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  • 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.

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 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?

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

 

Answers (1)

best_answer

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

Mathematical formulation of given problem is as  follows:

Minimize : 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 the feasible region are A(140,50),C(40,100),B(20,140)

The value of Z at corner points is as shown :

 corner points 

z=3x+3.5y

  
   A(140,50)             595  

B(20,140)

             550  
  C(40,100)             470 minimum
                 

Therefore 470 is the minimum value of Z.

Hence, Z has a minimum value 470 at point  C(40,100)

Posted by

seema garhwal

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