# The population of a county is growing at a rate of 9% per year, compounded continuously. How many...

## Question:

The population of a county is growing at a rate of 9% per year, compounded continuously. How many years will it take for the population to quadruple according to the exponential growth function?

## Exponential Growth

An exponential growth function has a rate of change that is proportional to its value at all times. If the initial quantity is Q(0), the rate of growth is r per year and the number of years is t, the function is:

$$P(t)=P(0)e^{rt}$$

## Answer and Explanation:

Let the initial population be {eq}x {/eq} so that it quadruples to {eq}4x {/eq} in {eq}t {/eq} years at the rate of {eq}0.09 {/eq} per year. We have:

\begin{align} xe^{0.09t}&=4x\\ e^{0.09t}&=4\\ 0.09t&=\ln 4\\ t&\approx 15.40 \end{align}\\