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 to the given linear programming problem (LPP) involves determining the feasible region defined by the intersection of the constraints. We have the following constraints for the LPP: 

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

To find the feasible region, we need to plot these constraints on a coordinate plane and look for the intersection that satisfies all the conditions:

1. For the constraint \(x + y \geq 10\), rewrite as \(y \geq -x + 10\). This represents the region above the line \(y = -x + 10\).
2. The inequality \(2x + 2y \leq 25\) simplifies to \(x + y \leq 12.5\). This represents the region below the line \(y = -x + 12.5\).
3. Constraints \(x \geq 0\) and \(y \geq 0\) restrict the region to the first quadrant.

Combining all these, the feasible region is where these shaded areas overlap in the first quadrant. This region is bounded by the intersection of the lines \(y = -x + 10\), \(y = -x + 12.5\), the y-axis (\(x = 0\)), and the x-axis (\(y = 0\)).
The attached image  accurately represents this feasible region, which shows the feasible area created by these constraints.

Answer 2

To solve this problem, we need to first plot the given constraints and identify the feasible region:

Constraint 1: \( x + y \geq 10 \)

This represents a straight line with a slope of -1, and the feasible region is above the line. To understand the boundary, when \( x = 0 \), we have \( y = 10 \); and when \( y = 0 \), we have \( x = 10 \). The region above the line (including the line itself) is where the inequality holds true.

Constraint 2: \( 2x + 2y \leq 25 \)

Simplifying the inequality gives \( x + y \leq 12.5 \). This is also a straight line with a slope of -1. To understand the boundary, when \( x = 0 \), we get \( y = 12.5 \); and when \( y = 0 \), we get \( x = 12.5 \). The region below the line (including the line itself) is where the inequality holds true.

Constraint 3: \( x \geq 0 \)

This constraint restricts the feasible region to the right of the y-axis, ensuring that \( x \) cannot be negative.

Constraint 4: \( y \geq 0 \)

This constraint restricts the feasible region to above the x-axis, ensuring that \( y \) cannot be negative.

Thus, the feasible region is the intersection of the regions defined by these four constraints:

The feasible region is formed by the area where the conditions for all the inequalities overlap. In other words, the area that satisfies all four constraints simultaneously. To visualize this, you would plot each of the lines corresponding to the constraints and shade the appropriate region on the graph based on whether the inequality is less than or greater than the line.

The graph that correctly represents this feasible region is shown in Option (3).