Question

The total cost​ (in dollars) of manufacturing x auto body frames is ​C(x)=60,000+400x. ​(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.

Answer 1

a) Average price per unit = 700

b) average marginal cost = -1.5 dollars/frame

c) For 201 frames are produced = $698.5

Step-by-step explanation:

Given Data:

C(x) = 60000 + 400x

a) Average cost for 200 units:

Total Cost of 200 units will be:

C(200) = 60000 + 400(200)

C(200) = 140,000

Total Cost of 200 units = 140,000

Average price per unit = Total Cost / number of units

Average price per unit = 140,000/200

Average price per unit = 700

b) Marginal Average Cost:

We know that marginal cost =
`C^(')`(x)

And the average marginal cost =
`(d)/(dx)`
`(C(x))/(x)`

So,

= the derivative of (
`(60,000 + 400(x))/(x)`)

= the derivative of (
`((60,000))/(x) + 400`)

=
`(-60,000)/(x^(2) )` + 0

So,

average marginal cost =
`(-60,000)/(x^(2) )`

at x = 200 units

average marginal cost =
`(-60,000)/(200^(2) )`

average marginal cost = -1.5 dollars/frame

c) Average cost per frame if 201 frames are produced:

We already seen in the part b that, the average marginal cost is decreasing by 1.5 dollars /frame. So,

if 201 frames are produced the instead of 200 then the difference is of 1 frame and we discussed that average marginal cost of 1 frame is decreasing at 1.5 dollars per frame.

So,

As we know the average cost from part a = 700 dollars.

Therefore, for 201 item = 700 dollars -1.5 dollars

For 201 frames are produced = $698.5