Question

Based on the census data, the population of whoville was 43,230 in the year of 2000 and has been growing at a rate of 3.5%

A. write an expontional modle for the population P(t), where t represents the number of years since 2000.
B. using your model, what is the estimated population in 2015?

Answer 1

The model would take the form of


`P(t)=P(0)e^(0.035t)`

So in 2015, when
`t=15`, the population would be


`P(15)=43230e^(0.035*15)\approx73079`

On the other hand, you could use the following discrete model for the population, which gives a similar result. If
`t=0` corresponds to the year 2000, then in 2001, when
`t=1`, you have


`P(1)=P(0)*1.035^1\approx44743`

In the next year,


`P(2)=P(1)*1.035^1=P(0)*1.035^2\approx46309`

And so on, making the general pattern for the
`t`th year to be


`P(t)=P(t-1)*1.035^1=P(t-2)*1.035^2=\cdots=P(0)*1.035^t`

This means in the year 2015, or when
`t=15`, the population should be approximately


`P(15)=P(0)*1.035^(15)\approx72425`