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Samantha B.
Engineering student at The George Washington University
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SAT II Mathematics Level 2
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Question:

The diameter and height of a right circular cylinder are equal. If the volume of the cylinder is 2, what is the height of the cylinder?

Samantha B.

We know that the volume of a right circular cylinder is given by V=π(r^2)h If the diameter and height are equal, then r=h/2 Thus, V=2= π((h/2)^2)h 2 = π(h^2/4)h 2 = (π/4)h^3 8/π = h^3 h=∛(8/π) h=1.37

Physics (Newtonian Mechanics)
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Question:

You are standing on the roof of a twelve story building. You reach your hand out and drop a ball straight down. Assuming that each story is 5 meters, and ignoring the effects of air resistance, how long does it take the ball to hit the ground?

Samantha B.

We start with the equation ∆x=vo∆t + ½ a∆t^2 We know that the original velocity has to be 0, so the equation becomes ∆x= ½ a∆t^2 Solving for ∆t, we get ∆t = √(2∆x/a) We know that a is the acceleration due to gravity, or 9.8 m/s^2 We also know that ∆x is distance to the ground, or 12 x 5meters = 60 meters Plugging these numbers into the equation, we get that ∆t = √((2(60meters))/((9.8 m/(s^2)))) = 3.5 seconds

Geometry
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Question:

Suppose you have four circles, each of radius 5 feet, lined up in a row so that each is barely touching the last, and then inscribed in a rectangle. Find the area of the space outside of the circles but within the rectangle.

Samantha B.

Four circles of radius 5 feet give us a total length of 40 feet, and a height of 10 feet. Thus, the area of the rectangle is 40 feet x 10 feet = 400 square feet. The area of each circle will be pi(radius)^2 or 25pi square feet Thus, the total area of all four circles is 100pi square feet, or approximately 314 square feet. Subtracting this from 400 square feet, we get that the area of this space is approximately 86 square feet.

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