A truck can carry at most 2000 kg and at most 24 cubic metres of goods. A crate of type A weighs 200 kg, takes 2 cubic metres and earns Rs. 300 for carrying it. A crate of type B weighs 100 kg, takes 3 cubic metres and earns Rs. 400. The maximum amount the truck can earn in one trip is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Rs. 3500
Let the truck carry x crates of type A and y crates of type B. Then
Weight: \(200x + 100y \le 2000\), that is, \(2x + y \le 20\)
Volume: \(2x + 3y \le 24\)
Maximise \(Z = 300x + 400y\), with \(x \ge 0\), \(y \ge 0\)
The two boundary lines meet where
\((2x + 3y) - (2x + y) = 24 - 20\)
\(y = 2, \; x = 9\)
The corner points are (0, 0), (10, 0), (9, 2) and (0, 8). Values of Z:
\(Z(0, 0) = 0\)
\(Z(10, 0) = 3000\)
\(Z(9, 2) = 2700 + 800 = 3500\)
\(Z(0, 8) = 3200\)
So the maximum earning is Rs. 3500, with 9 crates of type A and 2 of type B.
Why the other options are incorrect
Option (1) Rs. 3200 — Incorrect. This is Z at (0, 8), with only type B crates.
Option (3) Rs. 3000 — Incorrect. This is Z at (10, 0), with only type A crates.
Option (4) Rs. 3800 — Incorrect. Z is at most 3500 at every corner point, so Rs. 3800 cannot be earned.