Maximum Profit A commodity has a demand function modeled by...

Maximum Profit A commodity has a demand function modeled by $p=100-0.5 x,$ and a total cost function modeled by $C=40 x+37.5$
(a) What price yields a maximum profit?
(b) When the profit is maximized, what is the average cost per unit?

Key Concepts

Average Cost

Average cost is computed by dividing the total cost by the number of units produced. It provides insight into the cost efficiency per unit and is important in understanding the cost behavior as production scale changes.

Profit Maximization

Profit maximization involves finding the production level or price at which the profit function is at its highest. This is often achieved by taking the derivative of the profit function with respect to quantity and setting it equal to zero to find critical points.

Profit Function

A profit function is derived by subtracting the total cost function from the revenue function. It quantifies the earnings from production and sales, and maximizing this function is key in determining optimal production levels.

Revenue Function

A revenue function calculates the total income generated from selling a product, typically found by multiplying the selling price (which may depend on demand) by the quantity sold. This function is a fundamental component in profit analysis.

Total Cost Function

A total cost function represents the overall cost incurred in producing a certain quantity of goods, including fixed costs and variable costs. It is essential for evaluating profitability and making decisions on production levels.

Demand Function

A demand function expresses the relationship between the price of a good and the quantity demanded by consumers. It typically reflects how changes in price will impact purchasing behavior and is crucial for linking market price with the quantity sold.

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