# If the population doubled in size over in the 18-month period and the current population is...

## Question:

If the population doubled in size over in the 18-month period and the current population is 10,000, what will be the population 2 years from now?

## Exponential Growth:

When we do not have the exponential growth constant of an exponential function, we must use two values of the function to find the constant. After the constant we will have the exponential function to find any future value from the initial value.

## Answer and Explanation:

Data:

First Time: {eq}t=18\:\textrm{months} {/eq}

Present Value: {eq}P=10,000 {/eq}

First Future Value: {eq}F=20,000 {/eq}

Second Time: {eq}t=2\:\textrm{years}=24\:\textrm{months} {/eq}

Equation:

{eq}F=Pe^{kt} {/eq}

Then,

{eq}\begin{align*} F&=Pe^{kt} \\ \\ 20,000&=10,000e^{18k} \\ \\ 10,000e^{18k}&=20,000 \\ \\ e^{18k}&=\frac{20,000}{10,000} \\ \\ e^{18k}&=2 \\ \\ \ln \left(e^{18k}\right)&=\ln \left(2\right) \\ \\ 18k&=0.69 \\ \\ k&=\frac{0.69}{18} \\ \\ k&=0.0383 \end{align*} {/eq}

Now, the population in 2 years will be:

{eq}\begin{align*} F&=Pe^{0.0383t} \\ \\ F&=10,000e^{0.0383 \cdot 24} \\ \\ F&=\boxed{25,072.84} \end{align*} {/eq}