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 determine the quarterly deposit needed to save **\$23,000** over **6 years** with an interest rate of **5.9%**, compounded quarterly, we can use the **Future Value of an Ordinary Annuity** formula. ### **Given:** - **Future Value (FV):** \$23,000 - **Annual Interest Rate (r):** 5.9% or 0.059 - **Number of Years (t):** 6 - **Compounding Periods per Year (m):** 4 (quarterly) ### **Calculations:** 1. **Determine the Quarterly Interest Rate (i):** \[ i = \frac{r}{m} = \frac{0.059}{4} = 0.01475 \text{ (or 1.475\% per quarter)} \] 2. **Calculate the Total Number of Quarterly Payments (n):** \[ n = m \times t = 4 \times 6 = 24 \text{ quarters} \] 3. **Use the Future Value of an Ordinary Annuity Formula:** \[ FV = PMT \times \left( \frac{(1 + i)^n - 1}{i} \right) \] where: - **FV:** Future Value (\$23,000) - **PMT:** Quarterly Payment 4. **Rearrange the Formula to Solve for PMT:** \[ PMT = \frac{FV \times i}{(1 + i)^n - 1} \] 5. **Plug in the Known Values:** \[ PMT = \frac{23,000 \times 0.01475}{(1 + 0.01475)^{24} - 1} \] 6. **Calculate \((1 + i)^n\):** \[ (1 + 0.01475)^{24} \approx 1.42105 \] 7. **Compute the Denominator:** \[ 1.42105 - 1 = 0.42105 \] 8. **Calculate the Quarterly Payment (PMT):** \[ PMT = \frac{339.25}{0.42105} \approx 805.69 \] ### **Conclusion:** You need to deposit approximately **\$805.69** **quarterly** into the account to reach your savings goal of **\$23,000** in **6 years**. ### **Final Answer:** \[ \boxed{\$805.69} \]