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Nick N.

Harvard Graduate 2017

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Pre-Calculus

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Question:

A second degree polynomial of the form $$3x^2+Ax+B$$ where A and B are constants is equal to 0 when $$x=3$$ and $$x=\frac{-2}{3}$$. What are the values of A and B?

Nick N.

Answer:

To solve this problem we will begin by representing our second degree polynomial as the product of two first degree polynomials. Based on the information we are given, we can write: $$(x-3)(x+\frac{2}{3})=0$$ We can expand using the FOIL method to get: $$x^2-2\frac13x-2=0$$ We're almost done, but our new polynomial's leading coefficient does not match the one we were given. We can simply multiply our equation by 3 to find: $$3x^2-7x-6=0$$ So if $$3x^2+Ax+B=0$$ when $$x=3$$ and $$x=\frac{-2}{3}$$, then A = -7 and B = -6.

Trigonometry

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Question:

What is the tangent of 45 degrees?

Nick N.

Answer:

Here we can use our knowledge of right triangles to work out the answer without a calculator. In a right triangle the tangent of an angle is equal to $$\frac{\text{Opposite Side Length}}{\text{Adjacent Side Length}}$$. Our geometry knowledge tells us that the legs in a 45º - 45º - 90º triangle are of equal size, so we can use our ratio $$\frac{\text{Opposite Side Length}}{\text{Adjacent Side Length}}$$ to show that the tangent of 45º is equal to 1.

ACT

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Question:

You are searching for artifacts on the ocean floor using a remotely operated submarine. You want to view a shipwreck that is 12,000 ft below the surface, and your submarine descends at a constant rate of 16 ft every 3 seconds. About how long will it take your submarine to reach the wreck, assuming it begins at the surface? A) 4 minutes B) 12 minutes C) 38 minutes D) 46 minutes

Nick N.

Answer:

There are multiple approaches you could use to solve this problem. One method would be to divide 16 ft by 3 seconds, getting a rate of 5 1/3 ft per second. To find the time it would take to travel 12,000 ft at this rate we divide 12,000 ft by 5 1/3 ft per second and get 2,250 seconds, which can be converted to 37.5 minutes. You could also reach this answer by setting $$\frac{16 ft}{3 seconds}$$ = $$\frac{12,000 ft}{X seconds}$$ and using cross products. To solve for X you would multiply 3 seconds by 12,000 ft to get 36,000 ft * seconds, which you then divide by 16 ft to get 2,250 seconds. The answer is C!

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