A farmer sows x hectares of wheat and y hectares of mustard on at most 12 hectares of land. He must sow at least 3 hectares of wheat, and he can sow at most 6 hectares of mustard. The profit is Rs. 20,000 per hectare of wheat and Rs. 30,000 per hectare of mustard. Consider the following statements:
(A) The feasible region is bounded and has four corner points.
(B) Sowing 2 hectares of wheat and 5 hectares of mustard is a feasible plan.
(C) (6, 6) is a corner point of the feasible region.
(D) The maximum profit is Rs. 3,00,000.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
The problem is
Maximise \(Z = 20000x + 30000y\)
subject to
\(x + y \le 12\) (land)
\(x \ge 3\) (wheat)
\(y \le 6\) (mustard)
\(y \ge 0\)
The line \(y = 6\) meets \(x + y = 12\) at (6, 6), and the line \(x = 3\) meets \(x + y = 12\) at (3, 9), which breaks \(y \le 6\). The region is a closed bounded quadrilateral with corner points (3, 0), (12, 0), (6, 6) and (3, 6). Values of Z:
\(Z(3, 0) = 60000\)
\(Z(12, 0) = 240000\)
\(Z(6, 6) = 120000 + 180000 = 300000\)
\(Z(3, 6) = 60000 + 180000 = 240000\)
So the region is bounded with corners (3, 0), (12, 0), (6, 6), (3, 6), and the maximum profit is Rs. 3,00,000 at (6, 6).
Checking the statements
(A) The feasible region is bounded and has four corner points. — Correct. Its corners are (3, 0), (12, 0), (6, 6) and (3, 6).
(B) Sowing 2 hectares of wheat and 5 hectares of mustard is a feasible plan. — Incorrect. \(x = 2 \lt 3\), so the wheat condition fails.
(C) (6, 6) is a corner point of the feasible region. — Correct. It is where \(y = 6\) meets \(x + y = 12\).
(D) The maximum profit is Rs. 3,00,000. — Correct. \(Z(6, 6) = 300000\) is the largest corner value.