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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 AVitamin BVitamin C
Food X123
Food Y221
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 ?


Solution

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.
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Mathematics
RS Agarwal
Standard XII

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