The minimum value of \(Z = 2x + 5y\) subject to the constraints \(x + 2y \ge 4\), \(x \le 4\), \(y \le 3\), \(x \ge 0\), \(y \ge 0\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 8
The region lies inside the rectangle \(0 \le x \le 4\), \(0 \le y \le 3\) and on the far side of the line x + 2y = 4 from the origin. It is bounded.
The line x + 2y = 4 meets the x-axis at (4, 0) and the y-axis at (0, 2). The corner points are (4, 0), (4, 3), (0, 3) and (0, 2).
\(Z(4, 0) = 8\)
\(Z(4, 3) = 8 + 15 = 23\)
\(Z(0, 3) = 15\)
\(Z(0, 2) = 10\)
The minimum is 8, at (4, 0).
Why the other options are incorrect
Option (1) 10 — Incorrect. This is Z at the corner (0, 2), which is larger than 8.
Option (3) 0 — Incorrect. This is Z at (0, 0), which is not feasible: there x + 2y = 0 < 4.
Option (4) 15 — Incorrect. This is Z at the corner (0, 3), which is not the smallest corner value.