A carpenter makes chairs and tables. A chair sells for Rs. 1000; the wood for it costs Rs. 300 and the labour Rs. 400. A table sells for Rs. 1500; the wood for it costs Rs. 600 and the labour Rs. 200. He can spend at most Rs. 9000 on wood and at most Rs. 7200 on labour, and he must make at least 4 chairs. If he sells everything he makes, his maximum profit is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Rs. 10300
The profit on one item is its selling price minus the cost of wood and labour:
Chair: \(1000 - 300 - 400 = 300\)
Table: \(1500 - 600 - 200 = 700\)
Let him make x chairs and y tables. Then
Wood: \(300x + 600y \le 9000\), that is, \(x + 2y \le 30\)
Labour: \(400x + 200y \le 7200\), that is, \(2x + y \le 36\)
Chairs: \(x \ge 4\), and \(y \ge 0\)
Maximise \(Z = 300x + 700y\)
The wood and labour lines meet where
\(2(x + 2y) - (2x + y) = 60 - 36\)
\(y = 8, \; x = 14\)
The line \(x = 4\) meets \(x + 2y = 30\) at (4, 13), and there \(2(4) + 13 = 21 \le 36\).
The corner points are (4, 0), (18, 0), (14, 8) and (4, 13). Values of Z:
\(Z(4, 0) = 1200\)
\(Z(18, 0) = 5400\)
\(Z(14, 8) = 4200 + 5600 = 9800\)
\(Z(4, 13) = 1200 + 9100 = 10300\)
So the maximum profit is Rs. 10300, with 4 chairs and 13 tables.
Why the other options are incorrect
Option (2) Rs. 9800 — Incorrect. This is Z at (14, 8), where both money limits are used up, but it is less than Z at (4, 13).
Option (3) Rs. 10500 — Incorrect. This is Z at (0, 15), with no chairs, which breaks the condition of at least 4 chairs.
Option (4) Rs. 26000 — Incorrect. This is the money received from sales at (14, 8), not the profit.