Fertiliser F1 contains 10% nitrogen and 5% phosphate, and fertiliser F2 contains 5% nitrogen and 10% phosphate. A farmer needs at least 12 kg of nitrogen and at least 9 kg of phosphate for a field. F1 costs Rs. 8 per kg and F2 costs Rs. 6 per kg. The minimum cost of meeting the needs is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Rs. 1040
Let x kg of F1 and y kg of F2 be used. Then
Minimise \(Z = 8x + 6y\)
subject to
\(0.10x + 0.05y \ge 12\), that is \(2x + y \ge 240\) (nitrogen)
\(0.05x + 0.10y \ge 9\), that is \(x + 2y \ge 180\) (phosphate)
\(x \ge 0,\ y \ge 0\)
The two lines meet where, putting \(y = 240 - 2x\),
\(x + 480 - 4x = 180\)
\(x = 100,\ y = 40\)
The feasible region is unbounded, with corner points (0, 240), (100, 40) and (180, 0). Values of Z:
\(Z(0, 240) = 1440\)
\(Z(100, 40) = 800 + 240 = 1040\)
\(Z(180, 0) = 1440\)
The half-plane \(8x + 6y \lt 1040\), that is \(4x + 3y \lt 520\), has no point in common with the feasible region, so the minimum cost is Rs. 1040 (100 kg of F1 and 40 kg of F2).