Question

\(\text{The feasible region represented by the constraints } 4x + y \geq 80, \; x + 5y \geq 115, \; 3x + 2y \leq 150, \; x, y \geq 0 \; \text{of an LPP is:}\)

• A. Region A
• B. Region B
• C. Region C
• D. Region D

Hint:

When solving linear programming problems, plotting the constraints is a crucial step. Each constraint will represent a line, and the feasible region is the intersection of all regions that satisfy the inequalities. Remember, the feasible region must satisfy all constraints, and regions outside of it do not meet the required conditions. Always check the boundaries and verify that all constraints are satisfied to determine the correct feasible region.

Answer 1

The Correct Option is C

To establish the feasible region defined by the constraints, the subsequent steps are executed:

Constraint Plotting:

The specified constraints are:

\[4x + y \geq 80\]

\[x + 5y \geq 115\]

\[3x + 2y \leq 150\]

\[x, y \geq 0 \text{ (implying the feasible region is confined to the first quadrant)}\]

Each constraint delineates a line on the \( xy \)-plane.

Feasible Region Identification:

The area that adheres to all constraints is the shaded zone formed by the intersection of these lines.

As depicted in the accompanying plot, Region C is delineated by these lines and signifies the viable solution set for the linear programming problem (LPP).

Validation:

Region C satisfies all constraints, notably the inequality \( 3x + 2y \leq 150 \), which defines its upper boundary.

Alternative regions fail to meet all constraints concurrently.

Consequently, Region C represents the feasible region for the stated LPP.