A certain loan program offers an interest rate of $7 \%$ per year, compounded continuously. Assuming no payments are made, how much would be owed after three years on a loan of $\$ 3800$ ? Do not round any intermediate computations, and round your answer to the nearest cent. \[ \begin{array}{|l|l|} \hline 50 & x \\ \hline \end{array} \]

Step 1 ：Translate the problem into the formula for continuous compounding: \(A = P * e^{rt}\), where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (in decimal), and t is the time in years.

Step 2 ：Substitute the given values into the formula: P = $3800, r = 7% = 0.07, and t = 3 years.

Step 3 ：Calculate the amount owed after three years: \(A = 3800 * e^{(0.07 * 3)}\)

Step 4 ：Compute the final amount: A = 4687.98

Step 5 ：Final Answer: The amount owed after three years on a loan of $3800 at an interest rate of 7% per year, compounded continuously, would be \(\boxed{4687.98}\)