The formula for continuously compounded interest is
[tex]\begin{gathered} A=Pe^{rt} \\ \text{ Where }A\text{ is the Amount or future value} \\ P\text{ is the Principal, or initial value} \\ r\text{ is the interest rate, and} \\ t\text{ is the time} \end{gathered}[/tex]
So, in this case, we have
[tex]\begin{gathered} A=\text{ \$}1,000 \\ P=\text{ \$}200 \\ r=4\text{\% }=\frac{4}{100}=0.04 \\ t=\text{ ?} \end{gathered}[/tex][tex]\begin{gathered} A=Pe^{rt} \\ \text{ Replace the know values} \\ \text{\$}1,000=\text{\$}200\cdot e^{0.04t} \\ \text{ Divide by \$200 from both sides of the equation} \\ \frac{\text{\$}1,000}{\text{\$}200}=\frac{\text{\$}200\cdot e^{0.04t}}{\text{\$}200} \\ 5=e^{0.04t} \\ \text{ Apply natural logarithm to both sides of the equation} \\ \ln (5)=\ln (e^{0.04t}) \\ \ln (5)=0.04t \\ \text{ Divide by 0.04 from both sides of the equation} \\ \frac{\ln(5)}{0.04}=\frac{0.04t}{0.04} \\ \boldsymbol{40.2\approx t} \end{gathered}[/tex]
Therefore, it will take approximately 40 years for the account to reach $1,000.
