Question

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 1kg of food is given below:

**Food **	Vitamin A	Vitamin B	Vitamin C
X	1	2	3
Y	2	2	1

One kg of food X costs Rs16 and one kg of food Y costs Rs20. Find the least cost of the mixture that will produce the required diet.

Answer 1

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.

Corner point| Z=16x+20y|
---|---|---
A(10,0)| 160|
B(2,4)| 112| $\rightarrow$Minimum
C(1,5)| 116|
D(0,8)| 160|

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

For this, we draw a graph for 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 2kg of food X and 4kg of food Y. The minimum cost of the food is Rs112.