An animal shelter mixes x kg of food P and y kg of food Q. Each kg of P contains 1 unit of protein and 3 units of minerals, and each kg of Q contains 2 units of protein and 1 unit of minerals. The mixture must contain at least 8 units of protein and at least 9 units of minerals, and at most 10 kg of the mixture can be stored. Food P costs Rs. 40 per kg and food Q costs Rs. 30 per kg. The minimum cost of the mixture is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Rs. 170
The problem is:
Minimise \(Z = 40x + 30y\)
\(x + 2y \ge 8\), \(3x + y \ge 9\), \(x + y \le 10\), \(x, y \ge 0\)
The lines \(x + 2y = 8\) and \(3x + y = 9\) meet where, putting \(y = 9 - 3x\),
\(x + 18 - 6x = 8\)
\(x = 2, \; y = 3\)
On the y-axis the region starts at \(y = 9\) (from \(3x + y \ge 9\)) and on the x-axis at \(x = 8\) (from \(x + 2y \ge 8\)); the storage limit \(x + y \le 10\) closes it. Its corner points are (0, 9), (2, 3), (8, 0), (10, 0) and (0, 10).
\(Z(0, 9) = 270\), \(Z(2, 3) = 80 + 90 = 170\)
\(Z(8, 0) = 320\), \(Z(10, 0) = 400\)
\(Z(0, 10) = 300\)
The minimum cost is Rs. 170, with 2 kg of P and 3 kg of Q.
Why the other options are incorrect
Option (1) Rs. 270 — Incorrect. This is the cost at the corner (0, 9), using food Q only.
Option (2) Rs. 320 — Incorrect. This is the cost at the corner (8, 0), using food P only.
Option (3) Rs. 300 — Incorrect. This is the cost at the corner (0, 10), which is higher than Rs. 170.