There are two types of fertilisers `F_(1)` and `F_(2)`.`F_(1)` consists of 10% nitrogen and 6% phosphoric acid and `F_(2)` consists of 5% nitrogen and 10% phosphoric acid. After testing he soil conditions a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If `F_(1)` costs Rs. 6/kg and `F_(2)` costs Rs. 5/kg, determine how much of each type of fertiliser should be used to that nutrient requirements are met at a minimum cost. What is the minimum cost?

Explanation

Let `x` kg and `y` kg of `F_(1)` and `F_(2)` respectively are used.

According to the problem,
Minimum cost `Z=6x+5y`
and constraints `xge0,yge0`
`(10x)/100+(5y)/100ge14implies2x+yge280`
`(6x)/100+(10y)/100ge14implies3x+5yge700`
First draw the graph of the lines `2x+y=280` and `3x+5y=700`.

Now, we find the feasible region using constraints `2x+yge280,3x+5yge700,xge0,yge0` and shade it.
It vertices are `A(250,0),B(100,80)` and `C(0,280)`. We find the value of `Z` at these vertices.

Therefore, the minimum cost is Rs. 100 using 100 kg `F_(1)` and `80 kg F_(2)` fertiliser.