Question

The population of a city since the year 2000 can be modeled as P(t)=800t²−700t+70000 where t=0 corresponds to the year of 2000. In which year is the population increasing at the rate of 8900 people per year?

Answer 1

The population of the city modeled by the equation P(t)=800t²−700t+70000 is increasing at the rate of 8900 people per year in the year 2006.

Step-by-step explanation:

The question involves finding the year when the population of a city is increasing at the rate of 8900 people per year. To do this, we need to calculate the derivative of the population model P(t) = 800t² - 700t + 70000, which gives the rate of population change with respect to time, and then solve for t when the derivative equals 8900.

First, let's take the derivative of P(t) with respect to t:

P'(t) = d/dt[800t² - 700t + 70000] = 1600t - 700.

Next, we set P'(t) equal to 8900 and solve for t:

1600t - 700 = 8900

1600t = 8900 + 700

1600t = 960

t = 9600 / 1600

t = 6

Since t=0 corresponds to the year 2000, t=6 corresponds to the year 2006. Therefore, the population of the city is increasing at the rate of 8900 people per year in 2006.