Suppose that \( \$ 14,000 \) is deposited for six years at \( 5 \% \) APR. Calculate the interest earned if interest is compounded quarterly. Round your answer to the nearest cent.

To calculate the interest earned on a deposit when compounding is involved, we can use the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] 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 (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for, in years. In this case: - \( P = 14,000 \) - \( r = 0.05 \) - \( n = 4 \) (quarterly) - \( t = 6 \) Now, substitute the values into the formula: \[ A = 14000 \left(1 + \frac{0.05}{4}\right)^{4 \times 6} \] Calculate \( \frac{0.05}{4} \): \[ \frac{0.05}{4} = 0.0125 \] Now plug that back into the formula: \[ A = 14000 \left(1 + 0.0125\right)^{24} \] \[ A = 14000 \left(1.0125\right)^{24} \] Now calculate \( \left(1.0125\right)^{24} \): \[ (1.0125)^{24} \approx 1.348850024 \] Now calculate \( A \): \[ A \approx 14000 \times 1.348850024 \approx 18883.90 \] The interest earned is: \[ \text{Interest} = A - P = 18883.90 - 14000 = 4883.90 \] So, the interest earned is approximately \( \$4883.90 \).