The point at which the maximum value of x + y, subject to the constraints x + 2y ≤ 70, 2x+ y ≤ 95, x, y ≥ 0 is obtained, is
Options
(30, 25)
(20, 35)
(35, 20)
(40, 15)
Solution
(40, 15)
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.
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 \[\left( 40, 15 \right)\] .