Table 4.2 shows cost, C(q), and revenue, R(q) (a) At...

Table 4.2 shows cost, $C(q),$ and revenue, $R(q)$ (a) At approximately what production level, $q$, is profit maximized? Explain your reasoning. (b) What is the price of the product? (c) What are the fixed costs? $$\begin{array}{l|r|r|r|r|r|r|r}\hline q & 0 & 500 & 1000 & 1500 & 2000 & 2500 & 3000 \\\hline R(q) & 0 & 1500 & 3000 & 4500 & 6000 & 7500 & 9000 \\\hline C(q) & 3000 & 3800 & 4200 & 4500 & 4800 & 5500 & 7400 \\\hline\end{array}$$

Table 4.2 shows cost, $C(q),$ and revenue, $R(q)$
(a) At approximately what production level, $q$, is profit maximized? Explain your reasoning.
(b) What is the price of the product?
(c) What are the fixed costs?
$$\begin{array}{l|r|r|r|r|r|r|r}\hline q & 0 & 500 & 1000 & 1500 & 2000 & 2500 & 3000 \\\hline R(q) & 0 & 1500 & 3000 & 4500 & 6000 & 7500 & 9000 \\\hline C(q) & 3000 & 3800 & 4200 & 4500 & 4800 & 5500 & 7400 \\\hline\end{array}$$

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Key Concepts

Cost Function and Fixed Costs

The cost function indicates the total cost of producing q units, which generally includes fixed costs and variable costs. Fixed costs are the expenses that remain constant regardless of the production level, commonly identified as the cost incurred when no units are produced. Understanding fixed costs is essential for analyzing overall profitability and break-even points.

Revenue Function and Unit Price

In this context, the revenue function represents the total income generated from selling q units of the product. When the relationship is linear, the unit price can be determined by dividing the total revenue by the number of units sold. A constant slope in the revenue function indicates that each unit is sold at the same price.

Profit Maximization

This concept involves determining the production level at which the difference between total revenue and total cost is the greatest. It requires analyzing profit as a function of quantity (profit = revenue minus cost) and identifying the quantity that yields the highest profit value.

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