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A dietician wishes to mix together two k...

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 contents of 1 kg food is given below:

1 kg of food `X` costs Rs. 16 and 1 kg of food `Y` costs Rs. 20. Find the least cost of the mixture which will produce the required diet?

Text Solution

Verified by Experts

Let `x` kg and `y` kg of food `X` and food `Y` respectively are in the mixture.
According to the problem,
Minimum cost `z=16x+20y`.
and constraints `xge0`
`yge0, x+2yge10`
`2x+2yge12, 3x+yge8`
First, draw the graph of lines `x+2y=10,`2x+2y=12` and `3x+y=8`

Now, we find the feasible region by the constraints `x+2yge10,2x+2yge12,3x+yge8,xge0,yge0` and shade it. The vertices of this shaded region are `A(0,8),B(1,5),C(2,4),D(10,0)` at which we find the value of `Z`.

Therefore, the minimum cost of food is Rs. 112.
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