(Diet problem) A dietician has to develop a special diet using two foods `P` and `Q`. Each packet (containing `30g`) of food `P` contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food `Q` contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires atleast 240 units of calciums atleast 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin A in the diet? What is the minimum amount of vitamin A? How many packets of each food should be used to maximise the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?

Explanation

Let the dietician used `x` packets of food `P` and `y` packets of food `Q`. Then


Minimise `Z=6x+3y`…………………1
and constraints
`12x+3yge240implies4x+yge80`………………2
`4x+2yge460impliesx+5yge115`……………….3
`6x+4yle300implies3x+2yle150`..................4
`xge0,yge0`....................5
First draw the graph of the line `4x+y=80`.

Put `(0,0)` in the inequation `4x+yge80`,
`4xx0+0ge80implies0ge80` (False)
Thus, the half plane does not contains origin.
Now, draw the graph of the line `x+5y=115`

Put `(0,0)` in the equation `x+5y+115`,
`0+5xx0ge115implies0ge115` (False)
Thus, the half plane does not contain the origin.
Now draw the graph of the line `3x+2y=150`

Put `(0,0)` in the inequation `3x+2yle150`
`3xx0+2xx0le150implies0le150` (True)
Therefore, half plane contains the origin. Since, `x,yge0`, so the feasible region lies in first quadrant.
From equations `4x+y=80` and `x+5y=115`. The point of intersection is `A(15,20)`.
Similarly from equations `3x+2y=150` and `x+5y=115` the point of intersection is `B(40,15)`. Thus the feasible region is ABCA.
Its vertices are `A(15,20),B(40,15)` and `C(2,72)`
We find the value of `Z` at these vertices.

Thus, the maximum value of `Z` is 285 at point `B(40,15)`. Therefore, to obtain the maximum vitamin A, 40 packages of food `P` and 15 packages of food `Q` should be produced. THus the maximum quantity of vitamin `A` is 285.