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Let's break down the problem and solve it step by step.

### Given:
The total cost function \( C(x) \) to produce \( x \) units of perfume is:

\[
C(x) = \frac{8x - 1}{6x + 7}
\]

#### Part (a): Find the average cost function.

The average cost function \( \bar{C}(x) \) is given by:

\[
\bar{C}(x) = \frac{C(x)}{x}
\]

Substitute the given \( C(x) \):

\[
\bar{C}(x) = \frac{\frac{8x - 1}{6x + 7}}{x} = \frac{8x - 1}{x(6x + 7)}
\]

Thus, the average cost function is:

\[
\bar{C}(x) = \frac{8x - 1}{x(6x + 7)}
\]

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Do you want to continue solving parts (b) and (c)? Or do you have any questions about this part so far?

Here are five questions related to this topic:
1. How is the marginal cost related to the total cost function?
2. Why do we divide by \( x \) to get the average cost?
3. How would the shape of the average cost function change as \( x \) increases?
4. What real-life scenarios would benefit from knowing the average cost function?
5. Can you explain the relationship between fixed costs and average costs?

**Tip:** To differentiate a fraction function, use the quotient rule or rewrite it as a product of two terms and apply the product rule.

The total cost (in hundreds of dollars) to produce x units of perfume is C(x) = (8x - 1)/(6x + 7). Find the average cost function, the marginal average cost function, and the average and marginal average costs for a production level of 10 units.

Learn how to find the average and marginal average cost functions for a given total cost function C(x) = (8x - 1)/(6x + 7) for perfume production. This step-by-step guide will help you understand how to compute the costs for a production level of 10 units, suitable for advanced high school or college-level students.

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