A dietician 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 content of one kg of food is given below:

One kg of food X costs Rs.16 and one kg of food Y costs Rs.20. Find the least cost of the mixture which will produce the required diet?

112

Let the mixture contain x kg of food X and y kg of food Y

The mathematical formulation of the given problem is as follows.

Minimize $z=16x+20y$......(1)

subject to the constraints,

$x+2y≥10$......(2)

$x+y≥6$.........(3)

$3x+y≥8$.......(4)

$x,y≥0$..........(5)

The feasible region determined by the system of constraints is as follows:

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

The values of z at these corner points are as follows.

As the feasible region is unbounded, therefore, 112 may or may not be the minimum value of z

For this, we draw a graph of the inequality, $16x+20y<112$ or $4x+5y<28$ and check whether the resulting half-plane has points in common with the feasible region or not.

It can be seen that the feasible region has no common point with $4x+5y<28$

Therefore, the minimum value of z is 112 at (2,4).

Thus, the mixture should contain 2 kg of food X and 4 kg of food Y. The minimum cost of the mixture is Rs.112