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Ann-janette L.

PhD Mathematician turned stay-at-home mom/Math Tutor for 7 years

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Pre-Calculus

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Question:

What is the cotangent of $$\pi$$?

Ann-janette L.

Answer:

You can use the unit circle to determine values of the $$sin x$$ and $$cos x$$ functions, but cot is not identified on the unit circle, so you must use an identity to link $$cot x$$ to the values that you CAN find on the unit circle. Use the identity: $$\cot x = \frac{\cos x}{\sin x}$$, and use the unit circle to find $$\cos \pi = -1$$ and $$\sin \pi = 0$$. The plug in these values: $$\cot \pi = \frac{\cos \pi}{\sin \pi} = \frac{-1}{0}$$. But division by zero is not defined, so $$\cot \pi$$ is undefined.

Statistics

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Question:

Problem: A teacher has 33 students. She gives a test and gives each student a grade that is a percentage. For example, Ann received a 97%, indicating that she answered 97% of the questions correctly. You are given the 33 grades and told to determine whether the average grade was significantly higher than 70%. Should you perform a one-sample t-test or a test of one proportion?

Ann-janette L.

Answer:

Since the variable values "student grades" are stated as percentages, you might think at first that you need to perform a test of one proportion. However, you need to look both at the type of data given and the question that is being asked. Since you are being asked a question about the AVERAGE grade, you will calculate the mean (average) of the grades and compare that to the hypothesized grade of 70, so you will perform a one-sample t-test, as long as all other assumptions for that test are satisfied. Suppose that the question was worded this way: You are given the 33 grades and told to determine whether a significant majority got a passing grade (a grade of 60 or more). In this case, you are being asked a question about the MAJORITY, which means more than 50%, or 0.50, which is a proportion. So you would take the original variable, "student grade", and construct a new variable from it, "Pass", which would be "yes" or "no", based on whether each grade was above 60 or not. Then, you would calculate the percent (or proportion) of grades that was considered passing, and use this statistic to perform a test of one proportion.

Algebra

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Question:

Find two consecutive, positive integers whose product is 72.

Ann-janette L.

Answer:

Success with word- or story-problems requires thinking carefully about what each word means. "find two consecutive, positive integers..." means: 1. you will find 2 numbers that are integers (integers are whole number that can be positive or negative), and you also know that they are "consecutive", meaning one comes right after the other in the list of integers, e.g., 3 and 4, or 21 and 22. So we have unknown numbers (variables) that are consecutive integers, so if we label one of them x, then the next one can be labeled x+1. 3. Next, we need to work toward putting these two numbers, x and x+1, into an equation that can be solved for x. Go back to the words and see what you haven't used yet. You are told "...whose product is 72". 4. "product" means multiply, "is" means equals, so this means if we multiply the two numbers together, we get 72. Use this to write an equation: (x)(x+1)=72. 5. Now this equation can be solved for x as follows: (x)(x+1)=72 x^2 + x = 72 x^2 + x - 72 = 0 This is a quadratic equation, so it must be factored to be solved: (x+9)(x-8)=0 Either factor can =0 for the product to be zero, so set each factor to zero: x+9=0 or x-8=0 x=-9 or x=8. Two possible solutions, but you were given one more piece of information: the two integers are "positive", so x=8 is the correct solution. Then the two integers are x=8 and x+1=9. Check your solution: The product of the two consecutive, positive integers is supposed to be 72, so (x)(x+1)=8*9=72.

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