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Matt G.
Biology Student at Vanderbilt University
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SAT II Mathematics Level 2
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Question:

There are 15 marbles in a bag, 6 red and 9 blue. 5 marbles are randomly drawn from the bag without replacement. What is the probability that there are more blue than red marbles drawn? The marbles are equal size, and otherwise identical except for color.

Matt G.
Answer:

The marbles are identical except for color so order doesn't matter, and we are looking for combinations. Recall that for a combination of $$B$$ items from a set of $$A$$, we can abbreviate it as $$(_AC_B)$$, and this is equal to $$\frac{A!}{B!*(A-B)!}. $$ There are $$(_{15}C_5)$$, or 3003, combinations of 5 marbles from 15. This is the denominator. There can be 5 blue and 0 red, 4 blue and 1 red, or 3 blue and 2 red marbles. Thus, we can express the total number of valid combinations as $( (_9C_5)(_6C_0) + (_9C_4)(_6C_1)+(_9C_2)(_6C_2) = (126)(1)+(126)(6)+(84)(15)=2124$) And the total probability of valid combinations versus the total is $$\frac{2124}{3003}$$

Geometry
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Question:

Explain how the unit circle is used in calculating the sine, cosine, and tangent values of a given angle A.

Matt G.
Answer:

Sine of A is defined as the length of the opposite leg divided by the length hypotenuse of a right triangle with angle A. Similarly, cosine is the length of the adjacent leg divided by the length of the hypotenuse. Abbreviated, sinA = o/h, and cosA = a/h. If we set the hypotenuse as 1, then sinA = o, and cosA = a, simplifying things. Let us imagine a circle with radius 1, centered on the origin. As the radius line sweeps around all 360 degrees of the circle, a triangle can be formed with one line on the x-axis, and one line parallel to the y-axis. The angle A is defined as the corner on the origin, with the angle A going from the "radius line" to the positive side of the x-axis. Therefore, the height of the end of the "radius line" is equal to the length of the opposite side, or sinA. Likewise, the distance of the end of the "radius line" from the y-axis is equal to cosA. Tangent is equal to the opposite length divided by the adjacent length. This is also equal to sinA/cosA. Since we know sinA and cosA from the unit circle, we can find tanA as well.

Biology
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Question:

Explain how a drug that made the mitochondrial membrane permeable to protons would promote weight loss.

Matt G.
Answer:

In normal conditions and function, a proton gradient is created across the inner membrane of a mitochondrion with the higher concentration in the intermembrane space. A drug that made the membrane permeable to protons would destroy this gradient, and stop or greatly inhibit the ability of ATP synthase to produce ATP. Thus, burning calories would be disconnected from ATP production and more calories would need to be burned in order to produce the same amount of ATP.

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