https://file.aiquickdraw.com/mathbot/file/1730527304356suoxi7w8.jpg

Let's solve this question in parts based on the image provided.

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### Given Information
- **Principal (Initial Deposit), $ a = 4000 $**
- **Annual Interest Rate, $ r = 3.8\% = 0.038 $**
- **Compounded Quarterly, $ k = 4 $**

### Formula for Compounded Interest
The formula for an account balance that compounds interest is:
\[
A(t) = a \left(1 + \frac{r}{k}\right)^{kt}
\]
where:
- $ A(t) $ is the account balance after $ t $ years.
- $ a $ is the principal.
- $ r $ is the annual interest rate (as a decimal).
- $ k $ is the number of times the interest is compounded per year.
- $ t $ is the number of years.

---

### Part A: What values should be used for $ a $, $ r $, and $ k $?
- **$ a = 4000 $**
- **$ r = 0.038 $**
- **$ k = 4 $**

These values are correctly provided in the question.

---

### Part B: How much money will Gianna have in the account in 8 years?

#### Solution
Using the compound interest formula, substitute the values:
\[
A(8) = 4000 \left(1 + \frac{0.038}{4}\right)^{4 \times 8}
\]
1. **Calculate the periodic rate**: $ \frac{0.038}{4} = 0.0095 $.
2. **Calculate the exponent**: $ 4 \times 8 = 32 $.
3. **Calculate inside the parentheses**: $ 1 + 0.0095 = 1.0095 $.
4. **Raise to the power**: $ 1.0095^{32} \approx 1.349353 $.
5. **Multiply by the principal**: $ 4000 \times 1.349353 \approx 5397.41 $.

So, Gianna should have approximately **$5397.41** in the account after 8 years (rounded to the nearest penny).

---

### Part C: What is the Annual Percentage Yield (APY)?

The Annual Percentage Yield (APY) is given by:
\[
\text{APY} = \left(1 + \frac{r}{k}\right)^k - 1
\]
1. **Substitute the values**: $ \text{APY} = \left(1 + \frac{0.038}{4}\right)^4 - 1 $.
2. **Calculate the periodic rate**: $ \frac{0.038}{4} = 0.0095 $.
3. **Calculate inside the parentheses**: $ 1 + 0.0095 = 1.0095 $.
4. **Raise to the power**: $ 1.0095^4 \approx 1.038678 $.
5. **Subtract 1**: $ 1.038678 - 1 = 0.038678 $.

Converting to a percentage, the APY is approximately **3.868%** (rounded to three decimal places).

---

### Final Answers:
- **B. Amount after 8 years**: $5397.41
- **C. APY**: 3.868%

Would you like further clarification on any part of this solution? Here are some related questions you might find helpful:

1. What is the difference between APR (annual percentage rate) and APY?
2. How does changing the compounding frequency affect the final balance?
3. What happens to the balance if the interest rate is increased?
4. How is the APY calculated for monthly compounding instead of quarterly?
5. How does compounding quarterly differ from compounding annually in terms of growth?

**Tip:** For accurate results in compound interest problems, ensure that you use enough decimal places in intermediate steps before rounding the final answer.

A bank features a savings account with an annual percentage rate of r = 3.8% compounded quarterly. Gianna deposits $4,000 into the account.

(A) What values should be used for a, r, and k?
(B) How much money will Gianna have in the account in 8 years?
(C) What is the annual percentage yield (APY) for the savings account? Round to three decimal places.

Learn how to calculate the compound interest and annual percentage yield (APY) for a savings account with an annual interest rate of 3.8%, compounded quarterly. This example involves an initial deposit of $4,000 with a target duration of 8 years. Suitable for high school students in Grades 10-12.

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