I am so lost for this homework question, can anyone give me some help, thanks. T

ID: 1209212 • Letter: I

Question

I am so lost for this homework question, can anyone give me some help, thanks.

The average cost per item to produce q items is given by a(q) = 0.01q^2 0.6q + 13, for q > 0.

(a) What is the total cost, C(q), of producing q goods?

(b) What is the minimum marginal cost? What is the practical interpretation of this result?

(d) Compute the marginal cost at q = 30. How does this relate to your answer to part (c)? Explain this relationship both analytically and in words.

The average cost per item to produce q items is given by a(q) = 0.01q^2 0.6q + 13, for q > 0.

(a) Average total cost is the total cost divided by the number of units produced. This implies AC = TC/q. To get TC from AC, multiply AC by q.

The total cost, C(q), of producing q goods = AC*q

C(q) = a(q)*q

=(0.01q2 0.6q + 13)q, for q > 0

=0.01q3 0.6q2 + 13q, for q > 0.

Hence total cost is c(q) = 0.01q3 0.6q2 + 13q, for q > 0.

(b) Marginal Cost: It is the first derivative of the total cost with respect to the output. So Marginal cost is:

MC(q) = dc(q)/q

=d(0.01q3 0.6q2 + 13q)/q

=0.03q2 1.2q + 13

The minimum level of marginal cost is achieved when the derivative of marginal cost is set equal to zero.

MC'(q) = 0

0.06q - 1.2 = 0

q = 20

At q = 20, MC(20) = 0.03(20)2 1.2(20) + 13 = 25 - 24 = 1

Hence the minimum of marginal cost is achieved when q = 20 and minimum MC is $1

AC'(q) = 0

0.02q 0.6 = 0

q = 30. At q = 30, AC(30) = 0.01(30)2 0.6*30 + 13 = 4

Hence the lowest average cost is $4 at q = 30

(d) The value of marginal cost at q = 30 is given by:

MC(30) = 0.03(30)2 1.2(30) + 13 = 4

Note that MC = AC when AC is at its minimum level. This provides the general economic result:

When MC>AC, AC rises

When MC<AC, AC falls

When MC = AC, AC is at its minimum.

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