Question

Starting with an initial population of P(0)=17, the population of a group of squirrels grows at a rate of

P′(t)=18−1/2t squirrels per month, for 0≤t≤36.
How much does the population change in the first 36 months? Round your answer to the nearest integer.

Answer 1

To determine the squirrel population change over 36 months with a growth rate of P'(t)=18-½t, we integrated the rate function from 0 to 36, yielding a population change of 324 squirrels.

Step-by-step explanation:

To find out how much the population of squirrels changes in the first 36 months given the initial population P(0)=17 and the growth rate P'(t)=18-½t squirrels per month, we integrate the rate of change over the given time interval.

The population change ΔP over time from 0 to 36 months is the integral of P'(t) from 0 to 36:

ΔP = ∫036 (18 - ½t) dt

Perform the integration:

1. ΔP = [18t - ½ * (½t2)]036
2. ΔP = [(18*36) - ½ * (½*362)] - [0]
3. ΔP = 648 - ½ * (½*1296) = 648 - 324 = 324

Therefore, the population increases by 324 squirrels over the period. Adding this to the initial population gives us the final population:

P(36) = P(0) + ΔP = 17 + 324 = 341