Cyrus deposits a fixed quarterly amount into an annuity account for his child's college fund. He wishes to accumulate a future value of $80,000$⁢80,000 in 1717 years. Assuming an APR of 3.3%3.3% compounded quarterly, how much of the $80,000$⁢80,000 will Cyrus ultimately deposit in the account, and how much is interest earned? Round your answers to the nearest cent, if necessary.

To solve this problem, we need to calculate the quarterly deposit amount and break down the future value into the total deposits made by Cyrus and the interest earned. This is a future value of an annuity problem, where an annuity is a series of equal payments made at regular intervals.

### Formula for Future Value of an Annuity (Compounded Quarterly):
The formula for the future value of an annuity is:

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

Where:
- $FV$ is the future value of the annuity (in this case, $80,000),
- $P$ is the regular payment (quarterly deposit) we are solving for,
- $r$ is the interest rate per period,
- $n$ is the total number of periods.

### Step 1: Identify the values

1. **APR** = 3.3%, compounded quarterly.
   - The quarterly interest rate $r$ is $ \frac{3.3\%}{4} = \frac{0.033}{4} = 0.00825$.
   
2. **Time** = 17 years.
   - The number of quarters (periods) $n$ is $17 \times 4 = 68$.

3. **Future Value** = $80,000.

### Step 2: Solve for $P$ (the quarterly deposit)

We rearrange the future value formula to solve for $P$:

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

Substituting the known values:

\[
P = \frac{80,000}{\frac{(1 + 0.00825)^{68} - 1}{0.00825}}
\]

Let me calculate this.The quarterly deposit Cyrus needs to make is approximately **$881.89**.

### Step 3: Calculate the total amount deposited

To find out how much Cyrus will ultimately deposit over 17 years, we multiply the quarterly deposit by the total number of periods:

\[
\text{Total Deposits} = P \times n = 881.89 \times 68
\]

Let me calculate this.The total amount Cyrus will ultimately deposit over 17 years is approximately **$59,968.29**.

### Step 4: Calculate the interest earned

The interest earned is the difference between the future value and the total deposits:

\[
\text{Interest Earned} = FV - \text{Total Deposits} = 80,000 - 59,968.29
\]

Let me calculate that.The total interest earned over 17 years is approximately **$20,031.71**.

### Summary:
- Quarterly deposit: **$881.89**
- Total amount deposited: **$59,968.29**
- Interest earned: **$20,031.71**

Would you like any more details or have any questions?

Here are five related questions:
1. How would the result change if the APR were 4% instead of 3.3%?
2. What if Cyrus wanted to accumulate $100,000 instead of $80,000?
3. How would switching from quarterly to monthly compounding affect the deposit?
4. How does the future value formula adjust if payments are made at the beginning of each quarter?
5. What is the impact on the deposit amount if the investment period is reduced to 10 years?

**Tip**: Compounding frequency greatly influences the growth of an investment. The more frequent the compounding, the more interest accrues.

Cyrus deposits a fixed quarterly amount into an annuity account for his child's college fund. He wishes to accumulate a future value of $80,000 in 17 years. Assuming an APR of 3.3% compounded quarterly, how much of the $80,000 will Cyrus ultimately deposit in the account, and how much is interest earned? Round your answers to the nearest cent, if necessary.

Learn how to calculate the future value of an annuity for a college fund, where a fixed quarterly deposit is made into an account with a 3.3% APR compounded quarterly. This step-by-step guide includes the calculation of total deposits and interest earned over 17 years, suitable for students in Grades 11-12.

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