Exponential Growth and Decay

The current version of the SAT gives problems on exponential growth and decay. Here are the basics that you should know if you want to get a perfect SAT score:

A general exponential function has the form f(t) = a(1 + r)^ct, where a = f(0) is the initial amount and r is the growth rate. If r > 0, then we have exponential growth and if r < 0 we have exponential decay.

Examples: (1) The exponential function  f(t) = 300(2)^t  can be used to model a population with a growth rate of 1 = 100% each year that begins with 300 specimens. The growth rate of 100% tells us that the population doubles each year.

(2) The exponential function  f(t) = 50(3)^2t  can be used to model a population with a growth rate of 2 = 200% every 6 months that begins with 50 specimens. The growth rate of 200% tells us that the population triples. Since c = 2, the tripling occurs every  1/2  year or 6 months.

(3) The exponential function  f(t) = 120(0.75)^t/3  can be used to model a substance which is decaying at a rate of 1 – 0.75 = 0.25 = 25% every 3 years. The initial amount of the substance might be 120 grams. Since c = 1/3, the 25% decay occurs every 3 years.

(4) A quantity that continually doubles over a fixed time period can be modeled by the exponential function f(t) = a(2)^t/d  where a is the quantity at time t = 0, and d is the doubling time in years.

Now try the following SAT math problem. I will post a solution after Thanksgiving, but feel free to post your own solutions in the comments meanwhile.

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}

More SAT and ACT Math Problems with Explanations

If you are preparing for the SAT or ACT, you may want to take a look at the Get 800 collection of test prep books.

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