Question

The population of a village is 6400. The population is decreasing exponentially at a rate of r% per year. After 22 years, the population will be 2607. Find the value of r.

Answer 1

`\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill & \$2607\\ P=\textit{initial amount}\dotfill &6400\\ r=rate\to r\%\to (r)/(100)\\ t=\textit{elapsed time}\dotfill &22\\ \end{cases} \\\\\\ 2607=6400(1 - (r)/(100))^(22) \implies \cfrac{2607}{6400}=\left( \cfrac{100-r}{100} \right)^(22) \implies \sqrt[22]{\cfrac{2607}{6400}}=\cfrac{100-r}{100}`

`\cfrac{r-100}{100}=-\sqrt[22]{\cfrac{2607}{6400}}\implies r-100=-100\sqrt[22]{\cfrac{2607}{6400}} \\\\\\ r=100-100\sqrt[22]{\cfrac{2607}{6400}}\implies {\Large \begin{array}{llll} r\approx 4 \end{array}}`