AP Precalculus Exponential Functions and Modeling

Question 1

An arithmetic sequence has a_3 = 11 and a_7 = 27. What is a_20?

Accepted Answer

79

Answer

From a_7 − a_3 = 4d = 16, the common difference is d = 4. Then a_1 = 11 − 2(4) = 3, so a_20 = 3 + 19(4) = 79.

Question 2

A geometric sequence begins 6, 18, 54, ... What is the sixth term?

Accepted Answer

1458

Answer

The common ratio is 3, so g_n = 6·3^(n−1). Then g_6 = 6·3^5 = 6·243 = 1458.

Question 3

A town has 2400 residents, and its population decreases by 6% each year. Which function models the population P after t years?

Accepted Answer

P(t) = 2400(0.94)^t

Answer

A 6% decrease each year multiplies the population by 1 − 0.06 = 0.94 every year. That gives P(t) = 2400(0.94)^t.

Question 4

Which function is equivalent to f(t) = 5·2^(3t) written in the form a·b^t?

Accepted Answer

5·8^t

Answer

By the power rule for exponents, 2^(3t) = (2^3)^t = 8^t. So f(t) = 5·8^t.

Question 5

A bacteria culture starts at 200 cells and doubles every 3 hours, so N(t) = 200·2^(t/3). After how many hours will the culture reach 3200 cells?

Accepted Answer

12

Answer

Set 200·2^(t/3) = 3200, so 2^(t/3) = 16 = 2^4. Then t/3 = 4 and t = 12 hours.

AP Precalculus Practice Test