A small town in Indiana commissioned an actuarial firm to conduct a study to model its population growth. Based on current trends, the study found that the population (in thousands of people) for the next 7 years will be approximately [tex]P(t)=-8t^3+159t+2990[/tex], where [tex]t[/tex] is in years.

a. Find the rate of change function: [tex]P^{\prime}(t) = [/tex] [tex]\(\square\)[/tex]

b. [tex]P^{\prime}(1) = [/tex] [tex]\(\square\)[/tex]

c. [tex]P^{\prime}(2) = [/tex] [tex]\(\square\)[/tex]

d. [tex]P^{\prime}(3) = [/tex] [tex]\(\square\)[/tex]

e. [tex]P^{\prime}(4) = [/tex] [tex]\(\square\)[/tex]

What do these results mean for the town?
A. Its population will peak at about 135 people.
B. Its population will start decreasing within the next 3 years.
C. Its population will be zero within the next 3 years.
D. Its population will continue to increase for the next 4 years, but will do so more slowly.
E. None of the above.

To address the questions, we need to carefully go through the problem step-by-step.

a. Find the rate of change function, [tex]\( P'(t) \)[/tex]:

We start with the given population model:
[tex]\[ P(t) = -8t^3 + 159t + 2990 \][/tex]

To find the rate of change function [tex]\( P'(t) \)[/tex], we differentiate [tex]\( P(t) \)[/tex] with respect to [tex]\( t \)[/tex]:
[tex]\[ P'(t) = \frac{d}{dt}(-8t^3 + 159t + 2990) \][/tex]

Applying the rules of differentiation:
[tex]\[ P'(t) = -24t^2 + 159 \][/tex]

b. Calculate [tex]\( P'(1) \)[/tex]:
[tex]\[ P'(1) = -24(1)^2 + 159 = -24(1) + 159 = -24 + 159 = 135 \][/tex]

So, [tex]\( P'(1) = 135 \)[/tex].

c. Calculate [tex]\( P'(2) \)[/tex]:
[tex]\[ P'(2) = -24(2)^2 + 159 = -24(4) + 159 = -96 + 159 = 63 \][/tex]

So, [tex]\( P'(2) = 63 \)[/tex].

d. Calculate [tex]\( P'(3) \)[/tex]:
[tex]\[ P'(3) = -24(3)^2 + 159 = -24(9) + 159 = -216 + 159 = -57 \][/tex]

So, [tex]\( P'(3) = -57 \)[/tex].

e. Calculate [tex]\( P'(4) \)[/tex]:
[tex]\[ P'(4) = -24(4)^2 + 159 = -24(16) + 159 = -384 + 159 = -225 \][/tex]

So, [tex]\( P'(4) = -225 \)[/tex].

Finally, interpret the results:

Looking at the values of [tex]\( P'(t) \)[/tex]:

- When [tex]\( t = 1 \)[/tex], [tex]\( P'(1) = 135 \)[/tex]: The population is increasing.
- When [tex]\( t = 2 \)[/tex], [tex]\( P'(2) = 63 \)[/tex]: The population is still increasing but at a slower rate.
- When [tex]\( t = 3 \)[/tex], [tex]\( P'(3) = -57 \)[/tex]: The population is decreasing.
- When [tex]\( t = 4 \)[/tex], [tex]\( P'(4) = -225 \)[/tex]: The population is decreasing even more rapidly.

Based on the values of the derivative [tex]\( P'(t) \)[/tex], we can conclude:
- The population will start decreasing within the next 3 years.

So, the correct interpretation is:

Its population will start decreasing within the next 3 years.