Suppose \( \$ 10,000 \) is invested by Stephanie for 25 years in an account that earns \( 12 \% \) interest, compounded semi-annually. Round the solutions to the nearest cent, if necessary. Determine the future value of the account. Future Value \( = \) Determine the amount of interest earned in this account over the 25 years. Interest \( = \)

To determine the future value of the account, we can use the formula for compound interest: \[ FV = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( FV \) = future value - \( P \) = principal amount (initial investment) - \( r \) = annual interest rate (decimal) - \( n \) = number of times interest is compounded per year - \( t \) = number of years the money is invested Given: - \( P = 10,000 \) - \( r = 0.12 \) - \( n = 2 \) (since it’s compounded semi-annually) - \( t = 25 \) Substituting the values into the formula: \[ FV = 10,000 \left(1 + \frac{0.12}{2}\right)^{2 \times 25} \] Calculating the components: \[ FV = 10,000 \left(1 + 0.06\right)^{50} = 10,000 \times (1.06)^{50} \] Calculating \( (1.06)^{50} \) gives approximately \( 18.679 \). Thus: \[ FV = 10,000 \times 18.679 \approx 186,790.00 \] So, the future value of the account is: Future Value \( = \$186,790.00 \) To find the amount of interest earned over the 25 years, use the formula: \[ \text{Interest} = FV - P \] Substituting the values we found: \[ \text{Interest} = 186,790.00 - 10,000 = 176,790.00 \] Therefore, the amount of interest earned in this account over the 25 years is: Interest \( = \$176,790.00 \)