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 $38,000 apiece,how many should be supplied to each market to generate the largest possible profit?
A)x = 41,y = 643
B)x = 40,y = 500
C)x = 100,y = 8
D)x = 80,y = 100

Question

Quizplus · Accepted Answer

To maximize profit, we need to find the optimal values of x and y that maximize the total revenue minus the total cost. The revenue from the domestic market is $x(60 - \frac{x}{5} + \frac{y}{20})$ thousand dollars, and from the foreign market is $y(50 - \frac{y}{10} + \frac{x}{20})$ thousand dollars. The cost to produce x + y machines is $38(x + y)$ thousand dollars. Setting up the profit function and taking partial derivatives with respect to x and y, then setting them to zero to find the critical points, would lead to the conclusion that x = 80 and y = 100 maximizes the profit. This is a calculus-based optimization problem that requires knowledge of partial derivatives and critical point analysis to solve analytically.

A Manufacturer with Exclusive Rights to a Sophisticated New Industrial 60−x5+y2060 - \frac { x } { 5 } + \frac { y } { 20 }60−5x​+20y​

A Manufacturer with Exclusive Rights to a Sophisticated New Industrial $60 - \frac { x } { 5 } + \frac { y } { 20 }$