Involve the average cost of manufacturing a quantity q of a...

Involve the average cost of manufacturing a quantity $q$ of a good, which is defined to be. 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.

Involve the average cost of manufacturing a quantity $q$ of a good, which is defined to be.
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

Average Cost Function

This concept refers to the cost per unit of production calculated by dividing the total cost by the quantity produced. It is a critical measure in assessing production efficiency and is used to understand how costs behave as output changes.

Total Cost Function

The total cost function represents the entire expense incurred in the production process. It is derived by multiplying the average cost by the quantity produced or equivalently by integrating the marginal cost, thus giving the complete picture of expenses at any production level.

Marginal Cost

Marginal cost is the additional cost required to produce one more unit of a good. It is determined by taking the derivative of the total cost function with respect to quantity and is pivotal in decision making about scaling production.

Optimization Using Calculus

Calculus is used in production economics to find optimum production levels by identifying points where functions, such as average or marginal cost, reach their minimum or maximum. Techniques like setting the derivative to zero (critical points) and applying the second derivative test are key to this process.

Relationship between Marginal and Average Costs

A crucial economic principle is that at the production level where the average cost is minimized, the marginal cost is equal to the average cost. This relationship provides a deeper understanding of production efficiency and cost dynamics, guiding decisions on optimal output levels.

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