Which one of the following set of constraints does the given shaded region represent?

Correct Answer:
The set of constraints that represents the given shaded region is:
x + y ≤ 30 , x + y ≥ 15 , x ≤ 15 , y ≤ 20 , x , y ≥ 0

Step-by-Step Explanation:

To determine the set of linear inequalities representing the shaded feasible region, we examine each boundary line shown in the graph and test the region relative to it:

1. The vertical boundary line:
The graph displays a vertical line passing through 15 on the x-axis, which corresponds to the line:
x = 15
Since the shaded region lies entirely to the left of this vertical line, the coordinate values satisfy:
x ≤ 15

2. The horizontal boundary line:
There is a horizontal line passing through 20 on the y-axis, which corresponds to the line:
y = 20
The horizontal line and the vertical line intersect at the labeled point (15,20). Since the shaded region lies below this horizontal line, the y-coordinates satisfy:
y ≤ 20

3. The upper diagonal boundary line:
This line passes through the intercepts (30,0) and (0,30).
The equation of this line is given by:
x 30 + y 30 = 1 ⇒ x + y = 30
The shaded region lies below this line (towards the origin). Testing the origin (0,0) in the inequality:
0 + 0 ≤ 30 (which is True).
Thus, the corresponding inequality is:
x + y ≤ 30

4. The lower diagonal boundary line:
This line passes through the intercepts (15,0) and (0,15).
The equation of this line is given by:
x 15 + y 15 = 1 ⇒ x + y = 15
The shaded region lies above this line (away from the origin). Testing the origin (0,0) in the inequality:
0 + 0 ≥ 15 (which is False).
Thus, the corresponding inequality is:
x + y ≥ 15

5. Non-negativity constraints:
Since the shaded region lies entirely within the first quadrant, both variables must be non-negative:
x ≥ 0 , y ≥ 0

Combining all the derived inequalities, we get the set of constraints that represents the shaded region:
x + y ≤ 30 , x + y ≥ 15 , x ≤ 15 , y ≤ 20 , x , y ≥ 0