One kg of food X contains 2 units of vitamin A and 1 unit of vitamin B. One kg of food Y contains 1 unit of vitamin A and 3 units of vitamin B. A mixture of the two foods must contain at least 10 units of vitamin A and at least 15 units of vitamin B. Food X costs Rs. 30 per kg and food Y costs Rs. 20 per kg. The minimum cost of such a mixture is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Rs. 170
Let the mixture contain x kg of X and y kg of Y. Then the problem is:
Minimise \(Z = 30x + 20y\)
subject to \(2x + y \ge 10\), \(x + 3y \ge 15\), \(x, y \ge 0\)
The two lines meet where, putting \(y = 10 - 2x\),
\(x + 30 - 6x = 15\)
\(x = 3, \; y = 4\)
The region is unbounded, with corner points (0, 10), (3, 4) and (15, 0). Values of Z:
\(Z(0, 10) = 200\)
\(Z(3, 4) = 90 + 80 = 170\)
\(Z(15, 0) = 450\)
The half-plane \(30x + 20y \lt 170\) has no point in common with the region, because every feasible point satisfies
\(30x + 20y = 14(2x + y) + 2(x + 3y) \ge 140 + 30 = 170\)
So the minimum cost is Rs. 170, with 3 kg of X and 4 kg of Y.