The corner points of the feasible region with the constraints \(x + y \le 30,\ x+y \ge 15,\ y \le 20,\ x \le 15\), and \(x, y \ge 0\) are
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \((15,0), (15,15), (10, 20), (0, 20), (0,15)\)
The feasible region lies in the rectangle \(0 \le x \le 15,\ 0 \le y \le 20\), above the line \(x + y = 15\) and below the line \(x + y = 30\). The rectangle’s corner (15, 20) is cut off, since \(15 + 20 = 35 \gt 30\). The corner points are the intersections of the boundary lines:
\(x = 0\) and \(x + y = 15\): (0, 15)
\(x = 0\) and \(y = 20\): (0, 20)
\(y = 20\) and \(x + y = 30\): (10, 20)
\(x = 15\) and \(x + y = 30\): (15, 15)
\(x = 15\) and \(x + y = 15\): (15, 0)
Each of these points satisfies all the constraints, so the corner points are (15, 0), (15, 15), (10, 20), (0, 20) and (0, 15).
Why the other options are incorrect
Option (1) \((0,0), (0, 15), (0,20), (15,0), (10,20)\) — Incorrect. (0, 0) fails \(x + y \ge 15\), and the corner (15, 15) is missing.
Option (2) \((0,15), (20,0), (15, 0), (10,20)\) — Incorrect. (20, 0) fails \(x \le 15\).
Option (4) \((0,0), (20,0), (0,15), (15, 15)\) — Incorrect. (0, 0) fails \(x + y \ge 15\), and (20, 0) fails \(x \le 15\).