Question

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

• A. $x + y \le 30, x + y \ge 15, x \le 15, y \le 20, x, y \ge 0$
• B. $x + y \le 30, x + y \ge 15, y \le 15, x \le 20, x, y \ge 0$
• C. $x + y \ge 30, x + y \le 15, x \le 15, y \le 20, x, y \ge 0$
• D. $x + y \ge 30, x + y \le 15, y \le 15, x \le 20, x, y \ge 0$

Hint:

To quickly find the equation of a line from its intercepts, use the intercept form: $\frac{x}{a} + \frac{y}{b} = 1$, where 'a' is the x-intercept and 'b' is the y-intercept. To determine the inequality, pick a test point (the origin is easiest) and see if it satisfies the condition for the shaded region.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to determine the system of linear inequalities that defines the given shaded feasible region. We do this by finding the equation of each boundary line and then determining the direction of the inequality (e.g., $\le$ or $\ge$) by testing a point within the shaded region, like the origin (0,0) if it's not on the line.
Step 3: Detailed Explanation:
First, let's identify the boundary lines from the graph.

Line passing through (15, 0) and (0, 15): The equation is $\frac{x}{15} + \frac{y}{15} = 1$, which simplifies to $x + y = 15$. The shaded region is above this line (away from the origin). Testing the point (15, 20) which is in the region: $15 + 20 = 35 \ge 15$. So the inequality is $x + y \ge 15$.
Line passing through (30, 0) and (0, 30): The equation is $\frac{x}{30} + \frac{y}{30} = 1$, which simplifies to $x + y = 30$. The shaded region is below this line (towards the origin). Testing the origin (0,0): $0+0=0 \le 30$. So the inequality is $x + y \le 30$.
Vertical line passing through x = 15: The equation is $x = 15$. The shaded region is to the left of this line. So the inequality is $x \le 15$.
Horizontal line passing through y = 20: The equation is $y = 20$. The shaded region is below this line. So the inequality is $y \le 20$.
Non-negativity constraints: The shaded region is in the first quadrant, which means $x \ge 0$ and $y \ge 0$.
Combining all these inequalities, we get the set of constraints: \[ x + y \le 30, \quad x + y \ge 15, \quad x \le 15, \quad y \le 20, \quad x \ge 0, \quad y \ge 0 \] This set matches the constraints given in option (1).
Step 4: Final Answer:
The correct set of constraints is $x + y \le 30, x + y \ge 15, x \le 15, y \le 20, x, y \ge 0$.