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Question 102524: This one is set up like a word problem and I could use some assistance.
A company produces both large and small cabinets. A small cabinet requires 1 hour of labor and a large cabinet requires 4 hours of labor. The company has at most 80 hours of labor available each day. No more then 60 small cabinets and no more than 15 large cabinets can be produced in a day due to space limitations. If the company's profit is $120 per small cabinet and $250 per large cabinet, how many of each should be produced to maximize profit? What is the maximum profit?
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! A company produces both large and small cabinets. A small cabinet requires 1 hour of labor and a large cabinet requires 4 hours of labor. The company has at most 80 hours of labor available each day. No more then 60 small cabinets and no more than 15 large cabinets can be produced in a day due to space limitations. If the company's profit is $120 per small cabinet and $250 per large cabinet, how many of each should be produced to maximize profit? What is the maximum profit?
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Let x = no. of small cabinets
Let y = no. of large cabinets
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Write an equation for each constraint:
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The labor hrs equation
x + 4y =< 80
or
4y =< 80 - x
y =< 80/4 - x/4
y =< 20 - .25x; use this for graphing
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The Production space equation:
x =< 60; no more that 60 small cabinets
y =< 15: no more that 15 large cabinets
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Graph all these equations:
Here is the graph, you have to complete the graph by drawing a vertical line
thru x = 60
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The area of feasibility will be:
1.At or to the left of the vertical line that you draw in
2.At or below the horizontal line (purple) OR
3.At of below the green line, which ever is lowest.
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You can calculate the corner coordinate values which bound the feasibility area
Use this to find the profit
Use x = 60 and y = 20-.25x to find that corner:
Substitute x = 60 in y = 20=.25x
y = 20 - .25(60)
y = 20 - 15
y = 5
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Objective function for x=60, y=5
60(120) + 5(250) =
7200 + 1250 = $8450 profit, when you sell 60 small cab and 5 lg cab
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Use y = 15 to find x for the other corner
15 = 20 - .25x
.25x = 20 - 15
x = 5/.25
x = 20
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Objective function for x = 20, y = 15
20(120) + 15(250) =
2400 + 3750 = $6150 profit, when you sell 20 small cab and 15 lg cab
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The 3rd corner is x = 0, y = 15, obviously would not be max profit
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Did this make sense to you?
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