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

The feasible region is defined by the following constraints, determined from the graph:
1. Line 1: \( x + 2y = 76 \). The shaded region above this line establishes the constraint: \[ x + 2y \geq 76. \]
2. Line 2: \( 2x + y = 104 \). The shaded region below this line establishes the constraint: \[ 2x + y \leq 104. \]
3. Non-negativity constraints: The shaded region lies in the first quadrant, implying: \[ x \geq 0 \quad {and} \quad y \geq 0. \] 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: \( \boxed{ {(C)}} \)