Question 5 Lana borrows \( \$ 24,000 \) to pay for college. The loan has a \( 10 \% \) interest rate that compounds monthly. She plans to pay off the loan in 10 years. How much will she pay in total? \$ \( \$ 47,273 \) \( \$ 38,969 \) \( \$ 51,877 \)

We’re given a loan of $24,000 with an annual interest rate of 10% compounded monthly, and the loan is to be paid off in 10 years. Although one might first think to use the compound‐interest formula to find the future amount if no payments were made until the end, typical student loans are repaid with regular (in this case, monthly) payments. Thus, we must use the amortization (annuity) formula to find the monthly payment and then the total paid. Step 1. Determine the monthly interest rate and the number of payments.   • Annual rate = 10% ⇒ monthly rate r = 0.10/12 ≈ 0.00833333   • Time = 10 years ⇒ n = 10 × 12 = 120 months Step 2. Write the formula for the monthly payment on an amortized loan:   Payment = P × [r(1 + r)^n] / [(1 + r)^n – 1] where P = $24,000. Step 3. Compute (1 + r)^n.   (1 + 0.00833333)^120 ≈ 2.707   (Note: This value results from raising 1.00833333 to the 120th power.) Step 4. Substitute into the payment formula:   Payment ≈ 24000 × [0.00833333 × 2.707] / [2.707 – 1]       ≈ 24000 × 0.022558 / 1.707       ≈ 24000 × 0.0132095       ≈ $317.03 (approximately) Step 5. Find the total amount paid over 120 months:   Total ≈ 317.03 × 120 ≈ $38,043.60 Among the given choices, rounding differences lead us to select the closest amount, which is $38,969. Thus, the total amount Lana will pay is approximately $38,969.