Adam invests 6,854 in a retirement account with a fixed annual interest rate of 5% compounded continuously . What will the account balance be after 19 years? 18,445.72 17,722.45 16,858.12 19,488.65

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 (approximately 2.71828)<br /><br />Given:<br />- \( P = \$6,854 \)<br />- \( r = 5\% = 0.05 \)<br />- \( t = 19 \) years<br /><br />Now, plug these values into the formula:<br /><br />\[ A = 6854 \cdot e^{0.05 \cdot 19} \]<br /><br />First, calculate the exponent:<br /><br />\[ 0.05 \cdot 19 = 0.95 \]<br /><br />Next, calculate \( e^{0.95} \):<br /><br />\[ e^{0.95} \approx 2.5857 \]<br /><br />Now, multiply this by the principal amount:<br /><br />\[ A = 6854 \cdot 2.5857 \approx 17,722.45 \]<br /><br />Therefore, the account balance after 19 years will be approximately \( \$17,722.45 \).<br /><br />So, the correct answer is:<br /><br />\[ \$17,722.45 \]