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 one kg of food is given below :

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

One kg of food $X$ costs Rs. $6$ and one kg of food $Y$ costs Rs. $10$. Formulate the above problem as a linear programming problem and find the least cost of the mixture which will produce the diet graphically. What value will you like to attach with this problem ?

Answer 1

Let $x$ kg of food $X$ and $y$ kg of food $Y$ are mixed.
Let $z$ represents the cost function.
Min $z=6x+10y$
According to LPP,
$x+2y\ge 10$
$2x+2y\ge 12\Rightarrow x+y\ge 6$
$3x+y\ge 8$
We get two points in the solution set, $(1,5)$ and $(2,4)$
At $(1,5)$, $z=6+50=56$
At $(2,4)$, $z=12+40=52$
Hence, least cost of the mixture produced when $2$ kg of food $X$ and $4$ kg of food $Y$ are mixed.