Roberta wants to have \( \$ 15,000 \) in 1 year. Calculate how much she should invest now at \( 12 \% \) interest, compounded semiannually in order to reach this goal. Select one: a. \( \$ 13,350.00 \) b. \( \$ 13,200.00 \) c. \( \$ 14,225.00 \) d. \( \$ 12,800.00 \)

To determine how much Roberta should invest now, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] 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. Given: - \(A = 15,000\) - \(r = 0.12\) - \(n = 2\) (since it's compounded semiannually) - \(t = 1\) Rearranging the formula to solve for \(P\): \[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \] Now substituting the known values: \[ P = \frac{15000}{\left(1 + \frac{0.12}{2}\right)^{2 \times 1}} \] \[ P = \frac{15000}{\left(1 + 0.06\right)^{2}} \] \[ P = \frac{15000}{(1.06)^{2}} \] \[ P = \frac{15000}{1.1236} \approx 13332.28 \] The closest value among the options is: **a. \( \$ 13,350.00 \)**.