Question

A savings account starts with $231.25. After 8 years of continuous compounding at an interest rate, r, the account has $1850.

What is the interest rate percentage?

Round the answer to the nearest hundredth.

NO THE ANSWER IS NOT 87.7?!?!!? LIKE THE OTHER QUESTIONS.

Answer 1

`~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^(rt)\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$1850\\ P=\textit{original amount deposited}\dotfill & \$231.25\\ r=rate\to r\%\to (r)/(100)\\ t=years\dotfill &8 \end{cases}`

`1850=231.25e^{(r)/(100)\cdot 8}\implies \cfrac{1850}{231.25}=e^{(2r)/(25)}\implies 8=e^{(2r)/(25)} \\\\\\ \log_e(8)=\log_e\left( e^{(2r)/(25)} \right)\implies \ln(8)=\cfrac{2r}{25}\implies 25\ln(8)=2r \\\\\\ \cfrac{25\ln(8)}{2}=r\implies \stackrel{\%}{25.99}\approx r`