Question

part 2 of 2
in 2012, the population of a small town was 3,200. the population is decreasing at a rate of 3.7% per year. how can you rewrite an exponential decay function to find the quarterly rate?
how can you rewrite an exponential decay function?
○ a. multiply the annual growth rate by $\frac{1}{4}$ so it is a quarter of the rate.
○ b. multiply the exponent by $\frac{1}{4}$ so that the model compounds quarterly.
○ c. multiply the exponent by $\frac{1}{4}$ so that the model compounds quarterly.
○ d. multiply the annual growth rate by $\frac{1}{4}$ so it is a quarter of the rate.
what is the quarterly decay rate?
the quarterly decay rate is
(type an integer or decimal rounded to three decimal places as needed.)

Explanation:

Step1: Identify correct decay adjustment

For quarterly compounding of an annual decay model $P(t)=P_0(1-r)^t$, we adjust the exponent to $\frac{t}{4}$ (multiply exponent by $\frac{1}{4}$) to reflect 4 periods per year.

Step2: Calculate quarterly decay rate

Annual decay rate $r=3.7\%=0.037$. The quarterly rate is found by solving $(1-q)^4=1-0.037$:
$$1-q=(0.963)^{\frac{1}{4}}$$
$$q=1-(0.963)^{\frac{1}{4}}$$
Calculate $(0.963)^{\frac{1}{4}}\approx0.9902$, so $q\approx1-0.9902=0.0098=0.98\%$

Answer:

1. B. Multiply the exponent by $\frac{1}{4}$ so that the model compounds quarterly.
2. 0.98%