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Question
Choose the correct alternative :
The point at which the maximum value of z = x + y subject to the constraints x + 2y ≤ 70, 2x + y ≤ 95, x ≥ 0, y ≥ 0 is
Options
(36, 25)
(20, 35)
(35, 20)
(40, 15)
Solution
Z = x + y
The given inequalities are x + 2y ≤ 70, 2x + y ≤ 95.
Consider lines L1 and L2 where L1 : x + 2y = 70 and L2 : 2x + y = 95.
For line L1, plot A (0, 35) and B (70, 0)
For line L2, plot P (0, 95) and Q (47.5, 0).
Solving both lines we get x = 40, y = 15
The coordinates of origin O (0, 0) satisfies both the inequalities.
∴ The required region is on the origin side of both the lines L1 and L2.
As x ≥ 0, y ≥ 0, the feasible region is in the first quadrant.
OQRAO is the required feasible region.
At O (0, 0), z = 0
At Q (47.5, 0), Z = 47.5 + 0 = 47.5
At R (40, 15), Z = 40 + 15 = 55
At A (0, 35), Z = 0 + 35 = 35.
The maximum value of Z is 55 and it occurs at R (40, 15).
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