Choose the correct alternative : The point at which the maximum value of z = x + y subject to the constraints x + 2y $\leq$ 70, 2x + y $\leq$ 95, x $\geq$ 0, y $\geq$ 0 is

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Z = x + y The given inequalities are x + 2y $\leq$ 70, 2x + y $\leq$ 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. $\therefore$ The required region is on the origin side of both the lines L1 and L2. As x $\geq$ 0, y $\geq$ 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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