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Xin Suen O.
Undergraduate in UC Berkeley, some experience tutoring high school students
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Physics (Newtonian Mechanics)
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Question:

A ball is attached by a string to a post, and is being whirled around with some speed. When the post is held still, θ = 70º. When the post is accelerated upward with acceleration a, this angle decreases and is now θ' = 60º. What is a? You can assume that the string has negligible mass, that the speed of the ball does not change, that the ball does not bobble up and down, and that the post is still vertical to the ground.

Xin Suen O.
Answer:

Draw Free-Body Diagram for the case where 1) post is held still and 2) post is accelerating. Resolve the forces into horizontal and vertical components. Write equations relating the tension in the string to the centripetal force, for case 1) F*sin70º=mv^2/r, where r is given by the length of the string multiply by sin70º (simple geomtery) for case 2) F'*sin60º=mv^2/r, where r is given by the length of the string multiply by sin60º (simple geomtery) Solve the equation to obtain an expression for a.

Calculus
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Question:

Find all a>1 such that the integral with boundaries from 1 to a of x*ln(x) equals to 1/4.

Xin Suen O.
Answer:

Integrate the expression using integration by parts. Simplify the expression and equate to 1/4. Solve using a graphic calculator.

Statistics
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Question:

There are n apples in a bag. 7 of the apples are green. The rest of the apples are red. Tom takes a random apple from the bag and eats it. Tom then takes another apple from the bag at random. He eats the apple. The probability that the Tom eats 2 green apples is 1/5. Find n.

Xin Suen O.
Answer:

Probability that 1st apple is green = 7/n Probability that 2nd apple is green given that first apple is green = 6/(n-1) Probability that both apples are green = (7/n)*(6/(n-1)) = 1/5 Solve the equation, we get n=15 as the only positive solution.

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