Answered

Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

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.

Sagot :

GinnyAnswer
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.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.