Chapter 8.8, Problem 10PPS

(a)

To determine

To find: The equation of the form $y=a{e}^{kt}$ for Oregon.

Answer to Problem 10PPS

The equation is $y=a{e}^{0.01405t}$ .

Explanation of Solution

Given information:

The given function growth form is $3.4$ million $2000$ .

Calculation:

Calculate the equation.

  $\begin{array}{c}y=a{e}^{-kt}\\ 3.7=3.4e^{k6}\\ \frac{3.7}{3.4}=e^{6k}\\ 1.088=e^{6k}\\ \ln 1.088=\ln e^{k6}\\ 0.0843=6k\\ k=\frac{0.0843}{6}\\ =0.01405\end{array}$

Therefore the equation is $y=a{e}^{0.01405t}$ .

(b)

To determine

To find: The population of Oregon the equation is $2020$ .

Answer to Problem 10PPS

The population in $2020$ is $4.498$ million.

Explanation of Solution

Given information:

The given growth form is $3.4$ million $2000$ .

Calculation:

Calculate the equation.

  $\begin{array}{c}y=a{e}^{-kt}\\ putt=20\\ y=a{e}^{0.01405t}\\ =3.4e^{0.014*20}\\ =4.498\end{array}$

Therefore the population in $2020$ in $4.498$ million .

(c)

To determine

To find: The Oregon reach $6$ million people .

Answer to Problem 10PPS

The population will reach $6$ million after $t=40.5$ year.

Explanation of Solution

Given information:

The given growth form is $3.4$ million $2000$ .

Calculation:

Calculate the equation.

  $\begin{array}{c}y=a{e}^{-kt}\\ puty=6\\ y=a{e}^{0.0145t}\\ 6=3.4e^{0.014*t}\\ \frac{6}{3.4}=e^{0.014*t}\\ \ln 1.764=\ln e^{0.014*t}\\ 0.567=0.014t\\ t=\frac{0.567}{0.014}\\ =40.5\end{array}$

Therefore the population will reach $6$ million after $t=40.5$ year.