To offer scholarships to children of employees, a company invests $14,000 at the end of every three months in an annuity that pays 10.5% compounded quarterly. a. How much will the company have in scholarship funds at the end of ten years? b. Find the interest.

This problem involves calculating the future value of an annuity and the interest earned on the investment. Since the company is investing periodically in an account with interest compounded quarterly, we use the **future value of an ordinary annuity formula**:

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

Where:
- $FV$ = future value of the annuity
- $P$ = periodic payment (\$14,000 every three months)
- $i$ = interest rate per period
- $n$ = total number of periods

### Part a: Calculate the future value of the annuity
We are given:
- $P = 14,000$ (the amount invested at the end of each quarter)
- Annual interest rate = 10.5%, so the quarterly interest rate $i = \frac{10.5\%}{4} = 0.02625$
- Time = 10 years, so the total number of periods $n = 10 \times 4 = 40$ quarters

Now, we can substitute these values into the formula for $FV$:

\[
FV = 14,000 \times \frac{(1 + 0.02625)^{40} - 1}{0.02625}
\]

Let's calculate the future value.The future value of the company's scholarship fund at the end of ten years is approximately **$970,243.18**.

### Part b: Calculate the interest
To find the interest earned, we subtract the total amount of the payments made by the company from the future value.

The total amount invested by the company is:

\[
\text{Total investment} = P \times n = 14,000 \times 40 = 560,000
\]

The interest is:

\[
\text{Interest} = FV - \text{Total investment}
\]

Let's calculate the interest earned.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, you can manually calculate the interest using the following steps:

1. **Total investment** = $14,000 \times 40 = 560,000$
2. **Interest** = Future Value $-$ Total Investment  
\[
Interest = 970,243.18 - 560,000 = 410,243.18
\]

Thus, the interest earned is approximately **$410,243.18**.

Do you want any additional details or have any questions about these calculations?

Here are some related questions:
1. How would the future value change with monthly compounding instead?
2. What happens if the interest rate is increased to 12%?
3. How is the annuity formula derived?
4. What are other real-world applications of annuities?
5. How much does each quarterly contribution grow due to compounding?

**Tip**: Always double-check whether the interest is compounded monthly, quarterly, or annually when working with annuities.

Learn how to calculate the future value and interest of a company's $14,000 quarterly investment in an annuity at 10.5% compounded quarterly over 10 years. This problem involves calculating the future value using the annuity formula and determining the interest earned, suitable for finance or math students.

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