Question

The population of a town has grown at an annual rate of approximately 2.7%. How long will it take for its population of 14,450 people to double at this growth rate?

Answer 1

`\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \stackrel{ doubled }{28900}\\ P=\textit{initial amount}\dotfill &14450\\ r=rate\to 2.7\%\to (2.7)/(100)\dotfill &0.027\\ t=years \end{cases} \\\\\\ 28900 = 14450(1 + 0.027)^(t) \implies \cfrac{28900}{14450}=(1.027)^t\implies 2=1.027^t \\\\\\ \log(2)=\log(1.027^t)\implies \log(2)=t\log(1.027) \\\\\\ \cfrac{\log(2)}{\log(1.027)}=t\implies 26.02\approx t`