Question

The point at which the maximum value of x + y subject to the constraints x + 2y ≤ 70, 2x + y ≤ 95, x ≥ 0, y ≥ 0 is obtained, is ______.

पर्याय

•  (30, 25)

•  (20, 35)

•  (35, 20)

•  (40, 15)

Answer 1

 The point at which the maximum value of x + y subject to the constraints x + 2y ≤ 70, 2x + y ≤ 95, x ≥ 0, y ≥ 0 is obtained, is (40, 15).

Explanation:

We need to maximize the function Z = x + y

Converting the given inequations into equations, we obtain x + 2y = 70,  2x + y = 95,  x = 0z and y = 0

Region represented by x + 2y ≤ 70: The line x + 2y = 70 meets the coordinate axes at A(70, 0) and B(0, 35) respectively. By joining these points we obtain the line x + 2y = 70. Clearly (0, 0) satisfies the inequation x + 2y ≤ 70. So, the region containing the origin represents the solution set of the inequation x + 2y ≤ 70.

Region represented by 2x + y ≤ 95: The line 2x + y = 95 meets the coordinate axes at \[C\left( \frac{95}{2}, 0 \right)\]  respectively. By joining these points we obtain the line 2x + y = 95.

Clearly (0, 0) satisfies the inequation 2x + y ≤ 95. So, the region containing the origin represents the solution set of the inequation 2x + y ≤ 95.

Region represented by x ≥ 0 and y ≥ 0: Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations x ≥ 0, and y ≥ 0.

The feasible region determined by the system of constraints x + 2y ≤ 70, 2x + y ≤ 95, x ≥ 0, and y ≥ 0, are as follows.

The corner points of the feasible region are O(0, 0), \[C\left( \frac{95}{2}, 0 \right)\], E(40, 15) and B(0, 35). 

The values of Z at these corner points are as follows.

Corner point	Z = x + y
O(0, 0)	0 + 0 = 0
\[C\left( \frac{95}{2}, 0 \right)\]	\[\frac{95}{2}\] + 0 =  \[\frac{95}{2}\]
\[E\left( 40, 15 \right)\]	40 +15 = 55
B(0, 35)	0 + 35 = 35

We see that the maximum value of the objective function Z is 55 which is at  \[\left( 40, 15 \right)\].