A hostel plans a meal from two foods P and Q. One unit of P gives 8 g of protein and 2 g of fat and costs Rs. 20, and one unit of Q gives 5 g of protein and 3 g of fat and costs Rs. 10. At most Rs. 200 can be spent, and the meal may contain at most 30 g of fat. The greatest amount of protein the meal can give is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 85 g
Let the meal have x units of P and y units of Q. Here the protein is the objective, so it is maximised:
Maximise \(Z = 8x + 5y\)
subject to
\(20x + 10y \le 200\), that is \(2x + y \le 20\) (budget)
\(2x + 3y \le 30\) (fat)
\(x \ge 0,\ y \ge 0\)
The two lines meet where, subtracting,
\(2y = 10\)
\(y = 5,\ x = 7.5\)
The corner points are (0, 0), (10, 0), (7.5, 5) and (0, 10). Values of Z:
\(Z(0, 0) = 0\)
\(Z(10, 0) = 80\)
\(Z(7.5, 5) = 60 + 25 = 85\)
\(Z(0, 10) = 50\)
The greatest amount of protein is 85 g, with 7.5 units of P and 5 units of Q.