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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 ______.
Options
(30, 25)
(20, 35)
(35, 20)
(40, 15)
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Solution
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.
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 |
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