A farmer has 8 hectares of land to grow wheat and barley. Each hectare of wheat needs 3 units of water and each hectare of barley needs 1 unit of water, and only 18 units of water are available. The profit is Rs. 4000 per hectare of wheat and Rs. 3000 per hectare of barley. The maximum profit the farmer can make is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Rs. 29000
Let x hectares be under wheat and y hectares under barley. Then the problem is
Maximise \(Z = 4000x + 3000y\)
subject to \(x + y \le 8\) (land), \(3x + y \le 18\) (water), \(x \ge 0\), \(y \ge 0\)
On the axes, 3x + y = 18 gives (6, 0) and x + y = 8 gives (0, 8). The two lines meet where
\(2x = 18 - 8 = 10\)
\(x = 5,\ y = 3\)
The corner points are (0, 0), (6, 0), (5, 3) and (0, 8).
\(Z(0, 0) = 0\)
\(Z(6, 0) = 24000\)
\(Z(5, 3) = 20000 + 9000 = 29000\)
\(Z(0, 8) = 24000\)
The maximum profit is Rs. 29000, with 5 hectares of wheat and 3 hectares of barley.
Why the other options are incorrect
Option (1) Rs. 24000 — Incorrect. This is the profit at (6, 0) or at (0, 8), growing only one crop.
Option (2) Rs. 32000 — Incorrect. This is wheat on all 8 hectares, which needs 24 units of water, more than 18.
Option (3) Rs. 27000 — Incorrect. This is the profit at (3, 5), which is feasible but not a corner point, and gives less than 29000.