Chapter 3.2, Problem 14E

Reminder Round all answer to two decimal places unless otherwise indicated.

Total Revenue and Profit This is a continuation of Exercise 13. The total revenue $R$ for a manufacturer during a given time period is a function of the number $N$ of items produced during that period. In this exercise, we assume that the selling price per unit of the item is a constant, so it does not depend on the number of items produced. The profit $P$ for a manufacturer is the total revenue minus the total cost. If the profit is zero, then the manufacturer is at a break-even point. We consider again the manufacturer of widgets in Exercise $13$ with fixed costs of $\$1500$ pr month and a variable cost of $\$20$ per widget. Suppose the manufacturer sells $100$ widgets for $\$2300$ total.

a. Use a formula to express the total monthly revenue $R,$ in dollars, of this manufacturer in a month as function of the number $N$ of widgets produced in a month.

b. Use a formula to express the monthly profit $P,$ in dollars, of this manufacturer as function of the number of widgets produced in a month. Explain how the slope and initial of $P$ are derived from the fixed costs, variable cost, and price per widget.

c. What is the break-even point for this manufacturer?

d. Make graphs of total monthly cost and total monthly revenue. Include monthly production levels up to $1200$ widgets. What is the significance of the point where the graphs cross?