Question

Of the following, which group of constraints represents the feasible region given below?
\includegraphics[width=\linewidth]{latex.jpeg}

• 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

To determine the correct constraints, analyze the feasible region depicted in the graph: 
1. Line 1: \( x + 2y = 76 \) The region above this line is shaded, indicating the constraint: \[ x + 2y \geq 76. \] 
2. Line 2: \( 2x + y = 104 \) The region below this line is shaded, indicating the constraint: \[ 2x + y \leq 104. \] 
3. Non-negativity constraints: Since the shaded region is in the first quadrant: \[ x \geq 0 \quad {and} \quad y \geq 0. \] Thus, the group of constraints representing the feasible region is: \[ x + 2y \geq 76, \, 2x + y \leq 104, \, x \geq 0, \, y \geq 0. \]
Final Answer: \( \boxed{ {(C)}} \)