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The question asks to find the average amount earned per year between the 5th year and the 10th year for an investment of $5000 compounded continuously at 4% interest. Here's how we can solve it using the values provided on the graph.

We are provided the values of the function $ f(x) $ at two key points:

- At $ x = 5 $ years, $ f(5) = 6107 $ dollars.
- At $ x = 10 $ years, $ f(10) = 7459 $ dollars.

To find the average annual amount earned between the 5th and 10th years, we first calculate the total amount earned in that interval, then divide it by the number of years.

### Steps:

1. **Calculate the total amount earned:**
   \[
   \text{Total earned} = f(10) - f(5) = 7459 - 6107 = 1352 \, \text{dollars}
   \]

2. **Find the number of years between the 5th and 10th year:**
   \[
   \text{Number of years} = 10 - 5 = 5 \, \text{years}
   \]

3. **Average earnings per year:**
   \[
   \text{Average earnings per year} = \frac{1352}{5} = 270.4 \, \text{dollars/year}
   \]

Thus, the average amount earned per year between the 5th and 10th years is **$270.40**, which matches the value you have already entered.

Let me know if you need further clarification or details!

Here are 5 related questions you might want to explore:
1. How does continuous compounding differ from annual or quarterly compounding?
2. What is the formula for continuous compounding, and how can it be applied in this problem?
3. How would the average change if the interest rate were 5% instead of 4%?
4. Can you estimate the average annual growth between the 10th and 20th years based on the graph?
5. What would be the total value after 30 years at 4% continuous compounding?

**Tip:** When dealing with interest problems, remember that the formula for continuously compounded interest is $ A = P e^{rt} $, where $ A $ is the amount, $ P $ is the principal, $ r $ is the interest rate, and $ t $ is the time in years.

Find the average amount earned per year between the 5th year and the 10th year for an investment of $5000 compounded continuously at 4% interest.

Solve for the average amount earned per year between the 5th and 10th years for an investment of $5000 compounded continuously at 4%. This problem explores continuous compound interest and exponential growth in finance, suitable for students in Grades 9-12.

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