Chapter 3.5, Problem 29E

a.

To determine

Complete the table using given data.

Answer to Problem 29E

The populations are $\left(106.1,143.15,196.25,272.37\right)$ .

Explanation of Solution

Given:

The given model is $p=20.6+85.5e^{0.0360t}$ . $p$ denotes the populations of Horry County, South Carolina.

The given table is

  

Calculation:

  $p=20.6+85.5e^{0.0360t}$

Hence the complete table is given below.

Year	Value of ‘   $t$ ’	Population   $p=20.6+85.5e^{0.0360t}$
$\begin{array}{l}1980\\ 1990\\ 2000\\ 2010\end{array}$	$\begin{array}{l}0\\ 10\\ 20\\ 30\end{array}$	$\begin{array}{l}p=20.6+85.5e^{0.0360\cdot 0}=106.1\\ p=20.6+85.5e^{0.0360\cdot 10}\approx 143.15\\ p=20.6+85.5e^{0.0360\cdot 20}\approx 196.25\\ p=20.6+85.5e^{0.0360\cdot 20}\approx 272.37\end{array}$

b.

To determine

Find the year in which the populations of Horry County reach at the given number.

Answer to Problem 29E

The populations of Horry County reach at the given number in $38$ years.

Explanation of Solution

Given:

The given model is $p=20.6+85.5e^{0.0360t}$ and the given number of population is $350,000$ .

Calculation:

  $p=20.6+85.5e^{0.0360t}$

Substitute the value of $p=350$ in the equation.

  $350=20.6+85.5e^{0.0360t}$

Subtract $20.6$ from both sides.

  $\begin{array}{l}\Rightarrow 350-20.6=20.6+85.5e^{0.0360t}-20.6\\ \Rightarrow 329.4=85.5e^{0.0360t}\end{array}$

Divide both sides by $85.5$ .

  $\Rightarrow \frac{329.4}{85.5}=\frac{85.5}{85.5}{e}^{0.0360t}$

  $\Rightarrow \frac{329.4}{85.5}=e^{0.0360t}$

Taking natural log on both sides.

  $\Rightarrow \ln \frac{329.4}{85.5}=\ln e^{0.0360t}$

Use power rule.

  $\Rightarrow 1.348756=0.0360t$

Divide both sides by $0.0360$ .

  $\begin{array}{l}\Rightarrow \frac{1.348756}{0.0360}=\frac{0.0360}{0.0360}t\\ \Rightarrow 37.456=t\\ \Rightarrow t\approx 38\end{array}$

Hencethe populations of Horry County reach at the given number in $38$ years.

c.

To determine

Check whether the given model is valid for long term predictions.

Answer to Problem 29E

The right answer is ‘no’.

Explanation of Solution

Given:

The given model is $p=20.6+85.5e^{0.0360t}$ .

Calculation:

The given model is not valid for long term predictions because for large values of ‘ $t$ ’, the values of model $p=20.6+85.5e^{0.0360t}$ increase without bound.

Hence the right answer is‘no’.