Question

A housewife wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. 

The vitamin contents of 1 kg of each food are given below.

If 1 kg of food X cost ₹6 and 1 kg of food Y costs ₹10, find the minimum cost of the mixture which will produce the diet.

Answer 1

Let x and y be number of kilograms of food X and Y.

∴According to the question,

x + 2y ≥ 10, 2x + 2y ≥ 12, 3x + y ≥ 8, x ≥ 0, y ≥ 0

Minimize Z = 6x + 10y

The feasible region determined x + 2y ≥ 10, 2x + 2y ≥ 12, 3x + y ≥ 8, x ≥ 0, y ≥ 0 is given by

The feasible region is unbounded. The corner points of feasible region are A(0,8) , B(1,5) , C(2,4) , D(10,0).

The value of Z at corner points are

Corner Point	Z = 6x + 10y
A(0, 8)	80
B(1, 5)	56
C(2, 4)	52	Minimum
D(10, 0)	60

The minimum value of Z is 52 at point (2,4).

Hence, the diet should contain 2 kgs of food X and 4 kgs of food Y for the least cost of Rs. 52.