Stefan invests 7,690 in a retirement account with a fixed annual interest rate of 9% compounded continuously How long will it take for the account balance to reach 38,858.26

To solve this problem, we can use the formula for continuous compounding interest:<br /><br />\[ A = P \cdot e^{rt} \]<br /><br />Where:<br />- \( A \) is the final amount<br />- \( P \) is the principal amount (initial investment)<br />- \( r \) is the annual interest rate (as a decimal)<br />- \( t \) is the time in years<br />- \( e \) is the base of the natural logarithm<br /><br />Given:<br />- \( A = \$38,858.26 \)<br />- \( P = \$7,690 \)<br />- \( r = 9\% = 0.09 \)<br /><br />We need to find \( t \).<br /><br />First, substitute the given values into the formula:<br /><br />\[ 38,858.26 = 7,690 \cdot e^{0.09t} \]<br /><br />Next, solve for \( t \):<br /><br />1. Divide both sides by 7,690:<br /><br />\[ \frac{38,858.26}{7,690} = e^{0.09t} \]<br /><br />\[ 5.06 = e^{0.09t} \]<br /><br />2. Take the natural logarithm (ln) of both sides to solve for \( t \):<br /><br />\[ \ln(5.06) = \ln(e^{0.09t}) \]<br /><br />\[ \ln(5.06) = 0.09t \]<br /><br />3. Solve for \( t \):<br /><br />\[ t = \frac{\ln(5.06)}{0.09} \]<br /><br />Using a calculator to find the natural logarithm and perform the division:<br /><br />\[ \ln(5.06) \approx 1.624 \]<br /><br />\[ t = \frac{1.624}{0.09} \approx 18.02 \]<br /><br />Therefore, it will take approximately 18.02 years for the account balance to reach \$38,858.26.