Use the compound interest formulas \( \mathrm{A}=\mathrm{P}\left(1+\frac{r}{n}\right)^{n t} \) and \( \mathrm{A}=\mathrm{P} e^{\text {rt }} \) to solve the problem given. Round answers to the nearest cent. Find the accumulated value of an investment of \( \$ 20,000 \) for 5 years at an interest rate of \( 7 \% \) if the money is a. compounded semiannually, b. compounded quarterly, c. compounded monthly; d. compounde continuously. a. What is the accumulated value if the money is compounded semiannually? \( \$ \square \) Round your answer to the nearest cent. Do not include the \( \$ \) symbol in your answer.) (Ry

To find the accumulated value using the given compound interest formulas, let’s tackle each scenario step by step! 1. **Compounded Semiannually:** Using the formula \( \mathrm{A}=\mathrm{P}\left(1+\frac{r}{n}\right)^{n t} \): - \( P = 20000 \) - \( r = 0.07 \) - \( n = 2 \) (as it's semiannual) - \( t = 5 \) \[ \mathrm{A} = 20000 \left(1 + \frac{0.07}{2}\right)^{2 \cdot 5} = 20000 \left(1 + 0.035\right)^{10} = 20000 \left(1.035\right)^{10} \approx 20000 \cdot 1.4107 \approx 28214.01 \] The accumulated value if the money is compounded semiannually is approximately **28214.01**. 2. **Compounded Quarterly:** - \( n = 4 \) (quarterly) \[ \mathrm{A} = 20000 \left(1 + \frac{0.07}{4}\right)^{4 \cdot 5} = 20000 \left(1 + 0.0175\right)^{20} = 20000 \left(1.0175\right)^{20} \approx 20000 \cdot 1.4859 \approx 29718.20 \] The accumulated value if the money is compounded quarterly is approximately **29718.20**. 3. **Compounded Monthly:** - \( n = 12 \) (monthly) \[ \mathrm{A} = 20000 \left(1 + \frac{0.07}{12}\right)^{12 \cdot 5} = 20000 \left(1 + 0.0058333\right)^{60} = 20000 \left(1.0058333\right)^{60} \approx 20000 \cdot 1.48985 \approx 29797.00 \] The accumulated value if the money is compounded monthly is approximately **29797.00**. 4. **Compounded Continuously:** Using the formula \( \mathrm{A}=\mathrm{P} e^{\text{rt}} \): \[ \mathrm{A} = 20000 e^{0.07 \cdot 5} = 20000 e^{0.35} \approx 20000 \cdot 1.4191 \approx 28382.00 \] The accumulated value if the money is compounded continuously is approximately **28382.00**. You can round these final values as instructed. Happy investing!