At the Factory in Example 1, the cost of producing x can openers is given by y = 2.75x + 26,000.
(a) Write an equation that gives the average cost per can opener when x can openers are produced.
(b) How many can openers should be made to have an average cost of $3 per can opener?
EXAMPLE 14
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 isy = 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.
Figure 1
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