Hi, I\'d like help with the following question please. Thanks The average cost p

Question

Hi, I'd like help with the following question please. Thanks

The average cost per item to produce q items is given by a(q) = 0.01q^2 - 0.6q + 14, for q > 0. What is the total cost, C(q), of producing q goods? C(q) = What is the minimum marginal cost? minimum MC = (Be sure you can say what the practical interpretation of this result is!) At what production level is the average cost a minimum? q = What is the lowest average cost? minimum average cost = Compute the marginal cost at q = 30. MC(30) = How does this relate to your previous answer? Explain this relationship both analytically and in words.

Explanation / Answer

Average cost or unit cost is equal to total cost divided by the number of goods produced. Thus the total cost is

TC = a(q)*q

=(0.01q^2 -0.6q + 14)q
= 0.01q^3 - 0.6q^2 + 14q

b)
Marginal cost is the increase in cost that accompanies a unit increase in output; the partial derivative of the cost function with respect to output.

MC = 0.03q^2 - 1.2q + 14

Taking the derivative

MC' = 0.06q - 1.2
0.06q - 1.2 = 0
q = 1.2/0.06
= 20

The minimum marginal cost is

MC = 0.03(20)^2 - 1.2(20) + 14
= 2

The production level at the minimum average cost is
q = 20

Taking the derivative

a'(q) = 0.02q -0.6
0.02q - 0.6 = 0
q = 0.6/0.02
= 30

The lowest average cost is

a(30) = 0.01(30)^2 - 0.6(30) +14
=5

e)
The marginal cost at q=30 is

MC(30) = 0.03(30)^2 - 1.2(30) +14
= 5

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