Chapter 4.6, Problem 11E

a.

To determine

Find the population grows exponentially.

Answer to Problem 11E

  $\boxed{20000}$

Explanation of Solution

Given information:

The graph shows the deer population in a Pennsylvania county between $2003$ and $2007$ . Assume that the population grows exponentially.

What was the deer population in $2003$ ?

  

Calculation:

According to the graph the deer population in $2003$ was $\boxed{20000}$ .

Hence, the solution is, $\boxed{20000}$

b.

To determine

Find the population grows exponentially.

Answer to Problem 11E

  $\boxed{n\left(t\right)=20000e^{0.1096t}}$

Explanation of Solution

Given information:

The graph shows the deer population in a Pennsylvania county between $2003$ and $2007$ . Assume that the population grows exponentially.

  

Find a function that model the deer population $t$ years after $2003$ .

Calculation:

According to the graph the deer population after $4$ years in $2007$ was $31000$

So calculate the relative rate of growth.

  $\begin{array}{l}n\left(t\right)=n_0{e}^{rt}\\ 31000=20000e^{4r}\\ 31000/20000=e^{4r}\\ 1.55=e^{4r}\\ \ln 1.55=\ln e^{4r}\\ 0.43825493=4r\\ r=\frac{0.43825493}{4}\\ r=0.109563732\\ r\approx 0.1096\\ \boxed{n\left(t\right)=20000e^{0.1096t}}\end{array}$

Hence, the solution is, $\boxed{n\left(t\right)=20000e^{0.1096t}}$

c.

To determine

Find the population grows exponentially.

Answer to Problem 11E

  $\boxed{48000}$

Explanation of Solution

Given information:

The graph shows the deer population in a Pennsylvania county between $2003$ and $2007$ . Assume that the population grows exponentially.

  

What is the projected deer population in 2011?

Calculation:

Initial population in $2003$ was $20000$ ,

  $\begin{array}{l}{n}_0=20000\\ t=2011-2003=8years\\ put\\ t=8in\boxed{n\left(t\right)=20000e^{0.1096t}}\\ n\left(8\right)=20000e^{0.1096\times 8}\\ =20000\left(2.403197158\right)\\ =48063.94316\\ \approx 48000\end{array}$

Hence, the solution is, $\boxed{48000}$

d.

To determine

Find the population grows exponentially.

Answer to Problem 11E

  $\boxed{2017}$

Explanation of Solution

Given information:

The graph shows the deer population in a Pennsylvania county between $2003$ and $2007$ . Assume that the population grows exponentially.

  

In what year will the deer population reach $100,000$ ?

Calculation:

Initial population in $2003$ was $20000$ ,

  $\begin{array}{l}n\left(t\right)=100000\\ n\left(t\right)=20000e^{0.1096t}\\ 100000=20000e^{0.1096t}\\ e^{0.1096t}=\frac{100000}{20000}=5\\ \ln e^{0.1096t}=\ln 5\\ 0.1096t=1.609437912\\ t=14.68465249\\ t=14years\\ 2003+14=2017\end{array}$

Hence, the solution is, $\boxed{2017}$