A unit makes two types of lamps, P and Q. A lamp of type P needs 30 minutes on the lathe and 20 minutes of polishing, and a lamp of type Q needs 15 minutes on the lathe and 40 minutes of polishing. The lathe can run for 12 hours a day and polishing can be done for 12 hours a day. The profit is Rs. 50 on a lamp of type P and Rs. 60 on a lamp of type Q, and all lamps are sold. The maximum profit in a day is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Rs. 1480
Let x lamps of type P and y lamps of type Q be made in a day. The times are in minutes, so the capacities are changed to minutes: 12 hours = 720 minutes.
Lathe: \(30x + 15y \le 720\), that is \(2x + y \le 48\)
Polishing: \(20x + 40y \le 720\), that is \(x + 2y \le 36\)
\(x \ge 0,\ y \ge 0\)
Maximise \(Z = 50x + 60y\). The two lines meet where, doubling the second and subtracting the first,
\(3y = 24\)
\(y = 8,\ x = 20\)
The corner points are (0, 0), (24, 0), (20, 8) and (0, 18). Values of Z:
\(Z(0, 0) = 0\)
\(Z(24, 0) = 1200\)
\(Z(20, 8) = 1000 + 480 = 1480\)
\(Z(0, 18) = 1080\)
The maximum profit is Rs. 1480, with 20 lamps of type P and 8 of type Q.