A small press prints x books and y magazines a day. A book needs 2 machine-hours of printing and a magazine needs 1, and 40 machine-hours of printing are available. A book needs 1 hour of binding and a magazine needs 2, and 44 hours of binding are available. At most 24 items can be packed in a day. The profit is Rs. 30 on a book and Rs. 35 on a magazine, and all items are sold. The maximum daily profit is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Rs. 820
The linear programming problem is
Maximise \(Z = 30x + 35y\)
subject to
Printing: \(2x + y \le 40\)
Binding: \(x + 2y \le 44\)
Packing: \(x + y \le 24\)
\(x \ge 0, \; y \ge 0\)
Meeting points of the boundary lines:
\(2x + y = 40\) and \(x + y = 24\): \(x = 16, \; y = 8\)
\(x + 2y = 44\) and \(x + y = 24\): \(y = 20, \; x = 4\)
Each point satisfies the remaining constraint: for (16, 8), binding needs \(16 + 16 = 32 \le 44\); for (4, 20), printing needs \(8 + 20 = 28 \le 40\). The axis corners are (20, 0) from printing and (0, 22) from binding.
So the corner points are (0, 0), (20, 0), (16, 8), (4, 20) and (0, 22). Values of Z:
\(Z(20, 0) = 600\)
\(Z(16, 8) = 480 + 280 = 760\)
\(Z(4, 20) = 120 + 700 = 820\)
\(Z(0, 22) = 770\)
So the maximum daily profit is Rs. 820, with 4 books and 20 magazines.
Why the other options are incorrect
Option (1) Rs. 760 — Incorrect. This is the value at (16, 8), which is less than at (4, 20).
Option (2) Rs. 770 — Incorrect. This is the value at (0, 22), which is less than at (4, 20).
Option (4) Rs. 840 — Incorrect. This needs 24 magazines, which take 48 hours of binding, more than 44.