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The problem involves calculating the future value (FV) of an investment with daily compounding interest, as well as the interest earned. The information provided is:

- Principal ($ P $) = $4,000
- Annual interest rate ($ r $) = 7.9% (or 0.079 as a decimal)
- Time ($ t $) = 10 years
- Compounding frequency = daily (365 days per year)

### Step 1: Calculate the Future Value (FV)
The formula for future value with daily compounding interest is:

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

Where:
- $ P $ is the principal amount
- $ r $ is the annual interest rate
- $ n $ is the number of times the interest is compounded per year
- $ t $ is the number of years

Given values:
- $ P = 4,000 $
- $ r = 0.079 $
- $ n = 365 $
- $ t = 10 $

Substitute the values into the formula:

\[
FV = 4000 \left(1 + \frac{0.079}{365}\right)^{365 \times 10}
\]

\[
FV = 4000 \left(1 + 0.00021644\right)^{3650}
\]

\[
FV = 4000 \left(1.00021644\right)^{3650}
\]

Using a calculator to compute the exponentiation:

\[
FV \approx 4000 \times 2.22163 = 8886.52
\]

Rounding to the nearest dollar:

\[
FV \approx 8,887
\]

### Step 2: Calculate the Interest Earned
The interest earned is simply the future value minus the principal:

\[
\text{Interest} = FV - P
\]

\[
\text{Interest} = 8887 - 4000 = 4887
\]

### Final Answers:
- **Future Value (FV):** $8,887
- **Interest Earned:** $4,887

Would you like further details or have any questions?

### Related Questions:
1. How does the compounding frequency affect the future value?
2. What is the difference between simple interest and compound interest?
3. How would the future value change if the interest were compounded monthly?
4. What would happen if the time period were extended to 20 years instead of 10?
5. How do different interest rates affect the amount of interest earned?
6. Can you explain the concept of continuous compounding?
7. How do you calculate the effective annual rate (EAR) given the nominal rate and compounding frequency?
8. What real-world scenarios benefit from daily compounding interest?

**Tip:** When calculating compounded interest, always ensure you use the correct number of compounding periods per year for accuracy.

Learn how to calculate the future value (FV) of an investment with daily compounding interest. This problem involves a principal of $4,000, an annual interest rate of 7.9%, and a time period of 10 years. Get a detailed step-by-step solution including the interest earned and calculations for future value.

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