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 :
[tex]$\frac{dP(t)}{dt}=200e^{0.02t}$[/tex] in 1990.
Now integrating,
[tex]$\int_0^{20}\frac{dP(t)}{dt}dt=\int_0^{20}200e^{0.02t} \ dt$[/tex]
[tex]$=\frac{200}{0.02}\left[e^{0.02(20)}-1\right]$[/tex]
[tex]$=10,000[e^{0.4}-1]$[/tex]
[tex]$=10,000[0.49]$[/tex]
=4900
[tex]$\frac{dP(t)}{dt}=200e^{0.02t}$[/tex]
[tex]$\int1.dP(t)=200e^{0.02t}dt$[/tex]
[tex]$P=\frac{200}{0.02}e^{0.02t}$[/tex]
[tex]$P=10,000e^{0.02t}$[/tex]
[tex]$P=P_0e^{kt}$[/tex]
This is initial population.
k is change in population.
So in 1995,
[tex]$P=P_0e^{kt}$[/tex]
[tex]$=10,000e^{0.02(5)}$[/tex]
[tex]$=11051$[/tex]
In 2000,
[tex]$P=10,000e^{0.02(10)}$[/tex]
[tex]=12,214[/tex]
Therefore, the change in the population between 1995 and 2000 = 1,163.
