Matt invests 4,612 in a retirement account with a fixed annual interest rate of 4% compounded continuously How long will it take for the account balance to reach 8,403.61 12 years 13 years 15 years 14 years

To solve this problem, we will use the formula for continuous compounding interest, which is given by:<br /><br />\[ A = P e^{rt} \]<br /><br />where:<br />- \( A \) is the amount of money accumulated after time \( t \),<br />- \( P \) is the principal amount (the initial amount of money),<br />- \( r \) is the annual interest rate (in decimal form),<br />- \( t \) is the time the money is invested for,<br />- \( e \) is the base of the natural logarithm.<br /><br />Given:<br />- \( A = \$8,403.61 \)<br />- \( P = \$4,612 \)<br />- \( r = 4\% = 0.04 \)<br /><br />We need to find \( t \).<br /><br />First, we rearrange the formula to solve for \( t \):<br /><br />\[ t = \frac{\ln\left(\frac{A}{P}\right)}{r} \]<br /><br />Now, we plug in the given values:<br /><br />\[ t = \frac{\ln\left(\frac{8403.61}{4612}\right)}{0.04} \]<br /><br />Calculate the fraction inside the logarithm:<br /><br />\[ \frac{8403.61}{4612} \approx 1.83 \]<br /><br />Now, take the natural logarithm of this value:<br /><br />\[ \ln(1.83) \approx 0.6109 \]<br /><br />Finally, divide by the interest rate:<br /><br />\[ t = \frac{0.6109}{0.04} \approx 15.2275 \]<br /><br />Since we are looking for the time in years and the options are whole numbers, we round to the nearest whole number:<br /><br />\[ t \approx 15 \]<br /><br />Therefore, it will take approximately 15 years for the account balance to reach \$8,403.61.<br /><br />The correct answer is:<br />15 years