The Nguyens are saving up to go on a family vacation in 5 years. They invest \( \$ 2400 \) into an account with an annual interest rate of \( 1.46 \% \) compounded daily. Answer the questions below. Do not round any intermediate computations, and round your final answers to the nearest cent. If necessary, refer to the list of financial formulas. Assume there are 365 days in each year. (a) Assuming no withdrawals are made, how much money is in the Nguyens' account after 5 years? (b) How much interest is earned on the Nguyens' investment after 5 years? \( \$ \square \)

To solve the problem, we will use the formula for compound interest, which can be expressed as: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial 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. ### (a) Calculating the total amount in the account after 5 years: Given: - \( P = 2400 \) - \( r = 1.46\% = 0.0146 \) - \( n = 365 \) - \( t = 5 \) Substituting these values into the formula: \[ A = 2400 \left(1 + \frac{0.0146}{365}\right)^{365 \times 5} \] Calculating it step by step: 1. Calculate \( \frac{0.0146}{365} \) which is approximately \( 0.0000401 \). 2. Then, calculate \( 365 \times 5 = 1825 \). 3. The formula becomes: \[ A = 2400 \left(1 + 0.0000401\right)^{1825} \] 4. Calculate \( A \): \[ A = 2400 \times (1.0000401)^{1825} \approx 2400 \times 1.078898 \approx 2589.37 \] So, after 5 years, the amount in the Nguyens' account is approximately \( \text{\$2589.37} \). ### (b) Calculating the interest earned: To find the interest earned, subtract the principal from the total amount: \[ \text{Interest} = A - P = 2589.37 - 2400 \approx 189.37 \] Thus, the interest earned on the Nguyens' investment after 5 years is approximately \( \text{\$189.37} \).