Question

It is estimated that t years from now the population of a city will number P(t)=(0.6 t-6)(0.1 t+9)+5 thousand people. How fast will the population (in thousands) be growing in 7 years?

Answer 1

To find the growth rate of the population in 7 years, take the derivative of the population function and substitute t = 7 into the derivative.

Step-by-step explanation:

To find how fast the population is growing in 7 years, we need to find the derivative of the population function. The population function is given as P(t) = (0.6t - 6)(0.1t + 9) + 5. We can take the derivative of P(t) with respect to t to find the rate of change of the population. After taking the derivative, we can plug in t = 7 to find the growth rate at that point in time.

Finding the derivative:

P'(t) = (0.6)(0.1t + 9) + (0.1)(0.6t - 6) = 0.06t + 5.4 + 0.06t - 0.6 = 0.12t + 4.8

Calculating the growth rate at t = 7:

P'(7) = 0.12(7) + 4.8 = 0.84 + 4.8 = 5.64

Therefore, the population will be growing at a rate of 5.64 thousand people per year in 7 years.