Answer :
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.
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.