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Question
Minimize Z = 24x + 40y subject to constraints
6x + 8y ≥ 96, 7x + 12y ≥ 168, x ≥ 0, y ≥ 0
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Solution
To draw the feasible region, construct table as follows:
| Inequality | 6x + 8y ≥ 96 | 7x + 12y ≥ 168 |
| Corresponding equation (of line) | 6x + 8y = 96 | 7x + 12y = 168 |
| Intersection of line with X-axis | (16, 0) | (24, 0) |
| Intersection of line with Y-axis | (0, 12) | (0, 14) |
| Region | Non-origin side | Non-origin side |
Shaded portion XABY is the feasible region, whose vertices are A(24, 0) and B(0, 14).
Here, the objective function is
Z = 24x + 40y
∴ Z at A(24, 0) = 24(24) + 40(0) = 576
Z at B(0, 14) = 24(0) + 40(14) = 560
∴ Z has minimum value 560 at x = 0 and y = 14.
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