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:

To determine constraints from a graph, carefully analyze the shaded region relative to the boundary lines.

Answer 1

The Correct Option is C

Step 1: Analyze the boundary lines
The constraints for the shaded region are based on the lines: 
\[ x + 2y = 76 \quad {and} \quad 2x + y = 104. \] 
From the diagram: 
- The region is above the line \( x + 2y = 76 \), so \( x + 2y \geq 76 \). 
- The region is below the line \( 2x + y = 104 \), so \( 2x + y \leq 104 \). 
- The region is in the first quadrant, so \( x \geq 0 \) and \( y \geq 0 \). 
Step 2: Verify each option
Option (C) correctly represents the constraints as: 
\[ x + 2y \geq 76, \quad 2x + y \leq 104, \quad x, y \geq 0. \] 
Step 3: Conclude the result
The group of constraints representing the feasible region is: 
\[ x + 2y \geq 76, \quad 2x + y \leq 104, \quad x, y \geq 0. \]