Question 1105392: Maximize z=8x+12y subject to 40x+80y≤560, 6x+8y≤72, xs≥0, and y≥0.
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The constraint boundary lines are
 slope -1/2; y-intercept (0,7), x-intercept (14,0)
 slope -3/4; y-intercept (0,9), x-intercept (12,0)
The intersection of the constraint boundary lines is (8,3)
Since both constraints have the area below the boundary lines shaded, the feasibility region has vertices (0,0), (0,7), (8,3), and (12,0).
The objective function evaluated at each vertex is...
(0,0): 8(0)+12(0) = 0
(0,7): 8(0)+12(7) = 84
(8,3): 8(8)+12(3) = 100
(12,0): 8(12)+12(0) = 96
The maximum value of the objective function is 100, at (x,y) = (8,3).
NOTE:
With the standard method taught for finding the maximum value of the objective function, you need to evaluate the objective function at every vertex, except maybe (0,0).
However, you don't really need to.
The slope of the objective function is -2/3. Because -2/3 is between the slopes of the two constraint lines (-1/2 and -3/4), the maximum value of the objective function is going to be at the intersection of the two constraint boundary lines.
To understand why this is so, just look at the feasibility region, considering the slopes of the constraint lines and the objective function, and see at which vertex of the feasibility region the constraint line will just touch the feasibility region instead of passing through it.
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