Problem

Solve the applied problem.In Example 1, how many can openers must be produced to have an a...

Solve the applied problem.

In Example 1, how many can openers must be produced to have an average cost per can opener of $3?

EXAMPLE 1

A factory that makes can openers has fixed costs (for building, fixtures, machinery, etc.) of $26,000. The variable cost (materials and labor) for making one can opener is $2.75.

(a) Find the cost equation that gives the total cost y of producing x can openers and sketch its graph.
(b) At what rate does the total cost increase as more can openers are made?
(c) What is the total cost of making 1000 can openers? 20,000? 40,000?
(d) In part (c), what is the average cost per can opener in each case?

SOLUTION

(a) Since each can opener costs $2.75, the variable cost of making x can openers is 2.75x. The total cost y of making x can openers is

The graph of this equation is the line in Figure 1.
(b) The cost equation shows that y increases by 2.75 each time x increases by 1. That is, total cost is increasing at the rate of $2.75 per can opener. This rate of change is the slope the cost equation line y = 2.75x + 26,000.
(c) The cost of making 1000 can openers is
y = 2.75x + 26,000 = 2.75(1000) + 26,000 = $28,750.
Similarly, the cost of making 20,000 can openers is
y = 2.75(20,000) + 26,000 = $81,000.
and the cost of 40,000 is
y = 2.75(40,000) + 26,000 = $136,000.
(d) The average cost per can opener in each case is the total cost divided by the number of can openers. So the average cost per can opener is as follows.