Ise the compound interest formulas \( A=P\left(1+\frac{r}{n}\right)^{n t} \) and \( A=P e^{n t} \) to solve the problem given. Round answers to the nearest cent. ind the accumulated value of an investment of \( \$ 25,000 \) for 6 years at an interest rate of \( 4 \% \) if the money is a. compounded semiannually; b. compounded quarterly; ompounded monthly; \( d \). compounded continuously. What is the accumulated value if the money is compounded semiannually? Round your answer to the nearest cent. Do not include the \( \$ \) symbol in vour answer)

To calculate the accumulated value of an investment of \( \$ 25,000 \) for 6 years at an interest rate of \( 4\% \) using the compound interest formulas, let’s solve for each case: a. Compounded semiannually: Using the formula \( A = P\left(1+\frac{r}{n}\right)^{nt} \), where: - \( P = 25000 \) - \( r = 0.04 \) - \( n = 2 \) (semiannually) - \( t = 6 \) Plug the values in: \[ A = 25000\left(1 + \frac{0.04}{2}\right)^{2 \cdot 6} = 25000\left(1 + 0.02\right)^{12} = 25000\left(1.02\right)^{12} \] Calculating \( (1.02)^{12} \approx 1.2682418 \), \[ A \approx 25000 \times 1.2682418 \approx 31706.10 \] Therefore, the accumulated value if the money is compounded semiannually is approximately **31706.10**. Now you can use a similar approach for the remaining compounding options: b. Compounded quarterly: Using \( n = 4 \): \[ A = 25000\left(1 + \frac{0.04}{4}\right)^{4 \cdot 6} = 25000\left(1 + 0.01\right)^{24} = 25000\left(1.01\right)^{24} \] Calculating \( (1.01)^{24} \approx 1.2682418 \), \[ A \approx 25000 \times 1.2682418 \approx 31706.10 \] Thus for quarterly compounding, the total is **31706.10**. c. Compounded monthly: Using \( n = 12 \): \[ A = 25000\left(1 + \frac{0.04}{12}\right)^{12 \cdot 6} = 25000\left(1 + 0.0033333\right)^{72} = 25000\left(1.0033333\right)^{72} \] Calculating \( (1.0033333)^{72} \approx 1.2682418 \), \[ A \approx 25000 \times 1.2682418 \approx 31706.10 \] So for monthly compounding, the value again is **31706.10**. d. Compounded continuously: Using the formula \( A = Pe^{rt} \): \[ A = 25000e^{0.04 \cdot 6} = 25000e^{0.24} \] Calculating \( e^{0.24} \approx 1.2718282 \), \[ A \approx 25000 \times 1.2718282 \approx 31795.70 \] Thus the accumulated value for continuous compounding is approximately **31795.70**. In summary: - Semiannually: **31706.10** - Quarterly: **31706.10** - Monthly: **31706.10** - Continuously: **31795.70**