question_answer 29)

A fruit grower can use two types of fertilizers in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each mix 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 mix should be used. What is the minimum amount of nitrogen added?

	Kg. per bag
	Brand P	Brand Q
Nitrogen	3	3.5
Phosphoric acid	1	2
Potash	3	1.5
Chlorine	1.5	2

 

Answer:

Let the number of bags used for fertilizer brand P = x.       and the number of bags used for fertilizer brand Q = y       The contents of each unit of fertilizer brands P and Q are given below :

Fertilizer	Phosphoric acid	Potash	Chlorine	Nitrogen
Brand P	1	3	1.5	3
Bran Q	2	1.5	2	3.5
Minimum Requirement	240	270	310

       The above L.P.P. is given as       Z = 3x + 3.5y subject to the constraints x + 2y  240, 3x + 1.5y  + 2y  310, x, y       L1 : x + 2y = 240      L2 : 3x + 1.5y = 270                                   Here amount of nitrogen is minimum at H(40, 100)             From the table it is clear that             No. of bags of fertilizer bag P = 40             No. of bags of fertilizer bag Q = 100       Also, the minimum amount of nitrogen added = 470 kg.