Question

Which one of the options given represents the feasible region of the linear programming model: 
Maximize \(45X_1 + 60X_2\) 
\(X_1 \le 45\) 
\(X_2 \le 50\) 
\(10X_1 + 10X_2 \ge 600\) 
\(25X_1 + 5X_2 \le 750\) 

 

• A. Region P
• B. Region Q
• C. Region R
• D. Region S

Hint:

To quickly determine which side of a constraint line is feasible, pick a test point like the origin (0, 0). For \(10X_1 + 10X_2 \ge 600\), plugging in (0,0) gives \(0 \ge 600\), which is false. So the feasible region is on the side of the line that does NOT contain the origin (i.e., above it). For \(25X_1 + 5X_2 \le 750\), plugging in (0,0) gives \(0 \le 750\), which is true. So the feasible region is on the side that contains the origin (i.e., below it).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The feasible region of a linear programming problem is the set of all points \((X_1, X_2)\) that satisfy all the given constraints simultaneously. We need to identify which of the labeled regions on the graph corresponds to this set of points.
Step 2: Detailed Explanation:
Let's analyze each constraint and the corresponding region on the graph. We also have the implicit constraints \(X_1 \ge 0\) and \(X_2 \ge 0\), as the graph is in the first quadrant.
1. \(X_1 \le 45\): This means the feasible region must be to the left of the vertical line \(X_1 = 45\).
2. \(X_2 \le 50\): This means the feasible region must be below the horizontal line \(X_2 = 50\).
3. \(10X_1 + 10X_2 \ge 600\): This simplifies to \(X_1 + X_2 \ge 60\). The line \(X_1 + X_2 = 60\) has intercepts at (60, 0) and (0, 60). Since the inequality is \(\ge\), the feasible region must be on or above this line. Regions P and R are below this line, so they are not feasible.
4. \(25X_1 + 5X_2 \le 750\): This simplifies to \(5X_1 + X_2 \le 150\). The line \(5X_1 + X_2 = 150\) has intercepts at (30, 0) and (0, 150). Since the inequality is \(\le\), the feasible region must be on or below this line. Region Q is above this line, so it is not feasible.
Step 3: Identifying the Feasible Region:
The feasible region must satisfy all four conditions:
- To the left of \(X_1 = 45\)
- Below \(X_2 = 50\)
- Above or on the line \(X_1 + X_2 = 60\)
- Below or on the line \(5X_1 + X_2 = 150\)
Only Region S (the yellow shaded area) satisfies all these conditions simultaneously. It is bounded by the four constraint lines.
Step 4: Final Answer:
The feasible region is represented by Region S.
Step 5: Why This is Correct:
Region S is the unique intersection of all the half-planes defined by the constraints. Region P violates constraint 3. Region Q violates constraint 4. Region R violates multiple constraints, including constraint 3. Therefore, S is the only feasible region.