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:
The objective is to establish the system of linear inequalities that delineate the provided shaded feasible region. This involves determining the equation for each boundary line and subsequently defining the inequality direction (e.g., $\le$ or $\ge$) by evaluating a point within the shaded area. If the origin (0,0) is not on the line, it can serve as a test point. Step 3: Detailed Analysis:
Initially, we identify the boundary lines from the graphical representation.
Line through (15, 0) and (0, 15):
The equation is $\frac{x}{15} + \frac{y}{15} = 1$, simplifying to $x + y = 15$. The shaded region lies above this line (away from the origin). Testing point (15, 20) within the region: $15 + 20 = 35 \ge 15$. Thus, the inequality is $x + y \ge 15$.
Line through (30, 0) and (0, 30):
The equation is $\frac{x}{30} + \frac{y}{30} = 1$, simplifying to $x + y = 30$. The shaded region lies below this line (towards the origin). Testing the origin (0,0): $0+0=0 \le 30$. Therefore, the inequality is $x + y \le 30$.
Vertical line at x = 15:
The equation is $x = 15$. The shaded region is to the left of this line. Hence, the inequality is $x \le 15$.
Horizontal line at y = 20:
The equation is $y = 20$. The shaded region is below this line. Consequently, the inequality is $y \le 20$.
Non-negativity constraints:
The shaded region is situated in the first quadrant, implying $x \ge 0$ and $y \ge 0$.
The consolidated system of inequalities is:\
\[ x + y \le 30, \quad x + y \ge 15, \quad x \le 15, \quad y \le 20, \quad x \ge 0, \quad y \ge 0 \]
This set of constraints corresponds to the options presented in option (1). Step 4: Conclusion:
The definitive set of constraints is $x + y \le 30, x + y \ge 15, x \le 15, y \le 20, x, y \ge 0$.
