Use the formula for compound interest to find the compound amount and the interest. Interest is compounded quarterly.
Principal
Rate of Interest
Time
$ 15 comma 000
11 %
3 and one half
 years
Question content area bottom
Part 1
What is the compound​ amount?
​$
  
21,929.91 ​(Round to the nearest​ cent.)
Part 2
What is the amount of compound​ interest?
​$
  
enter your response here ​(Round to the nearest​ cent.)

To calculate the compound amount and the compound interest using the formula for compound interest, we use the formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- $A$ = the amount of money accumulated after n years, including interest.
- $P$ = the principal amount (the initial amount of money).
- $r$ = the annual interest rate (decimal).
- $n$ = the number of times that interest is compounded per year.
- $t$ = the number of years the money is invested or borrowed.

### Given Data:
- $P = 15,000$
- $r = 11\% = 0.11$
- $t = 3.5$ years
- $n = 4$ (since the interest is compounded quarterly)

### Part 1: Calculate the Compound Amount

1. Substitute the values into the formula:
   \[
   A = 15,000 \left(1 + \frac{0.11}{4}\right)^{4 \times 3.5}
   \]

2. Calculate $\frac{0.11}{4}$:
   \[
   \frac{0.11}{4} = 0.0275
   \]

3. Calculate $4 \times 3.5 = 14$.

4. Now plug in the values:
   \[
   A = 15,000 \left(1 + 0.0275\right)^{14}
   \]
   \[
   A = 15,000 \left(1.0275\right)^{14}
   \]

5. Calculate $(1.0275)^{14}$:
   \[
   (1.0275)^{14} \approx 1.478442
   \]

6. Now calculate $A$:
   \[
   A \approx 15,000 \times 1.478442 \approx 22,176.63
   \]

So, the compound amount is approximately **$22,176.63** (rounded to the nearest cent).

### Part 2: Calculate the Compound Interest

The compound interest is calculated as:
\[
\text{Compound Interest} = A - P
\]
\[
\text{Compound Interest} = 22,176.63 - 15,000 = 7,176.63
\]

Thus, the compound interest is approximately **$7,176.63** (rounded to the nearest cent).

### Summary:
- **Compound Amount:** $22,176.63
- **Compound Interest:** $7,176.63

Would you like more details or have any questions? Here are some related questions you might consider:

1. How does changing the interest rate affect the compound amount?
2. What would be the total amount if the interest were compounded annually instead?
3. How does the length of time affect compound interest?
4. Can you explain the concept of continuous compounding?
5. How would you calculate the future value of an investment with different compounding frequencies? 

**Tip:** Always keep track of the compounding frequency, as it significantly affects the growth of the investment!

Use the formula for compound interest to find the compound amount and the interest. Interest is compounded quarterly. Principal $15,000, Rate of Interest 11%, Time 3.5 years.

Explore how to calculate the compound amount and interest for a principal of $15,000 compounded quarterly at an 11% interest rate over 3.5 years. The compound amount is approximately $22,176.63, and the compound interest is approximately $7,176.63.

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