Suppose that 2000 is loaned at a rate of 18% compounded monthly. Assuming that no payments are made,find the amount owed after 5 years. Do not round any Intermediate computations, and round your answer to the nearest cent.

To problem, we will use the formula for compound interest:<br /><br />\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]<br /><br />where:<br />- \( A \) is the amount owed after \( t \) years,<br />- \( P \) is the principal amount (initial loan amount),<br />- \( r \) is the annual interest rate (as a decimal),<br />- \( n \) is the number of times the interest is compounded per year,<br />- \( t \) is the time the money is invested or borrowed for, in years.<br /><br />Given:<br />- \( P = \$2000 \)<br />- \( r = 18\% = 0.18 \)<br />- \( n = 12 \) (since the interest is compounded monthly)<br />- \( t = 5 \) years<br /><br />Now, plug these values into the formula:<br /><br />\[ A =0 \left(1 + \frac{0.18}{12}\right)^{12 \cdot 5} \]<br /><br />First, calculate the monthly interest rate:<br /><br />\[ \frac{0.18}{12} = 0.015 \]<br /><br />Next, add 1 to the monthly interest rate:<br /><br />\[ 1 + 0.015 = 1.015 \]<br /><br />Then, raise this to the power of the total number of compounding periods (12 months/year * 5 years):<br /><br />\[ 1.015^{60} \]<br /><br />Using a calculator to find \( 1.015^{60} \):<br /><br />\[ 1.015^{60} \approx 2.406619 \]<br /><br />Finally, multiply this result by the principal amount:<br /><br />\[ A = 2000 \times 2.406619 \approx 4813.24 \]<br /><br />So, the amount owed after 5 years, rounded to the nearest cent, is:<br /><br />\[ \boxed{4813.24} \]