Quiz: Quiz 6: (18) The total cost (in dollars) of manufacturing \( x \) auto body frames is \( C(x)=80,000+700 x \). (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 \( (\mathrm{A}) \) and (B) to estimate the average cost per frame if 501 frames are produced.

To find the average cost per unit if 500 frames are produced, we first substitute \( x = 500 \) into the cost function \( C(x) \). So we have \( C(500) = 80,000 + 700(500) = 80,000 + 350,000 = 430,000 \). The average cost per unit is then \( \frac{C(500)}{500} = \frac{430,000}{500} = 860 \) dollars per frame. Next, to find the marginal average cost at a production level of 500 units, we need the average cost function \( A(x) = \frac{C(x)}{x} \). The marginal average cost \( A'(x) \) is calculated from this. The average cost at \( x \) is \( A(x) = \frac{80,000 + 700x}{x} = \frac{80,000}{x} + 700 \). Therefore, \( A'(x) = -\frac{80,000}{x^2} \). At \( x = 500 \), \( A'(500) = -\frac{80,000}{500^2} = -\frac{80,000}{250,000} = -0.32 \). This means the marginal average cost at this level is approximately \( -0.32 \) dollars per frame, implying that producing an additional frame will reduce the average cost slightly. Finally, to estimate the average cost per frame if 501 frames are produced, we can use the average cost calculated in part A and add the marginal average cost from B. The estimated average cost at 501 frames is \( A(501) \approx 860 - 0.32 \approx 859.68 \) dollars per frame. So, producing an additional frame brings down the average cost nicely!