Question

A dietician wishes to mix two types of food, X and Y, in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food X contains 2 units/kg of vitamin A and 1 unit /kg of vitamin C, while food Y contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs ₹5 per kg to purchase the food X and ₹7 per kg to purchase the food Y. Determine the minimum cost of such a mixture.

Answer 1

Let x and y be number of units of X and Y.

∴According to the question,

2x + y ≥ 8, x + 2y ≥ 10, x ≥ 0, y ≥ 0

Minimize Z = 5x + 7y

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

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

The value of Z at corner points are

Corner Point	Z = 5x + 7y
A(0, 8)	56
B(2, 4)	38	Minimum
C(10, 0)	50

The minimum value of Z is 160 at point (4,2).

Hence, the dietician should mix 2 units of X and 4 units of Y to meet the requirements at minimum cost of Rs.38.