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 three years. What amount must be deposited quarterly into an account that earns 5.7% interest compounded quarterly in order to reach your goal round your answer to the nearest sent if necessary.

To solve this problem, we need to use the formula for the **future value of an ordinary annuity**, since the amount is deposited at regular intervals (quarterly) into an account that earns interest.

### Formula:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $ FV $ is the future value of the annuity (total amount to be saved)
- $ P $ is the payment amount deposited each period (what we're trying to find)
- $ r $ is the interest rate per period
- $ n $ is the total number of periods

### Given Information:
1. $ FV = 23,000 $ (the amount needed after 3 years)
2. The annual interest rate is 5.7%, which is compounded quarterly. So, the quarterly interest rate $ r $ is:
\[
r = \frac{5.7\%}{4} = \frac{0.057}{4} = 0.01425
\]
3. The total number of periods $ n $ is:
\[
n = 3 \text{ years} \times 4 \text{ quarters/year} = 12 \text{ quarters}
\]

### Plugging into the Formula:
\[
23,000 = P \times \frac{(1 + 0.01425)^{12} - 1}{0.01425}
\]

Now, let's solve for $ P $.The amount that must be deposited quarterly to reach the goal of $23,000 over the next three years is approximately **$1,771.05**.

Do you want more details or have any questions?

### Related Questions:
1. How would the solution change if the interest rate was compounded monthly instead of quarterly?
2. What would happen to the quarterly deposit amount if the annual interest rate increased to 7%?
3. How much interest would be earned over the three-year period?
4. What would be the impact on the deposit amount if the goal was to reach $30,000 instead of $23,000?
5. How would the solution differ if the deposit was made at the beginning of each period instead of at the end?

**Tip**: Always ensure that the compounding period for the interest rate matches the period of the deposits for accurate calculations.

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 three years. What amount must be deposited quarterly into an account that earns 5.7% interest compounded quarterly in order to reach your goal? Round your answer to the nearest cent if necessary.

Find out how much you need to deposit quarterly to save $23,000 in 3 years with a 5.7% interest rate compounded quarterly. Learn to apply the future value of annuity formula and the compound interest theorem to calculate the deposit amount.

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