Question

Of the following, which group of constraints represents the feasible region given below?
 

• A. \( x + 2y \leq 76, \, 2x + y \geq 104, \, x, y \geq 0 \)
• B. \( x + 2y \leq 76, \, 2x + y \leq 104, \, x, y \geq 0 \)
• C. \( x + 2y \geq 76, \, 2x + y \leq 104, \, x, y \geq 0 \)
• D. \( x + 2y \geq 76, \, 2x + y \geq 104, \, x, y \geq 0 \)

Hint:

For linear programming constraints, analyze the inequalities by checking the direction of the shaded region relative to the lines.

Answer 1

The Correct Option is C

The feasible region, illustrated in the graph, defines the following constraints: 1. For Line 1, represented by \( x + 2y = 76 \), the shaded area above it signifies the constraint: \[ x + 2y \geq 76. \] 2. For Line 2, represented by \( 2x + y = 104 \), the shaded area below it signifies the constraint: \[ 2x + y \leq 104. \] 3. The shaded region's placement in the first quadrant imposes non-negativity constraints: \[ x \geq 0 \quad {and} \quad y \geq 0. \] The complete set of constraints for the feasible region is: \[ x + 2y \geq 76, \, 2x + y \leq 104, \, x \geq 0, \, y \geq 0. \]
Final Answer: \( \boxed{{(C)}} \)