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 18 years 17 years 15 years 16 years

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 />- \( r = 0.09 \) (since 9% as a decimal is 0.09)<br /><br />We need to find \( t \).<br /><br />First, plug in 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 />\[ 5.05 = e^{0.09t} \]<br /><br />2. Take the natural logarithm (ln) of both sides to solve for \( t \):<br /><br />\[ \ln(5.05) = 0.09t \]<br /><br />3. Calculate \( \ln(5.05) \):<br /><br />\[ \ln(5.05) \approx 1.624 \]<br /><br />4. Solve for \( t \):<br /><br />\[ 1.624 = 0.09t \]<br /><br />\[ t = \frac{1.624}{0.09} \]<br /><br />\[ t \approx 18.02 \]<br /><br />So, it will take approximately 18 years for the account balance to reach \$38,858.26.<br /><br />Therefore, the correct answer is:<br /><br />\[ \boxed{18 \text{ years}} \]