The Gray family wants to save money to travel the world. They plan to invest in an ordinary annuity that earns \( 2.4 \% \) interest, compounded quarterly. Payments will be made at the end of each quarten. How much money do they need to pay into the annuity each quarter for the annulty to have a total value of \( \$ 13,000 \) after 13 years? Do not round intermediate computations, and round your final answer to the nearest cent. If necessary, refer to the list of financial formulas. Hill

To determine how much the Gray family needs to invest quarterly in their annuity, we can use the future value of an ordinary annuity formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) is the future value of the annuity (in this case, \( \$ 13,000 \)) - \( P \) is the payment amount per period (this is what we need to find) - \( r \) is the interest rate per period - \( n \) is the total number of payments Given that the annual interest rate is \( 2.4\% \) compounded quarterly, we convert it to a quarterly rate: \[ r = \frac{2.4\%}{4} = 0.006 \] Next, we calculate the total number of payments over 13 years, with payments made quarterly: \[ n = 13 \times 4 = 52 \] Now plug these values into the future value formula: \[ 13,000 = P \times \frac{(1 + 0.006)^{52} - 1}{0.006} \] Calculating \( (1 + 0.006)^{52} \): \[ (1.006)^{52} \approx 1.348850 \] So, substituting back gives: \[ 13,000 = P \times \frac{1.348850 - 1}{0.006} \] \[ 13,000 = P \times \frac{0.348850}{0.006} \] \[ 13,000 = P \times 58.14167 \] Now, solve for \( P \): \[ P = \frac{13,000}{58.14167} \approx 223.49 \] Thus, the Gray family needs to pay approximately \( \$ 223.49 \) into the annuity each quarter.