Question 1128132: Your art club wants to sell greeting cards using members’ drawings. Small blank cars cost $10 per box of 25. Large blank cards cost $15 per box of 20. You make a profit of $52.50 per box of small cards and $85 per box of large cars. The club can buy no more than 350 total cars and spend no more than $210.
A.) How can the art club maximize its profit?
B.) The card company has a minimum order requirement of 5 boxes of each per order. Does the art club meet this minimum requirement when it maximizes its profit?
E.) What is the maximum profit the art club can make and meet the minimum order requirement?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Let x be the number of 25-card boxes at $10 each and y be the number of 20-card boxes at $15 each.
The constraint inequality for the numbers of boxes of cards is
The constraint inequality for the cost of the boxes of cards is
Here is a graph of the constraint boundary lines:
The intersection point of the two boundary lines is (6,10).
The profit function is  .
The corners of the feasibility region, and the profit function evaluated at each corner, are
(0,0), P = 0
(0,14), P = 14*85 = 1190
(6,10), P = 6*52.50+10*85 = 1165
(14,0), P = 14*52.50 = 735
A) ANSWER: The club can maximize its profit at $1190 by buying 14 20-card boxes and 0 25-card boxes.
B) ANSWER: No, the club does not meet this requirement when maximizing profit.
Part C) introduces another constraint inequality:  . With this added constraint, the corners of the feasibility region are (5,0), (5,32/3), (6,10), and (14,0).
However, it is presumably not possible to order fractions of boxes of cards. So the new "corner" of the feasibility region is (5,10).
Clearly the profit at (5,10) is going to be less than at (6,10); so now the maximum possible profit is $1165 by ordering 6 20-card boxes and 10 25-card boxes.
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Dear tutor @ikleyn....
Please keep to your usually very good answers to students' questions, and stop posting these unwarranted, arrogant, and childish notes.
I have NEVER "criticized" you, and I never WOULD criticize you (or anyone else) for solving this problem by this method. It is the basic method shown in virtually all resources.
What I have done on occasion in the past with similar problems is to show that there is a refinement to the process (comparing the slope of the objective function to the slopes of the constraint boundary lines) which CAN be used to solve some problems like this more quickly.
In your post, you state "now he himself applies it (this method)... as a first choice method // I am very glad of it".
This is not my first choice method. I prefer to solve problems the easiest way I know; comparing slopes of the objective function and the constraint boundary lines leads, in many problems, to faster solutions, and with less work.
The fact that I did not use the refinement in my response to this question in no way means that I have concluded that your way is better; for you to state that you are "very glad of it" -- implying that I have conceded that your method is the right way to solve problems like this -- is childish.
Answer by ikleyn(52798)  (Show Source):
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