The maximum value of \(Z = 3x + 4y\) subject to the constraints \(x + y \le 7\), \(x \le 5\), \(x \ge 0\), \(y \ge 0\) is M. When the constraint \(y \le 4\) is added to these, the maximum value of Z becomes N. Then \(M - N\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3
Without \(y \le 4\), the line \(x + y = 7\) meets \(x = 5\) at (5, 2). The corner points are (0, 0), (5, 0), (5, 2) and (0, 7):
\(Z(5, 0) = 15\)
\(Z(5, 2) = 23\)
\(Z(0, 7) = 28\)
So \(M = 28\).
The added constraint \(y \le 4\) cuts off the corner (0, 7). The line \(y = 4\) meets \(x + y = 7\) at (3, 4), so the new corner points are (0, 0), (5, 0), (5, 2), (3, 4) and (0, 4):
\(Z(5, 2) = 23\)
\(Z(3, 4) = 9 + 16 = 25\)
\(Z(0, 4) = 16\)
So \(N = 25\), and
\(M - N = 28 - 25 = 3\)
Why the other options are incorrect
Option (1) 2 — Incorrect. \(M - N = 28 - 25 = 3\), not 2.
Option (3) 5 — Incorrect. This is \(28 - 23\), using (5, 2) as the new maximum point and missing the new corner (3, 4).
Option (4) 12 — Incorrect. This is \(28 - 16\), using (0, 4), which is not the new maximum point.