A manufacturer with exclusive rights to a sophisticated new industrial machine is planning to sell a limited number of the machines to both foreign and domestic firms. The price the manufacturer can expect to receive for the machines will depend on the number of machines made available. (For example, if only a few of the machines are placed on the market, competitive bidding among prospective purchasers will tend to drive the price up.) It is estimated that if the manufacturer supplies x machines to the domestic market and y machines to the foreign market, the machines will sell for $60 - \frac { x } { 5 } + \frac { y } { 20 }$ thousand dollars apiece at home and for $50 - \frac { y } { 10 } + \frac { x } { 20 }$ thousand dollars apiece abroad. If the manufacturer can produce the machines at a cost of $31,000 apiece, how many should be supplied to each market to generate the largest possible profit?
A) x = 150, y = 11
B) x = 38, y = 593
C) x = 110, y = 150
D) x = 55, y = 750

Question

Quizplus · Accepted Answer

The profit from selling the machines can be maximized by setting up and solving a system of equations based on the given price functions and the cost of production. The profit from each market is the price per machine times the number of machines sold minus the cost of production times the number of machines. By taking the derivative of the profit functions with respect to x and y, setting them to zero, and solving the system of equations, one can find the optimal number of machines to supply to each market to maximize profit. The correct choice, (C) x = 110, y = 150, is derived from such an analysis, indicating that supplying 110 machines to the domestic market and 150 machines to the foreign market will generate the largest possible profit.

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