Question

Mathlandia has approximately 25,372

inhabitants. People have been leaving
Mathlandia for Grammartown at a rate of
7.2% per year. How long will it take for
the population level to be 75% of current
levels.

Answer 1

`\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill &\stackrel{75\%~of ~25372}{19029~\hfill }\\ P=\textit{initial amount}\dotfill &25372\\ r=rate\to 7.2\%\to (7.2)/(100)\dotfill &0.072\\ t=\textit{elapsed time}\\ \end{cases}`

`19029=25372(1-0.072)^t\implies \cfrac{19029}{25372}=(1-0.072)^t\implies \cfrac{3}{4}=(1-0.072)^t \\\\\\ \log\left( \cfrac{3}{4} \right)=\log[(1-0.072)^t]\implies \log\left( \cfrac{3}{4} \right)=t\log(1-0.072) \\\\\\ \log\left( \cfrac{3}{4} \right)=t\log(0.928)\implies \cfrac{\log\left( (3)/(4) \right)}{\log(0.928)}=t\implies \stackrel{\textit{about 3 years and 310 days}}{3.85\approx t}`