Suppose that $\$ 3300$ is borrowed for six years at an interest rate of $7 \%$ per year, compounded continuously. Find the amount owed, assuming no payments are made until the end.
Do not round any intermediate computations, and round your answer to the nearest cent.
Answer
Final Answer: The amount owed after six years, assuming no payments are made until the end, is \(\boxed{\$5022.47}\).
Step 1 ：Let's denote the principal amount as \(P\), the annual interest rate as \(r\), the time the money is invested for in years as \(t\), and the amount of money accumulated after n years, including interest as \(A\).
Step 2 ：In this case, \(P = \$3300\), \(r = 7\% = 0.07\), and \(t = 6\) years. We need to find \(A\).
Step 3 ：The formula for continuous compound interest is \(A = P * e^{rt}\).
Step 4 ：Substituting the given values into the formula, we get \(A = 3300 * e^{(0.07 * 6)}\).
Step 5 ：Calculating the above expression, we find that \(A = \$5022.47\).
Step 6 ：Final Answer: The amount owed after six years, assuming no payments are made until the end, is \(\boxed{\$5022.47}\).
