Question 1114280: A manufacturer of small copiers makes a profit of $200 on a deluxe model and $250 on a standard model. The company wants to produce at least 70 deluxe models per week and at least 55 standard models per week. However, the weekly production is not to exceed 160 copiers. How many copiers of each kind should be produced in order to maximize the profit?
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
Before looking at the formal mathematical solution, let's do some logical reasoning.
The only constraints are a minimum number of each model and a maximum total number of the two models. Since the profit on a standard model is higher than on a deluxe model, common sense says we should only produce the required minimum number of deluxe models.
So our answer should be that the maximum profit is when the company produces only the required minimum of 70 deluxe models and 160-70=90 standard models.
Let x be the number of deluxe models and y the number of standard models. Then the constraints are
(1) 
(2) 
(3) 
Those constraints give a closed feasibility region with corners (70,90), (70,55), and (105,55).
The objective function (in this case, profit), is 200x+250y.
The method that is usually taught for maximizing the objective function is to evaluate it at all the corners of the feasibility region. For this problem, we find
(70,90): 200(70)+250(90) = 14000+22500 = 36500
(70,55): 200(55)+250(55) = 14000+13750 = 27750
(105,55): 200(105)+250(55) = 21000+13750 = 34750
The maximum value of the objective function is at (70,90). So the company will maximize profit if they produce 70 deluxe models and 90 standard models.
Just as our logical analysis told us....
Note it is in fact not necessary to evaluate the objective function at every corner of the feasibility region. You can determine which corner will provide the maximum value of the objective function by comparing the slope of the objective function with the slopes of the constraint lines.
Considering the objective function as a linear equation  where c is a constant, the slope of the objective function is -4/5 and the slope of the constraint line is -1. The maximum value of the objective function will be obtained where a line with slope -4/5 just touches a corner of the feasibility region; that is at (70,90).
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