Chapter 4.6, Problem 10E

a.

To determine

Find an exponential model.

Answer to Problem 10E

  $n\left(t\right)=\boxed{{3500002}^{t/25}}$

Explanation of Solution

Given information:

The bat population in a certain Midwestern county was $350,000$ in $2009$ , and the observed doubling time for the population is $25$ years.

Find an exponential model $n\left(t\right)=n_0{2}^{t/a}$ for the population $t$ years after $2006$ .

Calculation:

Consider, bat population in a certain Midwestern county in year, $2009=350000$

Doubling time $=25year$

  $n_0=$ the initial population size,

  $r$ =the relative growth rate,

  $t$ =time

  $n\left(t\right)=n_0{2}^{t/a}$

  $\begin{array}{l}{n}_0=350000\\ r=25\end{array}$

  $n\left(t\right)=\boxed{{3500002}^{t/25}}$

Hence, the solution is, $n\left(t\right)=\boxed{{3500002}^{t/25}}$

b.

To determine

Find an exponential model.

Answer to Problem 10E

  $\boxed{n\left(t\right)=350000e^{0.028t}}$

Explanation of Solution

Given information:

The bat population in a certain Midwestern county was $350,000$ in $2009$ , and the observed doubling time for the population is $25$ years.

Find an exponential model for the population t years after 2006.

Find an exponential model $n\left(t\right)=n_0{e}^{rt}$ for the population $t$ years after $2006$ .

Calculation:

Consider, bat population in a certain Midwestern county in year, $2009=350000$

Doubling time $=25year$

  $n_0=$ the initial population size,

  $r$ =the relative growth rate,

  $t$ =time

  $n\left(t\right)=n_0{e}^{rt}$

  $\begin{array}{l}n\left(25\right)=350000e^{25/25}\\ =350000\times 2\\ =700000\\ now\\ 700=350e^{25r}\\ 2=e^{25r}\\ \ln 2=\ln e^{25r}\\ .693=25r\\ r=0.028\\ n\left(t\right)=350000e^{0.028t}\end{array}$

Hence, the solution is, $\boxed{n\left(t\right)=350000e^{0.028t}}$

c.

To determine

Find an exponential model.

Answer to Problem 10E

  

Explanation of Solution

Given information:

The bat population in a certain Midwestern county was $350,000$ in $2009$ , and the observed doubling time for the population is $25$ years.

Sketch a graph of the population at time $t$ .

Calculation:

The graph is as shown,

  

Hence, the solution is, GRAPH

d.

To determine

Find an exponential model.

Answer to Problem 10E

  $n\left(t\right)=\boxed{{3500002}^{t/25}}$

Explanation of Solution

Given information:

The bat population in a certain Midwestern county was $350,000$ in $2009$ , and the observed doubling time for the population is $25$ years.

Estimate when the population will reach $2$ million.

Calculation:

How much time it will to reach $2$ million.

  $\begin{array}{l}2,000,000=350000e^{0.028t}\\ \frac{2000,000}{350,000}=e^{0.028t}\\ 5.71=e^{0.028t}\\ ln5.71=\ln e^{0.028t}\\ 1.74=0.028t\end{array}$

  $\boxed{t=62.25}$

Hence, the solution is, $\boxed{t=62.25}$