Chapter 5.5, Problem 15E

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

Profit The total cost $C$ 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 measure all monetary values in dollars.) To determine a formula for the total cost, we need to know the manufacturer’s fixed costs (covering such things as plant maintenance and insurance) as well as the cost for each unit produced, which is called the variable cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. The total revenue $R$ for a manufacturer during a given time period is also a function of the number $N$ of items produced during that period. To determine a formula for the total monthly revenue, we need to know the selling price $p$ per unit of the item, which in general is also a function of $N$. To find the total revenue, we multiply this selling price by the number of items produced. The profit $P$ for a manufacturer is the total revenue minus the total cost.

Suppose that a manufacturer of widgets has a fixed costs of $2000 per month and that the variable cost is $30 per widget. Further, the manufacturer has developed the following table showing the highest price $p$, in dollars, of a widget at which $N$ widgets can be sold.

Number $N$	Price $p$
$200$	$41.00$
$250$	$40.50$
$300$	$40.00$
$350$	$39.50$

a. Use a formula to express the total monthly cost $C$ of this manufacturer as a function of $N$.

b. Use the table to find a linear model of $p$ as a function of $N$.

c. Use your answer to part b to find a formula expressing the total monthly revenue $R$ as function of $N$.

d. Use your answers to part a and part c to find a formula expressing the monthly profit $P$ as a function of $N$. What type of function is the profit: linear or quadratic?

e. Find the two monthly production levels at which the manufacturer just breaks even (that is, where the profit is zero).