Question

Which one of the following represents the correct feasible region determined by the following constraints of an LPP?
\[ x + y \geq 10, \quad 2x + 2y \leq 25, \quad x \geq 0, \quad y \geq 0 \]

• A.
• B.
• C.
• D.

Hint:

In problems involving multiple constraints, always break down each inequality and plot the corresponding boundary line. Then, determine which side of the line satisfies the inequality. For inequalities of the form \( \geq \) or \( \leq \), the feasible region will be either above or below the line. By identifying where all the constraints overlap, you can determine the feasible region. This approach is commonly used in linear programming problems and geometric optimization tasks.

Answer 1

The Correct Option is C

The solution for the linear programming problem (LPP) requires defining the feasible region. This region is the intersection of the following constraints:

• \(x + y \geq 10\)
• \(2x + 2y \leq 25\)
• \(x \geq 0\)
• \(y \geq 0\)

To determine the feasible region, we plot these constraints:

1. \(x + y \geq 10\) implies \(y \geq -x + 10\), indicating the area above the line \(y = -x + 10\).
2. \(2x + 2y \leq 25\) simplifies to \(x + y \leq 12.5\), meaning the area below the line \(y = -x + 12.5\).
3. \(x \geq 0\) and \(y \geq 0\) confine the region to the first quadrant.

The feasible region is the overlapping area within the first quadrant that satisfies all conditions. It is bounded by the lines \(y = -x + 10\), \(y = -x + 12.5\), the y-axis (\(x = 0\)), and the x-axis (\(y = 0\)). The provided image illustrates this feasible region, showing the area defined by these constraints.