The total cost​ (in hundreds of​ dollars) to produce x units of perfume is Upper C left parenthesis x right parenthesis equals StartFraction 2 x minus 9 Over 8 x plus 3 EndFraction
. 
​(a) Find the average cost function.
​(b) Find the marginal average cost function.
​(c) Find the average cost and the marginal average cost for a production level of 20 units. Interpret your results.
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Part 1
​(a) The average cost function is Upper C overbar​(x)equals
  
enter your response here.

Given the cost function \( C(x) = \frac{2x - 9}{8x + 3} \), we need to find the following:

### (a) Average Cost Function

The average cost function, denoted as \( \overline{C}(x) \), is given by the total cost divided by the number of units \( x \):
\[
\overline{C}(x) = \frac{C(x)}{x}.
\]

Substitute \( C(x) = \frac{2x - 9}{8x + 3} \) into the formula:
\[
\overline{C}(x) = \frac{\frac{2x - 9}{8x + 3}}{x}.
\]

This simplifies to:
\[
\overline{C}(x) = \frac{2x - 9}{x(8x + 3)}.
\]

### (b) Marginal Average Cost Function

The marginal average cost function is the derivative of the average cost function \( \overline{C}(x) \). To find this, we need to differentiate \( \overline{C}(x) \) with respect to \( x \):
\[
\overline{C}'(x) = \frac{d}{dx} \left( \frac{2x - 9}{x(8x + 3)} \right).
\]

This is a bit complex since it's a quotient, so we'll use the quotient rule:
\[
\overline{C}'(x) = \frac{[x(8x + 3)] \cdot \frac{d}{dx}(2x - 9) - (2x - 9) \cdot \frac{d}{dx}[x(8x + 3)]}{[x(8x + 3)]^2}.
\]

### (c) Average Cost and Marginal Average Cost for 20 Units

1. **Average Cost** at \( x = 20 \):
\[
\overline{C}(20) = \frac{2 \times 20 - 9}{20(8 \times 20 + 3)}.
\]

2. **Marginal Average Cost** at \( x = 20 \):
\[
\overline{C}'(20) = \text{Substitute } x = 20 \text{ into the derivative } \overline{C}'(x).
\]

Let me calculate these values now.Here are the results:

### (a) Average Cost Function
\[
\overline{C}(x) = \frac{2x - 9}{x(8x + 3)}.
\]

### (b) Marginal Average Cost Function
\[
\overline{C}'(x) = \frac{-8(2x - 9)}{x(8x + 3)^2} + \frac{2}{x(8x + 3)} - \frac{2x - 9}{x^2(8x + 3)}.
\]

### (c) Average Cost and Marginal Average Cost for 20 Units
1. **Average Cost** at \( x = 20 \):
\[
\overline{C}(20) \approx 0.0095 \quad (\text{in hundreds of dollars, so } 0.95 \text{ cents per unit}).
\]

2. **Marginal Average Cost** at \( x = 20 \):
\[
\overline{C}'(20) \approx -0.00033 \quad (\text{in hundreds of dollars per unit change}).
\]

**Interpretation**:
- The average cost of producing 20 units is approximately 0.95 cents per unit.
- The marginal average cost is negative, indicating that producing one more unit beyond 20 will decrease the average cost slightly.

Do you want further details or have any questions? 

Here are some related questions for further exploration:

1. How does the average cost function behave as \( x \) increases?
2. What does the marginal cost function tell us about cost efficiency?
3. How can the marginal average cost be used in decision-making?
4. What happens to the average cost when production levels are very low or very high?
5. Can we find the minimum average cost using calculus?

**Tip**: Marginal costs are useful in understanding how costs change with production levels, which can aid in making efficient production decisions.

The total cost​ (in hundreds of​ dollars) to produce x units of perfume is C(x) = (2x - 9) / (8x + 3). (a) Find the average cost function. (b) Find the marginal average cost function. (c) Find the average cost and the marginal average cost for a production level of 20 units. Interpret your results.

Learn how to find the average cost function and marginal average cost function for the perfume production model C(x) = (2x - 9) / (8x + 3). Explore calculus concepts such as cost efficiency and marginal analysis with a detailed solution. Suitable for college-level students studying calculus.

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