Question

The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 25% in 10 years. What will be the population in 50 years? (Round your answer to the nearest person.)
How fast is the population growing at
t = 50?
(Round your answer to two decimal places.)

Answer 1

$\bf \textit{Amount of Population Growth}\\\\ A=Ie^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\\ I=\textit{initial amount}\\ r=rate\to r\%\to \frac{r}{100}\\ t=\textit{elapsed time}\\ \end{cases}\\\\ -------------------------------\\\\ \textit{now, it was 500, in increased by 25\% so }500+\stackrel{\textit{25\% of 500}}{125}\implies 625$

$\bf \textit{in 10 years}\implies \begin{cases} A=625\\ I=500\\ t=10 \end{cases}\implies 600=500e^{r10}\implies \cfrac{600}{500}=e^{10r} \\\\\\ \cfrac{625}{500}=e^{10r}\implies ln\left(\frac{625}{500} \right)=ln(e^{10r})\implies ln\left(\frac{625}{500} \right)=10r \\\\\\ \cfrac{ln\left(\frac{625}{500} \right)}{10}=r\implies 0.022314\approx r$

$\bf -------------------------------\\\\ A=500e^{0.0.022314t}\qquad \qquad \stackrel{t=50}{A}=500e^{0.022314\cdot 50}\implies A\approx 1526$

how fast is the population growing?  well, I'd think is "r", or the rate, which is  0.022314 * 100 or 2.2314%.