Chapter 4.6, Problem 5E

a.

To determine

Find the population grows exponentially.

Answer to Problem 5E

  $\boxed{n\left(t\right)=18000e^{0.08t}}$

Explanation of Solution

Given information:

The fox population in a certain region has a relative growth rate of $8\%$ per year. It is estimated that the population in $2005$ was $18000$ .

Find a function $n\left(t\right)=n_0{e}^{rt}$ that model the population $t$ years after $2005$ .

Calculation:

Consider exponentially growth model,

  $\begin{array}{l}n\left(t\right)=n_0{e}^{rt}\\ r=8\%=0.08\\ n_0=18000\\ n\left(t\right)=18000e^{0.08t}\end{array}$

Therefore, function that model the population $t$ years after $2005$ is,

  $\boxed{n\left(t\right)=18000e^{0.08t}}$

Hence, the solution is, $\boxed{n\left(t\right)=18000e^{0.08t}}$

b.

To determine

Find the population grows exponentially.

Answer to Problem 5E

  $\boxed{n\left(8\right)=34137}$

Explanation of Solution

Given information:

The fox population in a certain region has a relative growth rate of $8\%$ per year. It is estimated that the population in $2005$ was $18000$ .

.Use the function from part $\left(a\right)$ to estimate the fox population in the year $2013$ .

Calculation:

Estimate the fox population in the year $2013$ .

  $\begin{array}{l}now,\\ t=2013-2005=8\\ n\left(t\right)=18000e^{0.08t}\\ put,t=8\\ n\left(8\right)=18000e^{0.08\left(8\right)}\\ n\left(8\right)=18000e^{0.64}\\ n\left(8\right)=18000\left(1.896480879\right)\\ n\left(8\right)=34136.65583\\ n\left(8\right)\approx 34137\end{array}$

Therefore, the fox population in the year $2013$ is $\boxed{n\left(8\right)=34137}$ .

Hence, the solution is, $\boxed{n\left(8\right)=34137}$