Questions: The total cost (in dollars) of manufacturing x auto body frames is C(x)=40,000+900x. (A) Find the average cost per unit if 500 frames are produced. (B) Find the marginal average cost at a production level of 500 units. (C) Use the results from parts (A) and (B) to estimate the average cost per frame if 501 frames are produced. (A) If 500 frames are produced, the average cost is per frame.

Solution Steps

To solve these problems, we will use the given cost function $C(x) = 40,000 + 900x$.

(A) Calculate the average cost per unit by dividing the total cost by the number of units produced.

(B) Find the marginal average cost by differentiating the average cost function and evaluating it at 500 units.

(C) Estimate the average cost for 501 frames using the results from parts (A) and (B).

The average cost per unit when 500 frames are produced is given by:

$$\text{Average Cost} = \frac{C(500)}{500} = \frac{900 \times 500 + 40,000}{500} = 980$$

The marginal average cost is the derivative of the average cost function evaluated at 500 units:

$$\text{Average Cost Function} = \frac{C(x)}{x} = \frac{900x + 40,000}{x}$$

$$\text{Marginal Average Cost} = \left. \frac{d}{dx} \left( \frac{900x + 40,000}{x} \right) \right|_{x=500} = -\frac{4}{25}$$

Using the results from the previous steps, estimate the average cost for 501 frames:

$$\text{Estimated Average Cost for 501 Frames} = 980 + \left(-\frac{4}{25}\right) = \frac{24,496}{25}$$

Final Answer