Question

A) find a function that models the population t years after 2000 (t=0 for 2000). B) use the functions from part (a) to estimate the fox population in the year 2008.

Answer 1

The Solution.

The function that models the population after t years is

$p(t)=ae^{bt}$

In this case,

$\begin{gathered} b=7\text{ \%=0.07} \\ \text{when t=0, p(0)=14000} \end{gathered}$

To find the value of a:

$14000=ae^{(0.07\times0)}$$\begin{gathered} 14000=a\times1 \\ 14000=a \\ a=14000 \end{gathered}$

So, the required function is

$p(t)=14000e^{0.07t}$

To estimate the fox population in the year 2008, we have

$\begin{gathered} \text{ In this case, t=8 years} \\ p(t)=14000e^{0.07(8)} \\ p(t)=14000e^{0.56}=14000\times1.7506725 \\ p(t)=24,509.415\approx24509 \end{gathered}$

Hence, the correct answer is 24509.