Chapter 8, Problem 65SGR

To determine

To find: The annual rate of growth for the city.

Answer to Problem 65SGR

The annual growth rate of the city is $0.041$ or $4.1\%$ .

Explanation of Solution

Given:

The population of the city 20 years ago was $24,330$ .

The current population is $55,250$ .

Calculation:

Substitute $55,250$ for $y$ , $24,330$ for $a$ and $20$ for $t$ in the formula $y=a{e}^{kt}$ to find the time.

  $\begin{array}{c}55250=24330e^{20k}\\ e^{20k}=\frac{55250}{24330}\\ e^{20k}=2.27\end{array}$

Take $\ln$ on both sides of the equation $e^{20k}=2.27$ .

  $\begin{array}{c}\ln e^{20k}=\ln \left(2.27\right)\\ 20k=\ln \left(2.27\right)\\ k=\frac{\ln \left(2.27\right)}{20}\\ \approx 0.041\end{array}$

Therefore, the annual growth rate of the city is $0.041$ or $4.1\%$ .