Which one of the following set of constraints does the given shaded region represent?
Show 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.
$x + y \le 30, x + y \ge 15, x \le 15, y \le 20, x, y \ge 0$
$x + y \le 30, x + y \ge 15, y \le 15, x \le 20, x, y \ge 0$
$x + y \ge 30, x + y \le 15, x \le 15, y \le 20, x, y \ge 0$
$x + y \ge 30, x + y \le 15, y \le 15, x \le 20, x, y \ge 0$
Show Solution
The Correct Option isA
Solution and Explanation
Step 1: Concept Identification:
Determine the system of linear inequalities defining the shaded feasible region. For each boundary line, find its equation and then establish the inequality direction ($\le$ or $\ge$) by testing a point within the shaded region, such as the origin (0,0), provided it is not on the line.
Step 3: Explanation:
Identify the boundary lines from the graph.
Line through (15, 0) and (0, 15):
Equation: $\frac{x}{15} + \frac{y}{15} = 1$, simplified to $x + y = 15$. The region is above this line. Test point (15, 20): $15 + 20 = 35 \ge 15$. Inequality: $x + y \ge 15$.
Line through (30, 0) and (0, 30):
Equation: $\frac{x}{30} + \frac{y}{30} = 1$, simplified to $x + y = 30$. The region is below this line. Test point (0,0): $0+0=0 \le 30$. Inequality: $x + y \le 30$.
Vertical line at x = 15:
Equation: $x = 15$. The region is to the left. Inequality: $x \le 15$.
Horizontal line at y = 20:
Equation: $y = 20$. The region is below. Inequality: $y \le 20$.
Non-negativity constraints:
The region is in the first quadrant. Inequalities: $x \ge 0$ and $y \ge 0$.
Combining these inequalities yields the constraint set:
\[ x + y \le 30, \quad x + y \ge 15, \quad x \le 15, \quad y \le 20, \quad x \ge 0, \quad y \ge 0 \]
This set corresponds to option (1).
Step 4: Conclusion:
The correct set of constraints is $x + y \le 30, x + y \ge 15, x \le 15, y \le 20, x, y \ge 0$.
