G:dollars to manufacture x units of a device in an hour at one of their manufacturing centers. How many devices should be manufactured each hour to minimize average cost? .002x+160 units What is the resulting average cost of a device? $ How does the average cost compare with the marginal cost at the optimal production level? Find how much they differ.

- How many devices should be manufactured each hour to minimize average cost? **21000** units.

- What is the resulting average cost of a device? $ 441160.001

- How does the average cost compare with the marginal cost at the optimal production level? Find how much they differ. $ 0.00

The marginal cost and the average cost are absolutely the same at the optimal cost.

Explanation:

C(x) = 441000 + 160x + 0.001x²

Average cost = (Total cost)/(number of units)

Average cost = (441000/x) + 160 + 0.001x

At minimum average cost,

dAC/dx = 0 (Note: AC means average cost)

dAC/dx = - 441000/x² + 0.001

when it is equal to 0

- 441000/x² = - 0.001

x² = 441000/0.01 = 441000000

x = 21000 units

Average cost of a device.

Average cost = (441000/x) + 160 + 0.001x

For one device, x = 1

average cost = (441000/1) + 160 + 0.001(1)

Average cost of a device = 441000 + 160 + 0.001 = $ 441160.001

At optimal level, that is, x = 21000 units

Average cost = (441000/x) + 160 + 0.001x = (441000/21000) + 160 + 0.001(21000) = $202

Marginal cost = dC/dx (Rate of change of total cost with the number of units)

C(x) = 441000 + 160x + 0.001x²

Marginal cost = (dC/dx) = 160 + 0.002x

At x = 21000 units

Marginal cost = 160 + 0.002(21000) = $202

The marginal cost and the average cost are absolutely the same at the optimal cost.

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