Question

Suppose that $2000 is initially invested in an account at a fixed interest rate, compounded continuously. Suppose also that, after four years, the amount of

money in the account is $2411. Find the interest rate per year.
Write your answer as a percentage. Do not round any intermediate computations, and round your percentage to the nearest hundredth.

Answer 1

`~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^(rt)\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$2411\\ P=\textit{original amount deposited}\dotfill & \$2000\\ r=rate\to r\%\to (r)/(100)\\ t=years\dotfill &4 \end{cases}`

`2411=2000e^{(r)/(100)\cdot 4}\implies \cfrac{2411}{2000}= e^{(r)/(25)}\implies \log_e\left( \cfrac{2411}{2000} \right)=\log_e\left( e^{(r)/(25)} \right) \\\\\\ \log_e\left( \cfrac{2411}{2000} \right)=\cfrac{r}{25}\implies 25\cdot \ln\left( \cfrac{2411}{2000} \right)=r\implies \stackrel{\%}{4.67}\approx r`