Suppose you'd like to save enough money to pay cash for your next car. The goal is to save an extra \( \$ 23,000 \) over the next 6 years. What amount must be deposited quarterly into an account that earns \( 5.9 \% \) interest, compounded quarterly, in order to reach your goal? Round your answer to the nearest cent, if necessary. Formuia

To find the amount that must be deposited quarterly, you can use the future value of an ordinary annuity formula, which is: \[ FV = P \times \left(\frac{(1 + r/n)^{nt} - 1}{r/n}\right) \] Where: - \( FV \) is the future value (the amount you want to save, \( \$23,000 \)), - \( P \) is the payment (the amount to be deposited quarterly), - \( r \) is the annual interest rate (5.9% or 0.059), - \( n \) is the number of compounding periods per year (4 for quarterly), - \( t \) is the number of years (6). Rearranging the formula to solve for \( P \): \[ P = \frac{FV}{\left(\frac{(1 + r/n)^{nt} - 1}{r/n}\right)} \] Now, plug in the values: - \( FV = 23000 \) - \( r = 0.059 \) - \( n = 4 \) - \( t = 6 \) Calculate \( P \): 1. Calculate \( r/n \): \[ r/n = 0.059/4 = 0.01475 \] 2. Calculate \( nt \): \[ nt = 4 \times 6 = 24 \] 3. Calculate \( (1 + r/n)^{nt} \): \[ (1 + 0.01475)^{24} \approx 1.4082 \] 4. Now calculate the full formula: \[ \frac{(1.4082 - 1)}{0.01475} \approx 27.5584 \] 5. Finally, solve for \( P \): \[ P = \frac{23000}{27.5584} \approx 834.14 \] So, you need to deposit approximately \( \$834.14 \) quarterly to reach your goal of \( \$23,000 \) in 6 years!