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Let's solve the problem based on the information provided in the image.

### Given:
- Cost function: \( C(x) = 147 + 8.5x \)
- Revenue function: \( R(x) = 7x - 0.03x^2 \)

### To find:
The **marginal average cost function** \( \overline{C}'(x) \).

### Steps:
1. **Average Cost Function:**
   The average cost function \( \overline{C}(x) \) is given by the total cost function \( C(x) \) divided by \( x \):
   \[
   \overline{C}(x) = \frac{C(x)}{x}
   \]
   Using the given cost function \( C(x) = 147 + 8.5x \):
   \[
   \overline{C}(x) = \frac{147 + 8.5x}{x} = \frac{147}{x} + 8.5
   \]

2. **Marginal Average Cost Function:**
   The marginal average cost function is the derivative of the average cost function \( \overline{C}(x) \) with respect to \( x \). So, we differentiate \( \overline{C}(x) = \frac{147}{x} + 8.5 \):
   \[
   \overline{C}'(x) = \frac{d}{dx} \left( \frac{147}{x} + 8.5 \right)
   \]
   The derivative of \( \frac{147}{x} \) is \( -\frac{147}{x^2} \), and the derivative of the constant 8.5 is 0. Thus:
   \[
   \overline{C}'(x) = -\frac{147}{x^2}
   \]

### Conclusion:
The marginal average cost function is:
\[
\overline{C}'(x) = -\frac{147}{x^2}
\]

Let me know if you need more details on any part or have any questions.

Here are 5 related questions you could explore:
1. How do you find the marginal cost from the total cost function directly?
2. What is the relationship between average cost and marginal cost?
3. How would the result change if the cost function included a cubic term?
4. How does the marginal average cost behave as \( x \to \infty \)?
5. Can we interpret the revenue function \( R(x) \) in terms of marginal revenue?

**Tip:** In economics, the marginal cost represents the additional cost incurred from producing one more unit, while marginal revenue shows the additional revenue from selling one more unit.

Find the marginal average cost function if cost and revenue are given by C(x)=147+8.5x and R(x)=7x−0.03x^2.

Learn how to calculate the marginal average cost function given cost C(x)=147+8.5x and revenue R(x)=7x−0.03x^2. This problem involves differentiation and calculus, suitable for university-level Mathematics II or Economics students.

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