Question

Population Growth A major corporation is building a 4,325-acre complex of homes, offices, stores, schools, and churches in the rural community of Glen Cove. As a result of this development, the planners have estimated that Glen Cove's population (in thousands) t years from now will be given by P(t) = 20t2 + 160t + 160 t2 + 8t + 64 . (a) Find the rate (in thousands of people/year) at which Glen Cove's population is changing with respect to time. P'(t) =

Answer 1

The rate at which Glen Cove's population is changing with respect to time is 40t + 168 thousand people per year.

Step-by-step explanation:

To find the rate at which Glen Cove's population is changing with respect to time, we need to find the derivative of the population function:

P(t) = 20t^2 + 160t + 160t^2 + 8t + 64.

To do this, we can take the derivative of each term separately.

The derivative of 20t^2 is 40t, the derivative of 160t is 160, and the derivative of 8t is 8.

Since 160t^2 and 64 are constant terms, their derivatives are 0.

Putting it all together, the derivative of P(t) is P'(t) = 40t + 160 + 8

= 40t + 168.

Therefore, the rate at which Glen Cove's population is changing with respect to time is 40t + 168 thousand people per year.