9. Reference of Que 8 : 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 maximise the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?

Kg per bag
  Brand A Brand P
Nitrogen 3 3.5
Phosphoric Acid 1 2
Potash 3 1.5
Chlorine 1.5 2

 

Answers (1)
S seema garhwal

Let fruit grower use x bags of brand P and y bags of brand Q.

Mathematical formulation of given problem is as  follows:

Maximize : z=3x+3.5y

Subject to constraint ,

                           x+2y\geq 240

                           x+0.5y\geq 90

                         1.5x+2y\geq 310

                            x,y\geq 0

The feasible  region determined by constraints is as follows:

       

The corner points of feasible region are B(20,140),A(140,50),C(40,100)

The value of Z at corner points is as shown :

 corner points 

z=3x+3.5y

 
   A(140,50)             595 maximum

B(20,140)

            550  
  C(40,100)             470 minimum
                 

therefore 595 is maximum value of Z .

Hence , Z has minimum value 595 at  point  A(140,50)

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