Question

The population of a certain town was 10,000 in 1990. The rate of change of the population, measured in people per year, is modeled by , where t is measured in years since 1990. Discuss the meaning of . Calculate the change in population between 1995 and 2000. Do we have enough information to calculate the population in 2020

Answer 1

The question is incomplete. The complete question is :

The population of a certain town was 10,000 in 1990. The rate of change of a population, measured in hundreds of people per year, is modeled by P prime of t equals two-hundred times e to the 0.02t power, where t is measured in years since 1990. Discuss the meaning of the integral from zero to twenty of P prime of t, d t. Calculate the change in population between 1995 and 2000. Do we have enough information to calculate the population in 2020? If so, what is the population in 2020?

Solution :

According to the question,

The rate of change of population is given as :

`$(dP(t))/(dt)=200e^(0.02t)$` in 1990.

Now integrating,

`$\int_0^(20)(dP(t))/(dt)dt=\int_0^(20)200e^(0.02t) \ dt$`

`$=(200)/(0.02)\left[e^(0.02(20))-1\right]$`

`$=10,000[e^(0.4)-1]$`

`$=10,000[0.49]$`

=4900

`$(dP(t))/(dt)=200e^(0.02t)$`

`$\int1.dP(t)=200e^(0.02t)dt$`

`$P=(200)/(0.02)e^(0.02t)$`

`$P=10,000e^(0.02t)$`

`$P=P_0e^(kt)$`

This is initial population.

k is change in population.

So in 1995,

`$P=P_0e^(kt)$`

`$=10,000e^(0.02(5))$`

`$=11051$`

In 2000,

`$P=10,000e^(0.02(10))$`

`=12,214`

Therefore, the change in the population between 1995 and 2000 = 1,163.