Question

Kyle invests in an account earning 4.5% interest compounded continuously. How long will it take to double his investment?

Doubling time in years =

Answer 1

let's take from the basic of one dolla!!! how long for a buck to turn into two bucks at 4.5%?

`~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^(rt)\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$2\\ P=\textit{original amount deposited}\dotfill & \$1\\ r=rate\to 4.5\%\to (4.5)/(100)\dotfill &0.045\\ t=years \end{cases} \\\\\\ 2=1e^(0.045\cdot t) \implies \log_e(2)=\log_e(e^(0.045t))\implies \log_e(2)=0.045t \\\\\\ \ln(2)=0.045t\implies \cfrac{\ln(2)}{0.045}=t\implies 15.4\approx t\qquad \textit{about 15 years and 146 days}`