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

We are given the total cost function for manufacturing $ x $ auto body frames as:
\[
C(x) = 50,000 + 900x
\]
Where $ C(x) $ is the total cost in dollars and $ x $ is the number of frames produced.

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

The average cost per unit, $ A(x) $, is calculated by dividing the total cost $ C(x) $ by the number of units $ x $:
\[
A(x) = \frac{C(x)}{x} = \frac{50,000 + 900x}{x}
\]
For $ x = 200 $:
\[
A(200) = \frac{50,000 + 900 \times 200}{200}
\]
\[
A(200) = \frac{50,000 + 180,000}{200} = \frac{230,000}{200} = 1,150 \text{ dollars}
\]
So, the average cost per unit when 200 frames are produced is **$1,150**.

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

The marginal average cost is the derivative of the average cost function $ A(x) $ with respect to $ x $:
\[
A(x) = \frac{50,000}{x} + 900
\]
Now, differentiate $ A(x) $:
\[
A'(x) = \frac{d}{dx} \left( \frac{50,000}{x} + 900 \right) = -\frac{50,000}{x^2}
\]
For $ x = 200 $:
\[
A'(200) = -\frac{50,000}{200^2} = -\frac{50,000}{40,000} = -1.25
\]
Thus, the marginal average cost at a production level of 200 units is **$-1.25 per unit**.

### Part (C): Estimate the average cost per frame if 201 frames are produced using parts (A) and (B).

Using the result from part (A), the average cost for 200 units is $1,150. The marginal average cost tells us how the average cost changes as we produce an additional unit.

So, for 201 units, the average cost can be approximated as:
\[
A(201) \approx A(200) + A'(200) \times (201 - 200)
\]
\[
A(201) \approx 1,150 + (-1.25) \times 1 = 1,150 - 1.25 = 1,148.75
\]
Therefore, the estimated average cost per frame when 201 frames are produced is **$1,148.75**.

---

Would you like further clarification on any part of the solution?

### Related Questions:
1. How does the average cost behave as more units are produced?
2. What is the significance of the fixed cost in the cost function?
3. How does the marginal average cost relate to economies of scale?
4. Can you explain how to derive the total cost from the average cost?
5. How would the solution change if the fixed cost was lower or higher?

### Tip:
When estimating changes in cost, always check whether the marginal change is positive or negative to understand whether costs are increasing or decreasing per unit.

The total cost (in dollars) of manufacturing x auto body frames is C(x) = 50,000 + 900x.

(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.

Learn how to compute the average cost and marginal cost for manufacturing auto body frames using cost functions. This detailed example walks through finding the average cost at 200 units, computing the marginal average cost, and estimating the cost for 201 frames using calculus principles.

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