEK at Uni. Great to see you again and love the humour, hope for more soon in COVID times ๐ love from Australia
• Good lecture for deliver
• Sheโs back!
• ๐ซโฅ๏ธโฅ๏ธโฅ๏ธNancy we missed u so much
• Was waiting for you from a long time
• You are a God among mere mortals!
• Hey Nancy, I just passed my last math class (calc 2) and I just wanna thank you so much for all the help.
• Queen is back!
• Hi Nancy welcome back
Iam math teacher from iraq
Can you help me
And you studio
Thanks
Best wishes
• I just love โค๏ธ you Nancy
• All my skipped lectures ended up here XD. You saved my high school math classess! Ty
• My Teachers this Quarantine period:
– Mathematics: NancyPi
– English: English with Lucy
• I love to learn English with Trump. He is funny and I really learn things
• Same here!!!
• Woahhh wtf bro so same!!!
• -Psychology: Jordan Peterson
-Physics: Fermilab
• minutephysics
• So good to see you back! I took calculus through differential equations in college 35 years ago and promptly stopped studying math when I started Medical School. For quite a while I have been rediscovering the beauty of math by re-learning the calculus, but occasionally stumble upon a huge gap in my memory of some basic concept once second nature to me all those years ago. This upsets me greatlyโbut when you happen to address the concept in one of your videos, your calm assuredness and straightforward presentations are so much appreciated. You are a superb teacher which in my line of work which is Medicine is the greatest complement I have to offer. Thank you again for the videos and welcome back. Steve K.
• Hi Nancy i missed you a lot and i’am glad to see you again ๐
• Always a great explanation, good to see you are back.
• Thanks for posting videos again
We missed you
• You are so cute nancy.nice vodeo
• hallo NancyPi, i am relly enjoying your mini-lectures. If it is possible, could you explain more regarding the theory of differential equation? I will appreciate for that.
• Your teaching is just awesome,well come back
• I love the humor you’ve been letting into these new videos! You’ve really kicked the editing up a notch too. You haven’t aged a day either, fyi!
• I don’t have maths, yet here I am!
• Damn damn damn, math.
• what is the theatan of that heart thingy?
• LOVED THIS VIDEO NANCY! You have inspired me to create my OWN channel which I will be creating videos covering topics in 6th-12th grade math. So far I have two math card games, and a series on integers, but have high school content planned for later this week. I am SUPER excited about my video going up tomorrow, which I think will make math fun for A LOT of people! I will be posting 3 videos a week and I would LOVE if you checked out my channel ๐
• I have watched tons of Nancyโs videos and that has led me to the Homemade Mathematics page which has opened up a whole new batch of content ๐ฅณ๐คฏ๐คฏ
• @allan donahue Thank you so much! I will! New videos every Monday, Wednesday and Friday!! ๐
• Homemade Mathematics Awesome! LETS GOOOO!
• Nancy pie is back ๐๐๐๐๐โค๏ธ๐๐๐โค๏ธโค๏ธโค๏ธ๐๐
• 7:42 I loved that subtle sarcastic comment. <3
• that’s so great you are back
• OMG!! you’re back!! <3
• Now I can get a good sleep! ^^
• you look ravishing than before…
• Darling he
• I can’t describe my happiness, I was upset knowing you stopped releasing/ posting content to youtube.
• we are so happy cause your back๐๐
• Thanks for the great video
• you helped me pass calc. thank you nancy
• Hello!
• Can you do one on laws of cosines
• welcome back queen
• Hi Nancy
Welcome back. Keep up the great work
• Fans Only ?
• Thanks, like your new sereis. Your’e not so intense now and a little more relaxed. When you started out, I was super paranoid of you, as In a former existence in another star system, I was a highly placed official. A type of general. They made a clone that was a killer, betrayer, and how you would tell these part from normal people, is they had no nivea’s on their bodies. So I looked and looked for a beauty spot on you and found one. When I did, I could finally relax. Good sereis. You take the difficult in math making it more than one dimension and therefore digestable. Thank you, wish you continued success.
• Ah! Finally you’re back
• NancyPi? More like cutiePi. Keep up the good work!
• Yeah, so glad to see a new post!
• Is this old recording? Plzz tell idol
• You mean: Angle Side Side (bless you, Mrs. Reordan, wherever you are)
• How did you know that I needed this????
• What kind of pens are used in the explanation?
• ๐โค๏ธ๐น
• This women soo beauty ๐ญ๐ญ๐๐ฅ
• Our math queen is back<3
• I’m guessing Law of Cosines video is next?, I can’t wait!!!
• what play list belongs this videos
• ๐
• Mam i am from India thanks ๐
• Great video Nancy! Welcome back! Please more videos like these.
• I learn so much , thanks
• Is a series on Linear Algebra on the horizon or Diff Equations?
• Law of Sirensโค๏ธ … Would you make a list of Differential Equations? Please. Is so gratificant tรญo learn with you.
• We’ve missed you! So happy to see you doing the maths!
• Long time For a Break….But i am still glad u are back:))
• Yo Nancy, can you make a vid on significant figures please?
• You are a real head turner !!
• Iam so happy to watch your lecture thank you so much
• She should talk about calc 3! ๐๐
• How do you find reciprocals for fractions??
• Superlative explanation.
• She is alive!
• So sexy Nancy. ๐
• Yeeees! Best news of 2020 yet!
• Love your videos Nancy… As a math teacher, i started creating videos for my students during Quarantine, so a newbie to youtube…. I am still wondering how do you create your videos ?
• How could I focus on math with such a beauty in the video๐๐
• I love you
• Do you take request? I have a problem you could teach how to solve that would be good for engineering students to know.
• Hi Nancy,
Can you show me the math that allows you to have stopped the ageing process?
Your face and skin are apparently not following the mathematical norms of getting older.
You’re not only a great mathematician, but a rather attractive one to boot.
Which reminds me of a famous quote, “To boot or not to boot, that is the hardest drive.”
Just saying,
LG
• My favorite math tutor
• Its hard to study bcuz I’m sinking in your eyes ..
• if u taught finance and economics I would get a distinction
• I don’t like math. I just watch your video because I like to.
• Even though math is my favourite subject and I’m good at it, for some reason I’m binge-watching her videos lol. Hope you’ll do Law of Cosines soon <3
• Hey nancy.
Please make a vid on second order derivatives…
• Nice to see you. To see you nice.
• On behalf of my students trying to work through the Law of Sines during the “at home learning” portion of this very interesting school year, I want to say thank you for the excellent tutorial on what is often a confusing topic. I hope that more students can appreciate the work into creating these videos and take advantage of them, and others, to help support their education in the information age. – @Osterclass
• Yay! you so awsome, we missed you.
• Great video nancy.
• Hi Nancy. If I know all three angles of a triangle there will be infinite number of triangles. Correct?
• I think your videos are truly helpful and have helped me learn concepts better and I cannot thank you enough for that. It would be really nice if you explained the topic of quadratic equations because that is what I am doing in my school right now and I- have no idea what I am learning. I find your teaching style quite comfortable and would appreciate it if you considered quadratics as a tutorial.
• Hi Nancy Have you done anything with applications of the Fourier transform?
I am particularly interested in holography and lens less microscopy.
• Did you know that a hologram is a 2D Fourier transform? To view it you do the same transform. did you know that computers can do the transform now?
• God bless u. Nice vedio
• FIVE PEOPLE TURN DOWN THIS VIDEO, THEY MUST BE FROM THE TRUMP ADMINISTRATION.
• *THANK YOU FOR SAVING MY GRADE FOR CALCULUS! SERIOUSLY!*
• this made my life YOU’RE BACK
• I watched the whole video… I’m not even in school anymore
• How are you making your hand movements. I mean your writing on the screen on your front side & still we are watching it without mirror effect.
• Thank you so much
• is it me.. but there is no way I could learn any math from her.. not an indictment on her.. I am too superficial
• So, do you like, own a ballet studio? That’s the only place I can think of that…oh, nevermind. But that is truly some Chris Nolan-level cinematography.
• You’re back! ๐
• Thank you so much. I love math but I have been out of school for too long so this is very refreshing.
• I thought you stopped making videos! I used your channel so much when I was teaching myself calc to prepare for college. Love your channel!
• Long time no see!
• OMG wait if we all put in like a few dollars each month to NancyPiโs Patreon she could just stop whatever and only make videos for us everyday! and then everything will be ok and the world becomes better place forever
• Com essa professora eu consigo aprender rรกpido sobre duas coisas que eu nรฃo compreendo absolutamente nada: falar inglรชs e matemรกtica.
• First time I see a pretty math teacher๐
• Thanks for being back
• ๐.From Turkey
• Yeeeeeeeee u back
• Excellent, thanks for another fantastic video. It really has helped me out. I would love to see you do one on partial fractions, Eigenvectors/values, and functions of two variables.
• Respect โค๏ธ
• Great love from India ๐ฎ๐ณ๐๐ฎ๐ณ
• Please upload more often ma’am. Please make a series on India’s NCERT curriculum if you can. You’d get millions of viewers. Indian students need you! ๐
• I already know all this, but still watch your videos because you are cute
• Thank you for your help. โบ I do love you my teacher. โค
• Great to see from you after a long time.It would be great if you cover euclidean geometry for math olympiads.
• Trying my best to pay attention but I get distracted.
• Yo does she got a insta! I need help with math
• omg her voice is amazing…
• Your video is so nice mam…
Love from India.๐๐๐
• I’d like to keep the clockwise form , the Angle Side Side
• oh thank god your back, i was scared id have to actually attend lectures
• Nancy thank you for all the help you are doing for students over break who are struggling. Would it be possible for you to make a short video on lโhรดpitals rule. That would be great. Thank you.
• Your videos are awesome. Keep up the good work. I can’t help but wonder , what do you do to make your writing so clear. What cameras do you use? How do you capture your voice?
• nice video, i am also a mathematician, can you tell me that which software you are using in your videos
• It’s soo refreshing to see u
• You have come back !!!
• Yey…. your back Nancy-pi …….. not a math person but thereโs always something one can learn from a talented educator plus your sweet presence ๐
• You are smart and beautiful
• Marry me plz. ๐ ๐ ๐
• Great to see you!!! Hope all is well. Ready to have you back full time.
• She writes it all backwards left handed!
• ya abiii nerdesin geliyim buliyim seni ya
• The best side of having known english is YOUUUU
• ok i’ll say it…*writing backwards is more impressive than actually doing maths*
• I didnt notice that at the beginin…
Something is up
• Must be a mirror app thgh
• NancyPi is the best.
• after long time you uploaded a video
• everyone asking how she writes backwards, no one wondering how she talks from behind the glass
• Yay, you’re back!
• Nice
• I love u. U r awesome
• Please mey what board you use to write.
• Glad you’re back on and posting videos again!
• What she is using for writing.can anybody tell me pls. Is it any app or software?
• Mam i am also a teacher and it will be a great help if you let me know the application you use for such kind of video interaction. ..๐
• Nice lecture
• from where you are mam
• dont stop making videos you make more sense than my professor
• U r very pretty sis
• please make some classes on integral calculus!!!!!!๐๐๐๐๐๐
• Great to see you back Nancy. Hope to see more videos soon. You’re a fantastic educator.
• Can u teach applications of differentiation?
• I want to use LOVe Sign on u . how u write this ? please tell me
• well another video comes after that corno ends i guess๐๐๐
• I am Happy to see you back in action! Thank you,
• Sheโs back yessirrr
• What did you major in at MIT ?
• It would be nice to show a proof for the Law of Sines.
• thanks
• You voice is so cute
• yay, you’re back, have missed the knowledge that you’re willing to shear.
Your vids helped me get through Engineering Degree Maths.
kai mihi, te hauora pai, kia kaha
• I can’t focus,. Help me!!!!!
• i can solve it with two methord …
• low of sine….
• ms nancy pi happy teacher’s day
• hocam at gibisiniz maลallah
• Are you even real?
• nice
• Can you make vedios for jee mains
• You are so hot and sexy
• So, Are you married now ?
• I just love the title of ur videos! ๐
• Thank Ma’am ๐
• I’m not here to learn the Law of Sines if you know what I mean!
• I like 10000 times how u explain my world best
• No se que dice por que no entiendo el Inglรฉs JAJAJAJAJA pero, Galileo Galilei decia: “Las matemรกticas es un idioma universal” se puede ver video en otro idioma y entiendes perfectamente. Ya conozco estos temas, pero me los verรฉ todos por que se hizo mi crush la del vรญdeo
• Where were you? Nice to see you back!
• I am studying from India good teacher
• could you please please do more videos….i missed your classes..i’m a huge fan and you are a life saver.
• Me likie
• ยฟCan’t be subtitled with other languages?
• Upload videos on vector algebra
• You might not like maths, but you can like my videos and subscribe.
Done. ๐
• Thank you Nancy love from Africa ๐
• holy crap i thought this channel went defunct. GLAD TO SEE YOU BACK NANCY
• Beauty with brains. ๐ฅฐ
• MAM in such situation you are our support keep uploading videos and stay safe
• Delighted to see you! How are you doing? I wish you all the best! PS:…The words “Don’t panic.” coming from a mathematician is a bit alarming….;-) Though, that it is spoken with one of the most reassuring and calm voices I know, helps a lot!
• Love u Lucy my best teacher want your more beautiful videos like you
• Your channel is very beautifulโบ.
We also have a math channel in Spanish๐๐. What kind of blackboard do you use so that people can see you and what you write?
• Upload full video of properties of โ
• Is she real or is it a robot?
• I like Math. Join me. Thanks. New friend.
• Can we chat,dear
• HI NANCY. . GOOD JOB.
HOW CAN I USE TRANSPARENT BOARD LIKE YOURS ?
WITH THANKS. โบ
• i don’t like math or your videos
i like you
(jk your videos are awesome! sometimes, i have dreams about you being my math teacher; i never skip class <3)
• nice video
• You’re such a fucking beautiful…
• Nancy pi a qt3.14
• ๐ช๐ช๐ช๐ด๐ด๐ด๐ด๐ด๐ด๐ด thumbs down
• Thanks for the informative videos. Easily understandable. Will you make similar videos for cosines, tangent, secant, cosecant, cotangent?
• How can you write it that way?
• who knew math was so hot
• Wish u were my math teacher and smooth with the like
• if u need a bf look no further ๐ slide my Dm’s
• I would love to see your take on solving some of these so-called ‘Viral’ math problems running around Facebook these days like 6 / 2(1 + 2). They create arguments that go on for days. There’s another one with Horses, Horseshoes and boots. It’s a set of Simultaneous equations. Yet arguments ensue… Your thoughts? Thanks!
• can you please do videos on computer science or something related please!
• Looking very ๐๐๐๐
• ๐ฅฐ๐ฅฐ๐ฅฐ๐ฅฐ๐๐๐๐๐๐๐๐๐๐๐๐๐ฅฐ๐๐ฅฐ๐ฅฐ
• Perfeito! The best teacher, I liked it here in Brazil!
• only once in 6 months๐ข
• Tbh i was only watching her the entire time
• ๐
๐
๐Hermosa
๐
๐
• Want Video on INTEGRALS and APPLICATION OF INTEGRALS
• Your lectures are a treat to my eyes.
• #lrnwthdeepika
• No la entiendo del todo pero es hermosa.
• Nice and amazing
• ๅฏ๏ผๆฏๆ่ฏๅพทไธ่ง่คฒๆถตๆฐๆฎ๏ผ๏ผ๏ผSO๏ผ??ใ
• Just here to say Thank you because I passed Algebra 2 with a B.. but uh haha Iโm free Friday night.
• Good video ๐i like it ๐ฏ๐
• This video reminded me that I need to look for how to solve SSS which is law of cosines. Thank you so much and I’m glad I just discovered your channel.
• My name is Muammar Kamal
I am Iraqi Math teacher
• You are so beautiful and prettier , thanks for video
• Nice
• expontial how can calculated can you understood
• hi
• Are you satisfy why unit circle don’t have unit area but still it satisfy all the conditions of math and physics and so on ,but if we take 1/โฯ. as radius of circle and it’s still satisfied all the trigonometry conditions ????
• Use the bhagwati shakti trikon
• How I talk with you
• I noticed you are left handed ๐ You are also an excellent teacher. I wish you would have been around when I was taking differential equations ! Have a blessed day
• I was really waiting for this video
• Welcome back Nancy!
• Perfect combination Beauty+intelligence=Nancy
• I watch your video eve though I know all these stuff!!
• HOW YOU ARE WRITING ANY TOOLS PLS INFORM ME
• Liked!
• Hi Nancy! Can you please make videos on Sets
• Omg
How does she writes in inverted/mirror?
• You are great and your videos really useful! But honestly, I’m glad that the teenager version of myself didn’t study with a teacher like you but with the fleshless books. I could focus just the math ; )
I wonder, does people here notice the great work you’ve been doing to realize these simple, concise, yet thorough explanations? And did anybody realize that the video must have been flipped horizontally, to be read correctly, and that we are seeing the horizontally-reflected version of your figure? Just wondering.
• I just watched *most* of your q&a from 2018. Has anyone figured out your wizardry? I love to figure such things out. I will only mention that your dominant hand may not be the one it seems to be, and note that inverse and opposite are different transitions. Thank you for what you do here!! ๐ Signed, someone who only recently discovered engineering and maths are super exciting after dismissing the subjects for a long time, and whom has pretty much no friends from pre-this interest to talk with about this stuff. Thanks again!!!
• I love you mame
• ๐๐๐๐ท๐ท๐ท๐๐๐๐ท๐ท๐ท๐๐๐๐๐๐๐๐๐๐ท๐ท๐ท๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ท๐ท๐ท๐๐๐๐๐๐
• Hi,
• ๐ค๐ค๐ค
• I Love mam aap se pyar kr te haโคโคโคโคโคโคโค๐๐๐๐๐๐๐๐๐๐๐
• Ye hindi janti h dost
• ๐คฃ๐คฃ๐คฃ…
• Iam a Btech 2nd year student..and i don’t know what iam doing here..
• Yayyyyy, you’re back! ๐ Are you enjoying the Covid Apocalypse? I hope all is well in your world. Cheers.
• ููุทู ูุดุทู
• Interested vdo
• so f***ing beautiful, feminine and smart woman?! I can’t believe that!
• Woman are a complex maths equation complicated lol
• Hello, I demand that this beatiful teacher make an integration by partial fraction, I hope i can see how she would explain it? also liked and subbed though
• Mam you are so beautiful
• *Islam is the beuty for all humans specially for womans please read slam coz slam is perfect not muslims—allah(god) ordered for womans to take hijab to cover their beutiful body–this is good for your health and others!!!!!!please read slam!!!!*
• Excellent explain ๐
Brain with beauty ..
• Very nice explanation ma’am thank you so much you are great
• I thi
• Nancy you are a excellent teacher love you very much
• I am doing my graduation ..I don’t know why I am here !!
• Hottest Math teacher on youtube
• you’re pretty๐๐
• I got finals this week and studying during Quarantine was super hard! thank you for your videos
• Love u mam im from india
• so beautiful
• Love it
• Which board or software u r using to write in such a way?
• Hi Nancy what the name of application that you use to flip vedio please tell me send the name
• there’s lot of programs that can do it, I believe you can flip video with openshot, which is free and open source
• btw law of cosines could have also been nice <3
• Quarantine has all the content creators coming back.
• el tema es muy sencillo ,pero la maestra es muy guapa
• You are as beautiful as math.
• U are so beautiful ๐๐
• Hi. Hi. I would like to start reading math from the beginning ,. How do I fallow the path of mathematics videos
• I can’t learn trigonometry by this way, Not only because She is a beautiful girl also she has soft voice.
• Aw! back againโค๏ธโ๏ธ
• Where were u? I waited so much ๐ญ๐๐ญ to
• I think I pass my numerical test if she was teaching me cute lady omg
• Is it me or does she have very long arms
• Nice tuitor, I admiring you
• It’s amazing how you write everything backward so it appears right side up,very smart
• Just a simple video editing trick. Video is flipped
• Why don’t I have a beautiful teacher like you ?
• Nice classe
• Impressed
• โa+b=11
a+โb=7 then prove a=4 and b=9 miss Nancy please solve it
• Nice,,,common d: come to my YouTube channel
• You should do another Q&A video
• Non threatening Maths, great!
• NancyPi is Very Hot!!!
• hi,great videos,canyou helpwiththis In a race the probability that Tom wins is 0.3, the probability that Mathew
wins is 0.2 and the probability that Jerry wins is 0.4.
Find the probability that
(i) Tom or Jerry wins,
(ii) neither Tom or Mathew wins.
Assume that there are no dead heats
• Wow! Beautiful teacher. It is never too late to learn.
• รtimo . Adorei a explanaรงรฃo.
• NancyPi I love the wild child is inside you
• NancyPi, do you have any idea how many want to have the amount of subscribers that you currently do? We wish to see more videos of you explaining things, in ASMR style, as you always do. Teach us arithmetic, simple math things.
• Yay, welcome back!
• I am an electrical engineer here in Brazil. I’ve always been passionate about math. But, in love with the teacher is the first time. Thank you for your work.
• Thank you for helping me pass my FE exam. -someone that never took college level math. Your videos were my go-to for learning calculus, and I tried at least 3 books, but didn’t get the ah-ha moment until I watched your videos.
• WTF! Is she writing on glass from her right to left with flip words? I’m so confused. ๐ฒ
• how u do this on screen can u please explain it
i mean writing on the screen
• law of cosines? when?
• Muy bueno!
• Dang… fine and smart wifey material
• Maths girls are wife material!
• PLEASE how you are writing (on a screen or a board ) and why it’s not opposite ?
• How to calculate valu of sin34
• I saw you and I gave like
• Most attractive woman on YouTube….by far!
• Mam you are very cute๐๐๐๐๐
• Nice method
• Madam jee rauaa kaha se bani
• Intelligent and beautiful ๐๐
• Precious girl! And what a great teacher.
• Lo ando viendo el la bella cuarentena
• When did you become an expert in Trigonometry?
Me: Well, it’s complicated.
• I mean u are very tired and stressful lady.
Level of difficulty is zero in your content.
In India 5ths class student solve your question s by getter approach!!!
Jai Hind!!!!!
• which software do you use?
• Hey. Hii can i get a video on legendre polynomials . Plz its just a lot of pain.
• No big deal. Trigonometry is very easy. Every mathematician knows.
• you aer good teacher thank you
• Hola.
ยฟNo puedes explicar en espaรฑol?
En ingles no entiendo.
• Nc
• Professora linda๐
• Can you please tell which application u r using here to teach
It will be highly obliged
• You should make courses on Udemy. I would definitely buy them.
• Western culture is worst no respect interaction between teacher and students
• Can you provide me editing method like this vedio
• Hi Nancy, I am a teacher from Iraq. What kind of board do you write?
Thanks
• Teach us how to do Mathematical Induction & Strong Mathematical Induction. <3
• Now shes ruining my good night sleep ๐
• math=mental abuse to humans
• Wow thรฌ ra lร ฤang o chuรดng em bรฉ mat me nhแป hehehe ha ha ha
• ๐ฒ๐ท๐ฒ๐ท๐ฒ๐ท๐ฒ
• Fantastically delivered in such a simple and lovely manner
Hope to be like you someday
Mathematics Teacher from Nigeria
• Killer dress, needs a plunging neckline though.
• Thanks Nancy for this fine presentation. Pythagoras would be proud of you, but don’t go near the square root of 2, he will get all spastic. To work these type of problems, I use the web to find a table of trig. ratios. Looking forward to your next video.
• nancy you are so beautiful love you! i like the way you teach us!
• Iโm not even in school anymore but couldnโt resist watching a new Nancy video!
• she talks like Mark Ruffalo
• Oh My God, you are here again.
Thank You.
• From top to bottom u look like India’s Nora Fategi.
8 y ^ 2 = x ^ 2 – x ^ 4, 1/3 ^ (1/2)
• Love your videos, Nancy. Thanks to you, my master studies were not that hard. Hope we can see more of your videos in the future.
• I’m watching math for fun
• How you are doing this… I mean how you are making video…Tell me
• Thank you for shedding a little light!
• Nice๐ฅฐ
• I love how there like 50 dislikes yet thereโs 3.4k likes ๐
• Can you arrange a series of video on thomas calculus please
• how do you do the writing?
• woow prefect you calculating the math like alberto Einstein i liked ๐ฎ๐ฎ
• can got phone
• Why does it seems to like like LAMIS THEOREM…..
• How come everytime you say don’t panic I start to panic.
• Sorry for this type of comment, but I am simply a man.
It is awesome when such a beautiful woman speaks math ๐
This is the only semester I failed in all of highschool because of my teacher, went back and passed it in 2 days in summer school with a different teacher, wish I had you then!
• Nancy is the best!
• Hey madam how are you? ?
• W
hen you realise that she is writing backwards.
• Oh my the way she write backward is truely insane๐ฏ
• I like your way of teaching very much can you please make videos on parametric differentiation
• I felt sleepy with your voice ๐ด๐ด
• You’re so beautiful
• Assuming that all of the angles and length of the three sides are known or have been calculates using the law of sines, knowing that the usual formula has been calculated using the one-half of base times heights, without going through the very messy process of calculating the a height, is there a procedure or formula for calculating the area?
• im angry because i know you would reject me.
• I will watch any nancy pi video, even if I know how to do this already.
• Mam aap hot ho
• I love how you speak. It is charming and makes me feel relaxed and when I am doing any math I like to feel relaxed. Also, do you write on glass or what because I want to know? To add how hard is it to write backwards? I am also a lefty too and I hate writing in books because I have ink or lead on my hand afterwards? thanks for the awesome video and would love if you could answer my questions. thanks again.
• It’s the lightboard, actually she is not writing backwards she writes normally like us but flips the video during editing
• I am studying bsc maths . but I can’t see such a beautiful maths teacher in my whole education career.
• I like the way you teach Nancy, thank you for the hard work you put into your videos. You deserve more subscribers.
• Great seeing you again.
• Owhhh thank U for the explanation
• I want you to update every week. Please come back!!!
• I love your approach, you make maths more easy, keep up the good work.
๐
๐
• I love you
• Hey Nancy, could you also do how to factorize cubics, please?
• Hey Nancy, do you write backwards on glass? Or do you have special software and a pen? Thanks!
• nice way of teaching love it
• Nice…teaching #MasterMathematics
• Glad to see you back Nancy!
• Is she writing backwards? That would be crazy good lol
• I can’t focus cause she is too pretty.
• Nancy! What software do you use to write on your video?
• Hi Nancy can you make a video on complex number
• Please excuse me, I’m sure I’ve asked before but,… please do a post about the Fourier Transform,… I could really use LaPlace but I’m a menial electrical engineer,… I just need Fourier …
• Ma’am how do you shoot your video
• I’m falling in โค๏ธ love with Mathematics after seeing your videos. I guess you might be the reason behind how the term ‘The Beauty of Mathematics’ was coined.
-Love from Nepal ๐ณ๐ต
• I’m in love with math after seeing u….๐
• if I had such kind of teacher, I’d definitely become the best educationist in the world ๐
• Hi nancy
• You talk where well
• Gr8t
I can’t understand a thing I am in middle class I don’t know what are these identities for
However I have liked your video
And I will come again after I have grown up a little bit.
• ๐ can you make more Video for class 11th and 12th….
• thanks for holding my hand through math class
• I love this. I just discovered your page, and I want to say thank you. I am a computer science major, and I go against the stereotype. Some people dont believe in my dreams because of it. Thank you for going into what you love, no matter if you fit some mold that was set by society for every carrier option out there. I appreciate your mark on the world.
• These trig formulas are extremely valuable in the building trade. Not all angles are right triangles. Knowing this stuff saves you from holding up heavy pieces of lumber (or steel) and marking the angle, or taping the lengths while leaning out over a thirty foot roof. Give it a try, thanks for the video.
• You are a maths Angel, really…
• I like her, your voice reminds me of Dakota Johnson!!
• ุฎูุด ุดุฑุญ ุจุณ ู
ู ุงุญุณู ู
ู ุญูุฏุฑ ูููุฏ ๐
• I donโt get it
• Hi Nancy. Have you ever looked at Big O notation in regards to computer science? Everyone is soooo bad at explaining it. Basically it boils down to calculating runtime and memory cost of a function.
Not sure if you could do a series on it, but since you’re obviously good at math, there’s really not any “code” going on at all. Simply all theoretical math. I bet it would get a ton of views.
• Can we be friends ?
• ๐ธ๐๐ฅ
• How tf does she write in backward thats amazing af
• Can you do stats equations
• How the hell does she writes all the things in reflected form so that it appears correct to us…..
• How the hell did u learned to write backwards
• Hi Nancy,
I know my comment is a month late ๐
I really enjoy your math teachings.
But I’ve got this esoteric soft copy book I would like you to help me look out. It’s essentially logarithms, like advanced logarithm problems. I would love to get it across to you. I would love that you see it, because it could go a lonnng way to help me.
Cheers Nancy ๐
• Your explanations are always very clear.
• I love you nancy
• ุดุฑุญ ุฑุงุฆุน
• Darling kya bable he
• For every video you make i’m so very thankfull!!
• hey Nancy, great video. Me and my frieds love to watch your content. My current math teacher, mr edwards is great, however u are greater. Just wondering what your skin care routine? need info for assignement. Is ur last name actually Pi or is that a stage name?
• My crush is back.. welcome ๐๐ฅฐ
• Good video great material Iโm a maths teacher myself , use these questions for my students . And not bad looking either so Iโm all signed up !
• You should upload more videos
• Mam plz tell me from where are you this is my dream I tend to meet you
• I see now…
• GREAT work as usual.. love your videos. I am required to teacher some of my math classes online starting in September “thank you COVID 19” and i am interested in the writing system you are using. So far i think it’s the one that will work best for me. Can you share what it is and or how to set one up?
• Dear NancyPi,
I am taking Calculus 2. Can you please make a video about disks and washers and when to use either. Also can you please do 4 problems? Problem 1, disks about the x-axis. Problem 2 same problem, but this time about the y-axis. Problem 3, washers about the x-axis and Problem 4 the same one, but about the y-axis. You don’t have to solve them. I just have trouble setting up the problem. The integration is the easy part. The hard part is the setup. Please dumb it down as much as possible. This is the 2nd time I take Calculus 2 and I still do not understand it.
• Domanic Marcus check out the organic chemistry tutors video
• You make math beautiful ;]
• yoooo my g nancy
• You brought the humor on this one! Awesome video.
• Hot teacher beautifull
• Do you write on a glass in front of you!
It’s so amazing!
• what’s the system that u use in your system ?
• me im going to learn watch all the videos from math how does it work to solve the problem
• i think its called geometry law of sines
• because the shape and letters are familiar to me hahhaha
• which means when you using sines of letter and using to equal the sines and put in number of degree i think it is? do you because im not very good at math hahaha
• Helo mam
• No one:
Absolutely no one:
NancyPi: This is how you solve for non write triangles *while global pandemic*
• hey nancy pls make regular videos about advanced topic in maths , with a month that help me a lot ,
we are from kerala (india ) your videos help me a lot , and i find that same videos in different playlist appers
• You are the perfect example of beauty with brain ๐๐๐๐๐๐
• I wanted to thank you for your videos. Your videos on integration got me an A in physical chemistry! I loved the class, mostly because the math made sense because of your lessons. Thank you!
• Calculus 3 and differential equations please !!
• I have been following thus chanell, hoo my goodness, this is so steady and clear. I always check on this channel to see new uploads. I got inspired to create my channel too. You are really doing a great job. Thumbs up and looking out for more
• Very nice nancy. I impreesed with ur method. I wanna follow u plz guide me
• wOW Great to see you back ! Welcome back you are our hero and super helper indeed thank you so much
• Madam can you please tell me what kind of board that you are using and tools..?
• U re beautiful โค๏ธโค๏ธ
• Mam pls upload videos fast I really need it๐๐๐
• Make more videos and upload daily…
• WTF@#k amazing you can Write backwards ? I still dont get eventually i will learn thnx for your wisdom
• Please explain trigonometric ratios and identities
• Wow!! It’s worth watching this video…๐ค
• I’m here not for studie, for watching this beautiful face ๐คฃ๐๐
• Hello can you please explain finance( Simple interest )
• Beautiful ๐ wonderful ๐
• Nice class mam
• Is it wrong to like a teacher?
• Thank you<|endoftext|>
| 4.84375 |
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The research of Professor Bob McMahon is important for many reasons. Bob and his graduate students study molecules found in interstellar space. They begin this process by synthesizing stable precursor molecules in the lab. The precursor molecules are then "isolated" between argon atoms at 10 Kelvin under high vacuum. A rough example of isolated precursor molecules would be to think of Jello with fruit in it. The Jello represents the argon and the fruit represents the precursor molecules. It is important to isolate the molecules at low temperatures between solid argon atoms for two reasons. First, because the strange environment mimics that of space - cold, with no neighboring space molecules. Secondly, the precursor molecules must be isolated because after they are cooled and in the argon "matrix" light is shined on them and reactive molecules, the kind often found in space, are formed. If the molecules were near each other, or if there was much kinetic energy available to them, they would surely react, and the space environment would not be effectively mimicked. Because of the popular theory of the Big Bang, studying these molecules allows for clues as to the origin of life and also helps us to understand space itself. The data gained from the studies done in the lab can be compared to that obtained by astronomers from radio telescopes and other devices. If the data matches, molecules found in space can be positively identified. The more we know about space, the more we can surmise about the origin of the universe.<|endoftext|>
| 3.953125 |
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