SAT Exponential Growth Problem with Solution

Today I would like to give a solution to the SAT exponential growth problem presented in this post: Exponential Growth and Decay

Here is the problem once again, followed by a solution.

Level 4 SAT Exponential Growth 

On January 1, 2015, a family living on an island releases their two pet rabbits into the wild. Due to the short gestation period of rabbits, and the fact that the rabbits have no natural predators on this island, the rabbit population doubles each month. If P represents the rabbit population  years after January 1, 2015, then which of the following equations best models the rabbit population on this island over time?

A) P = 2^{t/12 + 1}
B) P = 2^{t + 1}
C) P = 2^12t
D) P = 2^{12t + 1}

Solution using the exponential growth model formula: As seen in example (4) from this post, a quantity that continually doubles over a fixed time period can be modeled by the exponential function P = a(2)^t/d  where a is the quantity at time t = 0, and d is the doubling time in years. In this case, there are initially 2 rabbits, so that a = 2, and the doubling time is every month, or every  1/12  year.

It follows that P = 2(2)^{t ÷ 1/12} = 2(2)^12t = 2¹2^12t = 2^{1 + 12t} = 2^{12t + 1}, choice (D).

Notes: (1) For a review of the laws of exponents used here, see the following post: Laws of Exponents

(2) For additional methods for solving this problem, you may want to take a look at 28 New SAT Math Lessons – Advanced Course

More SAT and ACT Math Problems with Explanations

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