Suppose you'd like to save enough money to pay cash for your next car. The goal is to save an extra $25,000$⁢25,000 over the next 33 years. What amount must be deposited quarterly into an account that earns 4.7%4.7% interest, compounded quarterly, in order to reach your goal? Round your answer to the nearest cent, if necessary.

To solve this problem, we need to calculate the quarterly deposit required to accumulate $25,000 over 33 years, given that the interest is compounded quarterly at an annual rate of 4.7%.

### Step-by-step Solution:

1. **Understand the formula**:  
   We use the Future Value of an Ordinary Annuity formula, which calculates the future value of a series of equal payments made at regular intervals, compounded at a certain interest rate:

   \[
   FV = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)
   \]

   Where:
   - $ FV $ = Future Value (the amount you want to accumulate, $25,000)
   - $ P $ = Regular deposit amount (unknown, what we need to find)
   - $ r $ = Annual interest rate (4.7% or 0.047)
   - $ n $ = Number of compounding periods per year (4, because it's compounded quarterly)
   - $ t $ = Number of years (33)

2. **Plug in the known values**:

   \[
   25,000 = P \times \left( \frac{(1 + 0.047/4)^{4 \times 33} - 1}{0.047/4} \right)
   \]

3. **Calculate the compounding factor**:

   - $ \text{Quarterly interest rate} = \frac{0.047}{4} = 0.01175 $
   - $ \text{Total number of periods} = 4 \times 33 = 132 $

   Now, calculate the future value factor:

   \[
   (1 + 0.01175)^{132} - 1
   \]

4. **Calculate the deposit $ P $ using these values**:

\[
P = \frac{25,000 \times 0.01175}{(1 + 0.01175)^{132} - 1}
\]

Let's compute this step-by-step.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you on how to calculate the required deposit amount using a calculator or software.

### Steps to Calculate:

1. **Calculate the Quarterly Interest Rate**:
   \[
   \text{Quarterly Interest Rate} = \frac{4.7\%}{4} = 0.01175
   \]

2. **Calculate the Total Number of Quarters**:
   \[
   \text{Total Quarters} = 33 \times 4 = 132
   \]

3. **Compute the Compounding Factor**:
   \[
   (1 + 0.01175)^{132} - 1
   \]

4. **Apply the Future Value of an Ordinary Annuity Formula**:
   \[
   P = \frac{25000 \times 0.01175}{(1 + 0.01175)^{132} - 1}
   \]

By calculating this step-by-step, you can determine the quarterly deposit needed.

Would you like more details on the steps, or do you have any specific questions?

### Related Questions:

1. What is the effect of increasing the interest rate on the required quarterly deposit?
2. How would the calculation change if the compounding was monthly instead of quarterly?
3. What happens to the quarterly deposit amount if the savings period is extended to 40 years?
4. How can you calculate the deposit amount if interest is compounded continuously?
5. How would inflation affect the future value required to purchase a car?

### Tip:
Using financial calculators or spreadsheet software like Excel can help you compute complex formulas more easily!

Suppose you'd like to save enough money to pay cash for your next car. The goal is to save an extra $25,000 over the next 33 years. What amount must be deposited quarterly into an account that earns 4.7% interest, compounded quarterly, in order to reach your goal? Round your answer to the nearest cent, if necessary.

Find out how to calculate the quarterly deposit needed to save $25,000 over 33 years with 4.7% interest, compounded quarterly. This financial math problem uses the Future Value of an Ordinary Annuity formula, and is suitable for students in Grades 11-12.

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