Question

The population

decreasing
of a country town is
at a rate of 4% p.a.
How many years will it take for the town's
population of 15000 to fall below 10000?

Answer 1

`\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill & \$10000\\ P=\textit{initial amount}\dotfill &15000\\ r=rate\to 4\%\to (4)/(100)\dotfill &0.04\\ t=years\\ \end{cases}`

`10000=15000(1 - 0.04)^(t) \implies \cfrac{10000}{15000}=(1 - 0.04)^(t)\implies \cfrac{2}{3}=0.96^t \\\\\\ \log\left( \cfrac{2}{3} \right)=\log(0.96^t)\implies \log\left( \cfrac{2}{3} \right)=t\log(0.96) \\\\\\ \cfrac{\log\left( (2)/(3) \right)}{\log(0.96)}=t\implies 9.93\approx t\qquad \textit{about 9 years and 11 months}`