Question

You are investing money at 10.5 percent annual interest, compounded continuously. It will take you ______ years to double your investment.

Answer 1

let's take it from the basic of one dolla!, how long will it take for $1 to become $2, namely double, at 10.5%?

`~~~~~~ \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 10.5\%\to (10.5)/(100)\dotfill &0.105\\ t=years \end{cases} \\\\\\ 2=1e^(0.105\cdot t) \implies \log_e(2)=\log_e(e^(0.105t))\implies \log_e(2)=0.105t \\\\\\ \ln(2)=0.105t\implies \cfrac{\ln(2)}{0.105}=t\implies 6.6\approx t\qquad \textit{about 6 years and 219 days}`