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Let's solve each part of the problem step-by-step.

### Given:
The total cost function is given by:
\[ C(x) = 50,000 + 600x \]
where \( x \) represents the number of auto body frames produced.

#### (A) Find the average cost per unit if 200 frames are produced.

The average cost per unit is calculated as:
\[ \text{Average Cost} = \frac{C(x)}{x} \]

Substitute \( x = 200 \):
\[ \text{Average Cost} = \frac{C(200)}{200} \]
\[ C(200) = 50,000 + 600 \times 200 \]
Calculating this will give the average cost.

#### (B) Find the marginal average cost at a production level of 200 units.

The marginal cost \( C'(x) \) is the derivative of \( C(x) \):
\[ C'(x) = 600 \]

The marginal average cost is given by the derivative of the average cost function:
\[ \text{Average Cost} = \frac{C(x)}{x} \]
Let's differentiate \(\frac{C(x)}{x}\) with respect to \( x \).

#### (C) Use the results from parts (A) and (B) to estimate the average cost per frame if 201 frames are produced.

The average cost of producing 201 frames can be approximated using the average cost at 200 units and the marginal average cost.

I'll perform these calculations now.### Solutions:
#### (A) Average cost per unit if 200 frames are produced:
The average cost per unit is:
\[ \text{Average Cost} = 850.0 \text{ dollars} \]

#### (B) Marginal average cost at a production level of 200 units:
The marginal average cost is:
\[ \text{Marginal Average Cost} = -1.25 \text{ dollars} \]

#### (C) Estimated average cost per frame if 201 frames are produced:
Using the results from parts (A) and (B), the estimated average cost per frame for 201 units is:
\[ \text{Estimated Average Cost} = 848.75 \text{ dollars} \]

This means that the average cost per unit slightly decreases as the production level increases from 200 to 201 units.

Would you like more details on any part of this solution? Here are some additional related questions:

1. How is the average cost per unit derived from the total cost function?
2. What does a negative marginal average cost indicate?
3. Can you explain the difference between marginal cost and marginal average cost?
4. How would the average cost change if the fixed cost increased?
5. What if we need the exact average cost for 201 units, rather than an estimate?

**Tip:** Understanding how marginal costs and average costs behave with production levels is crucial in optimizing manufacturing processes and setting pricing strategies.

The total cost (in dollars) of manufacturing x auto body frames is C(x) = 50,000 + 600x. (A) Find the average cost per unit if 200 frames are produced. (B) Find the marginal average cost at a production level of 200 units. (C) Use the results from parts (A) and (B) to estimate the average cost per frame if 201 frames are produced.

This problem involves determining the average cost per unit for 200 frames, finding the marginal average cost at 200 units, and using both to estimate the cost for 201 units. Learn how to apply cost functions, differentiation, and average cost calculations.

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