A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine m a bag of each brand are given m the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine. If the grower wants to minimise the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?

Explanation

Let's assume that bags of brand P needed be `X` and brand Q be `Y`
Since bag of brand P contains one kg and bag of brand Q contains two kg of Phosphoric acid.
Also, garden needs at least `240 kg` of phosphoric acid.
`therefore X+2Y≥240`
Since bag of brand P contains three kg and bag of brand Q contains 1.5 kg of potash.
Also, garden needs at least `270 kg` of potash.

`∴3X+1.5Y≥270`
`=>2x+y≥180`
Since bag of brand P contains 1.5 kg and bag of brand Q contains two kg of chlorine.
Also, garden needs at most 310 kg of chlorine.

`∴1.5X+2Y≤310 `
Since count of bags can't be negative.

`∴X≥0,Y≥0`
Now, Bag of brand P contains three kg of Nitrogen and Bag of brand Q contains 3.5 kg of Nitrogen.
So, total Nitrogen contents `Z=3X+3.5Y`
We have to minimise the amount of Nitrogen added to the garden.
After plotting all the constraints given by equations we conclude that to minimise the amount of nitrogen added to the garden, grower should use `40` bags of brand P and `100` bags of brand Q.
In this case minimum amount of nitrogen added to the garden will be `470 Kg`.