question archive A furniture manufacturer produces two types of display cabinets, type X and type Y
Subject:MathPrice:4.87 Bought7
A furniture manufacturer produces two types of display cabinets, type X and type Y. On a
weekly basis he must produce at least 2 of each type, but not more than 5 of type X or more
than 6 of type Y. It takes 4 hours to produce type X and 5 hours for type Y in a 40 hour
working week. At least 12 workers are needed with 2 working on type X and 3 on type Y at
any one time.
2.1 Represent the above information as a system of inequalities .
2.2 Draw a graph of the system and indicate the feasible region clearly.
2.3 If the profit (P) on type X is R800 and on type Y is R1000, write down the objective
function in the form P = ax + by .
2.4 Determine the number of each type that must be produced each week to make a
maximum profit. Determine the maximum profit
Answer: Let
?x= the number of units of cabinet type X?
?y= the number of untis of cabinet type Y?
Since our goal is to maximize profit, the object function is given by ( See the table above for a summary of how the constraint inequalities were obtained.
2.3
?Max P=800x+1000y?
2.1
subject to
?4x+5y≤40?
?2x+3y≥12?
?2≤x≤5?
?2≤y≤6?
?x , y≥0?
Sketch the graph of each of the linear equations obtained converting the constraint inequalities to equality
?1. 4x+5y=40 ?y=540−4x?=8−0.8x?
?2. 2x+3y=12 ?y=312−2x?=4−32?x?
2.4
The graph of the inequalities shows the the maximum profit will occur at point D(5,4)and the maximum profit is given by
?P(5,4)=800⋅5+1000⋅4=R 8000?
Note the point F(2.5,6) does not give maximum profit since a half cabinet can not be sold. The profit this point is R 8000 (P(2,6) round down 2.5 to 2).
2.2
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