Chapter 4.6, Problem 15E

a.

To determine

Find the population grows exponentially.

Answer to Problem 15E

  $\boxed{n\left(t\right)=29.76e^{.013t}}$

Explanation of Solution

Given information:

The population of California was $29.76$ million in $1990$ and $33.87$ million in $2000$ . Assume that the population grows exponentially.

Find a function that models the population $t$ years after $1990$ .

Calculation:

Consider the population growth model,

  $\begin{array}{l}n\left(t\right)=n_0{e}^{rt}\\ where,\\ n_0=initial-population\\ r=growth-rate\\ t=time\end{array}$

The population of California was $29.76$ million in $1990$ and $33.87$ million in $2000$ .

Now solve the growth rate per year,

  $\begin{array}{l}33.87=29.76e^{10r}\\ 1.14=e^{10r}\\ \ln 1.14=\ln e^{10r}\\ 0.13=10r\\ \boxed{r=.013}\\ \boxed{n\left(t\right)=29.76e^{.013t}}\end{array}$

Hence, the solution is, $\boxed{n\left(t\right)=29.76e^{.013t}}$

b.

To determine

Find the population grows exponentially.

Answer to Problem 15E

  $\boxed{t=53.5years}$

Explanation of Solution

Given information:

The population of California was $29.76$ million in $1990$ and $33.87$ million in $2000$ . Assume that the population grows exponentially.

Find the time required for the population to double.

Calculation:

Consider the population growth model,

  $\begin{array}{l}n\left(t\right)=n_0{e}^{rt}\\ where,\\ n_0=initial-population\\ r=growth-rate\\ t=time\end{array}$

The population of California was $29.76$ million in $1990$ and $33.87$ million in $2000$ .

Now solve the growth rate per year,

  $\begin{array}{l}33.87=29.76e^{10r}\\ 1.14=e^{10r}\\ \ln 1.14=\ln e^{10r}\\ 0.13=10r\\ \boxed{r=.013}\\ \boxed{n\left(t\right)=29.76e^{.013t}}\end{array}$

  $\begin{array}{l}2=e^{.013t}\\ \ln 2=\ln e^{.013t}\\ 0.69=0.013t\\ t=0.69/0.013\\ t=53.5years\end{array}$

Hence, the solution is, $\boxed{t=53.5years}$

c.

To determine

Find the population grows exponentially.

Answer to Problem 15E

  $\boxed{n\left(20\right)=38.60}$

Explanation of Solution

Given information:

The population of California was $29.76$ million in $1990$ and $33.87$ million in $2000$ . Assume that the population grows exponentially.

Use the function from part $\left(a\right)$ to predict the population of California in the year $2010$ . Look up California’s actual population in $2010$ , and compare.

Calculation:

Consider the population growth model,

  $\begin{array}{l}n\left(t\right)=n_0{e}^{rt}\\ where,\\ n_0=initial-population\\ r=growth-rate\\ t=time\end{array}$

The population of California was $29.76$ million in $1990$ and $33.87$ million in $2000$ .

Now solve the growth rate per year,

  $\begin{array}{l}33.87=29.76e^{10r}\\ 1.14=e^{10r}\\ \ln 1.14=\ln e^{10r}\\ 0.13=10r\\ \boxed{r=.013}\\ \boxed{n\left(t\right)=29.76e^{.013t}}\end{array}$

Put $20$ in above equation.

  $\begin{array}{l}n\left(20\right)=29.76e^{.013\times 20}\\ =29.76\times 1.3\end{array}$

  $\boxed{n\left(20\right)=38.60}$

Hence, the solution is, $\boxed{n\left(20\right)=38.60}$