Question

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 in a bag of each brand are given in 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 minimize 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? kg per bag Brand P Brand Q Nitrogen Phosphoric acid Potash Chlorine 3 1 3 1.5 3.5 2 1.5 2

Answer 1

Consider, Bags of brand P be x and bags of brand Q be y.

Now, table for the two types of different brand fertilizer is,

Brand	No. of bags	Nitrogen	Phosphoric Acid	Potash	Chlorine
P	x	3	1	3	1.5
Q	y	3.5	2	1.5	2
Total requirement			240 Kg	270 Kg	310 Kg

The required constraints are,

x+2y≥240 3x+1.5y≥270 1.5x+2y≥310 x≥0, y≥0

Minimize the amount of nitrogen,

Z=3x+3.5y

The line x+2y≥240gives the intersection point as,

x	0	240
y	120	0

The line 3x+1.5y≥270 gives the intersection points as,

x	0	90
y	180	0

Similarly the line 1.5x+2y≥310 gives the intersection point as,

x	0	206.67
y	155	0

Therefore, the graph for the minimum value of the Z is,

Substitute these points in the given objective function to find the minimize value of Z.

Corner points	Value of Z
( 20, 140 )	550
( 40, 100 )	470 (Minimum)
( 140, 50 )	595

Hence, the value of the Nitrogen will be minimum if the mixture contains,

No. of bags of brand P is equal to 40and no. of bags of brand Q is equal to the 100.

Thus, the minimum amount of the Nitrogen is 470 Kg.