Mathematics: 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.)

The population of a town grows at a rate proportional

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julie66491

11.03.2022

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

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

$\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%.

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