Final answer:
To determine the population 8 months from now, integrate the rate function r(t) and add the result to the initial population of 15,000. After integrating and applying the initial condition, the town's population will be 15,112 people 8 months from now.
Step-by-step explanation:
The student's question involves the use of calculus, specifically integrating a rate of change to find a function that represents the total population P(t) over time. To find the population 8 months from now, we need to integrate the rate function r(t) = 4 + 5t²/³ people per month. The integration will yield the population as a function of time P(t), which will then be added to the initial population of 15,000 people.
The population at any time t months from now is given by the integral of the rate of change of population, which is:
P(t) = ∫(4 + 5t²/³)dt
Performing the integration, we obtain:
P(t) = 4t + ¾5tµ/³ + C
Since the population at time 0 (current population) is 15,000, we set t = 0 to find C:
15,000 = C
Therefore, P(t) = 4t + ¾5tµ/³ + 15,000
Now we can find the population 8 months from now by substituting t = 8 into P(t):
Population after 8 months = 4(8) + ¾5(8)µ/³ + 15,000
P(8) = 32 + 20(4) + 15,000
P(8) = 32 + 80 + 15,000
P(8) = 15,112 people
Hence, the population of the small town in Michigan will be 15,112 people 8 months from now.