Chapter 12.6, Problem 43STP

To determine

To find the time in which the population becomes $10,000$ .

Answer to Problem 43STP

  $59$ years $4$ months

Explanation of Solution

Given:

The rate of population is modeled by $P\left(t\right)=24,000e^{-0.0064t}$ , where $t$ is time in years.

Calculation:

Given $P\left(t\right)=24,000e^{-0.0064t},$

If $P\left(t\right)=10000$ , then

  $\begin{array}{l}\Rightarrow P\left(t\right)=24,000e^{-0.0064t}\\ \Rightarrow 10000=24000e^{-0.0064t}\\ \Rightarrow \frac{10}{24}{e}^{0.0064t}=1\\ \Rightarrow e^{0.0064t}=\frac{24}{10}\\ \Rightarrow e^{0.0064t}=2.4\\ \Rightarrow 0.0064t=\log \left(2.4\right)\\ \Rightarrow 0.0064t=0.38\\ \Rightarrow t=\frac{10000}{64}\times 0.38\\ \Rightarrow t=59.4\end{array}$

Conclusion:

Therefore, the time in which the population becomes $10,000$ is $59$ years $4$ months.