A linear programming problem has two constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100, plus nonnegativity constraints on X and Y. Which of the following statements about its feasible region is true? A) There are four corner points including (50, 0) and (0, 12.5). B) The two corner points are (0, 0) and (50, 12.5). C) The graphical origin (0, 0) is not in the feasible region. D) The feasible region includes all points that satisfy one constraint, the other, or both. E) The feasible region cannot be determined without knowing whether the problem is to be minimized or maximized.

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Quizplus · Accepted Answer

The feasible region of a linear programming problem is defined by the intersection of all constraints, including nonnegativity constraints. The given constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100, along with nonnegativity constraints on X and Y, define a polygonal feasible region on the XY-plane. The origin (0,0) always satisfies the given constraints and nonnegativity, making it a part of the feasible region. The points (50, 0) and (0, 12.5) are found by setting Y and X to zero in the respective constraints, showing they are corner points of the feasible region. The feasible region is determined by the intersection of the constraints, not by the objective function (whether to minimize or maximize), making E incorrect. The statement that the feasible region includes all points that satisfy one constraint, the other, or both (D) is misleading because points must satisfy all constraints simultaneously to be in the feasible region. Therefore, the correct statement is that there are four corner points including (50, 0) and (0, 12.5), which is a characteristic of the feasible region defined by the given constraints and nonnegativity conditions.

A Linear Programming Problem Has Two Constraints 2X + 4Y