Chapter 8.8, Problem 12PPS

(a)

To determine

To find: the value of $k$ .

Answer to Problem 12PPS

The value of $k=0.0299$ .

Explanation of Solution

Given information:

The given function $p\left(t\right)=\frac{12.95}{1+2.4e^{-kt}}$ .

Calculation:

Calculate the equation.

  $\begin{array}{c}10=\frac{12.95}{1+2.4e^{-k70}}\\ \frac{10}{12.95}=\frac{1}{1+2.4e^{-k70}}]\\ 1+2.4e^{-k70}=\frac{12.95}{10}\\ 2.4e^{-70k}=1.295-1\\ e^{-70k}=0.1229\\ \ln e^{-70k}=\ln 0.1229\\ -70k=-2.096\\ k=0.0299\end{array}$

Therefore the value of $k=0.0299$ .

(b)

To determine

To find: The population of Ohio reach $12$ million .

Answer to Problem 12PPS

The population will reach $12$ million after $114$ year.

Explanation of Solution

Given information:

The given function $p\left(t\right)=\frac{12.95}{1+2.4e^{-kt}}$ .

Calculation:

Calculate the function.

  $\begin{array}{c}12=\frac{12.95}{1+2.4e^{-kt}}\\ 12\left(1+2.4e^{-kt}\right)=12.95\\ 12+28.8e^{-0.0299t}=12.95-12\\ e^{-0.0299t}=\frac{0.95}{28.8}\\ -0.0299t=\ln 0.0329\\ t=114.18\end{array}$ .

Therefore the population will reach $12$ million after $114$ year.