The average cost per item to produce q items is given by...

The average cost per item to produce $q$ items is given by $$ a(q)=0.01 q^{2}-0.6 q+13, \text { for } q>0 $$ (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? (c) At what production level is the average cost a minimum? What is the lowest average cost? (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
$$
a(q)=0.01 q^{2}-0.6 q+13, \text { for } q>0
$$
(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?
(c) At what production level is the average cost a minimum? What is the lowest average cost?
(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.

Key Concepts

Optimization of Cost Functions

Optimization in the context of cost functions involves finding the production level at which certain cost measures (like average cost) are minimized or maximized. This typically requires taking derivatives of the cost functions and setting them to zero to determine the critical points, ensuring that firms can operate efficiently while minimizing costs.

Relationship Between Marginal and Average Cost

There is a significant relationship between marginal cost and average cost in that the marginal cost curve reflects the cost of producing one more unit. When the marginal cost equals the average cost, this point often corresponds to the minimum of the average cost function. This relationship is key in economic analysis as it provides insight into production efficiency and cost control, illustrating how incremental changes affect overall cost metrics.

Total Cost Function

The total cost function represents the accumulated cost of producing a given number of items. In problems where the average cost per unit is known, the total cost can be calculated by multiplying the average cost function by the number of items produced. This concept is fundamental in economics and production analysis, linking the per-unit cost to overall expenditure.

Marginal Cost

Marginal cost is defined as the derivative of the total cost function with respect to the number of items produced, indicating the cost incurred for producing one additional unit. It is a crucial concept in microeconomics and production management, helping in decision-making about the optimal quantity to produce and understanding cost behavior as production scales.

Transcript

00:01 Okay, so we have here an equation of an average cost, which is a of q is equal to 0 .1 q squared minus 0 .6 q plus 13, wherein q is greater than 0.

00:13 So we are required to get a, the total cost, letter b, the minimum marginal cost, letter c, q or quantity at minimum cost, and letter d, marginal cost at q.

00:27 So this should be, sorry, i'm just going to rewrite this.

00:31 So this should be marginal cost at q is equal to 30.

00:36 Okay.

00:38 So solving for a, so we have here solution.

00:48 So solving for a, we know that the total cost is just equal to the average cost times the total number of quantity q.

01:00 So doing that, we have 0 .1.

01:03 Q squared minus 0 .6q plus 13 times q.

01:15 So solving, we now have 0 .1 qq minus 0 .6 q squared.

01:25 Oops, sorry, should be q squared plus 13 q.

01:34 So this is our total cost.

01:36 This is the equation for our total cost.

01:39 Now we want to find the marginal cost.

01:42 So to do that, we're just going to get the derivative of q, which is equal to 0 .03 q squared minus 1 .2 q plus 13.

01:58 Now to get the minimum marginal cost, we then get the derivative of this marginal cost, and we equate it to 0.

02:07 So we have d of mc over dq.

02:15 Over dq is just equal to 0 is equal to 0 .06 q minus 1 .2.

02:23 So solving for q, we have 0 .06 q is equal to 1 .2, and q is equal to 20.

02:34 So now to find the total, to find the value of the minimum marginal cost, we're just going to substitute this to our marginal cost equation, which is 0 .3 times, oh sorry, which should be 0 .03 times 20 squared minus 1 .2 times 20 plus 13...

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