Question

Gina puts $ 4500 into an account earning 7.5% interest compounded continuously. How long will it take for the amount in the account to grow to $ 5150?

Time in years =

Answer 1

`~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^(rt)\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$5150\\ P=\textit{original amount deposited}\dotfill & \$4500\\ r=rate\to 7.5\%\to (7.5)/(100)\dotfill &0.075\\ t=years \end{cases}`

`5150=4500e^(0.075\cdot t) \implies \cfrac{5150}{4500}=e^(0.075t)\implies \cfrac{103}{90}=e^(0.075t) \\\\\\ \log_e\left( \cfrac{103}{90} \right)=\log_e(e^(0.075t))\implies \log_e\left( \cfrac{103}{90} \right)=0.075t \\\\\\ \ln\left( \cfrac{103}{90} \right)=0.075t\implies \cfrac{\ln\left( (103)/(90) \right)}{0.075}=t\implies\stackrel{\textit{about 1 year and 291 days}}{ 1.8\approx t}`