Handout (Optimization)
(Schwartz; Math 157)
Work the problems on your own paper just like a normal homework assignment.
1. The cost ($) of producing x tons of Pleezm is given by () = 2 + 30 + 9,000 , and the
relationship between price and demand be given by = 500 − 1.25. Answer each of the
following assuming that everything that is produced is sold. Round answers to the nearest
hundredth.
(a) Find the production level that will minimize average cost.
(b) Find the production level that will maximize revenue.
(c) Find the production level that will maximize profit.
(d) Find the price that will maximize profit.
(e) Find the rate at which the profit is changing with respect to price.
(f) Find the production level at which the average cost is equal to the marginal cost. Is your
answer the same as any of the previous answers? Should it be? Explain your findings.
(g) Determine the optimal production level for the Pleezm producer.
2. For quality control, a manufacturer must inspect the product periodically in order to find
defective units. Making these inspections is costly, but so is having too many defective items
make it onto the shelves in the stores. Suppose that the total cost resulting from the
70
production of defective items is given by = + 3 , where x is the number of inspections,
c is a constant that represents the cost of doing one inspection, and k is a constant associated
with the cost of having too many defective units in retail circulation. Find the optimal
number of inspections for this manufacturer.

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