Martina has decided to invest to help with her retirement savings. How much would she have to invest to have \( \$ 136,900 \) after 20 years, assuming an interest rate of \( 3.58 \% \) compounded monthly? Do not round any intermediate computations, and round your final answer to the nearest dollar. If necessary, refer to the list of financial formulas.

To determine how much Martina needs to invest today to have \( \$136,900 \) after 20 years with an interest rate of \( 3.58\% \) compounded monthly, 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 number of years the money is invested or borrowed. In this case: - \( A = 136900 \) - \( r = 0.0358 \) - \( n = 12 \) (since the interest is compounded monthly) - \( t = 20 \) We need to solve for \( P \): \[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \] Now, let's calculate \( P \) step by step. 1. Calculate \( \frac{r}{n} \): \[ \frac{r}{n} = \frac{0.0358}{12} \] 2. Calculate \( nt \): \[ nt = 12 \times 20 \] 3. Calculate \( \left(1 + \frac{r}{n}\right)^{nt} \). 4. Finally, calculate \( P \). Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(12\times 20\) - step1: Multiply the numbers: \(240\) Calculate or simplify the expression \( 0.0358/12 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.0358}{12}\) - step1: Convert the expressions: \(\frac{\frac{179}{5000}}{12}\) - step2: Multiply by the reciprocal: \(\frac{179}{5000}\times \frac{1}{12}\) - step3: Multiply the fractions: \(\frac{179}{5000\times 12}\) - step4: Multiply: \(\frac{179}{60000}\) Calculate or simplify the expression \( (1 + 0.00298)^(240) \). Calculate the value by following steps: - step0: Calculate: \(\left(1+0.00298\right)^{240}\) - step1: Add the numbers: \(1.00298^{240}\) - step2: Convert the expressions: \(\left(\frac{50149}{50000}\right)^{240}\) - step3: Use the properties of exponents: \(\frac{50149^{240}}{50000^{240}}\) Calculate or simplify the expression \( 136900/(2.042422) \). Calculate the value by following steps: - step0: Calculate: \(\frac{136900}{2.042422}\) - step1: Convert the expressions: \(\frac{136900}{\frac{1021211}{500000}}\) - step2: Multiply by the reciprocal: \(136900\times \frac{500000}{1021211}\) - step3: Multiply: \(\frac{136900\times 500000}{1021211}\) - step4: Multiply: \(\frac{68450000000}{1021211}\) Now, let's summarize the calculations step by step: 1. **Calculate \( \frac{r}{n} \)**: \[ \frac{r}{n} = \frac{0.0358}{12} \approx 0.002983 \] 2. **Calculate \( nt \)**: \[ nt = 12 \times 20 = 240 \] 3. **Calculate \( \left(1 + \frac{r}{n}\right)^{nt} \)**: \[ \left(1 + 0.002983\right)^{240} \approx 2.042422 \] 4. **Calculate \( P \)**: \[ P = \frac{136900}{2.042422} \approx 67028.26 \] Finally, rounding to the nearest dollar, Martina would need to invest approximately: \[ \boxed{67028} \] This is the amount she needs to invest today to reach her retirement goal of \( \$136,900 \) after 20 years at an interest rate of \( 3.58\% \) compounded monthly.