Q. 155.0( 2 Votes )

# There are two types of fertilizers ‘A’ and ‘B’. ‘A’ consists of 12% nitrogen and 5% phosphoric acid whereas ‘B’ consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, the farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If ‘A’ costs ₹ 10 per kg and ‘B’ cost ₹ 8 per kg, then graphically determine how much of each type of fertilizer should be used, so that nutrient requirements are met at a minimum cost. [CBSE 2016]

Answer :

**Given:** Fertilizer ‘A’ consists of 12% nitrogen and 5% phosphoric acid whereas Fertilizer ‘B’ consists of 4% nitrogen and 5% phosphoric acid. ‘A’ costs Rs. 10 per kg and ‘B’ cost Rs. 8 per kg

**To find:** quantity of fertilizer should be used, so that nutrient requirements are met at a minimum cost

Let the quantity of fertilizer A and B be used as x and y respectively, and the total cost be z

⇒ z = 10x + 8y

We need to minimize the cost

Hence, the mathematical formulation of LPP is

Minimize z = 10x + 8y

subject to the constraints,

The feasible region determined by the system of constraints is as follows:

The corner points of the enclosed region are A (0, 300), B(30, 210), C(240, 0)

The value of z at these corners points is as follows:

Case 1: A(0, 300)

z = 10x + 8y

⇒ z = 10(0) + 8(300)

⇒ z = 0 + 2400

⇒ z = 2400

Case 2: B(30, 210)

z = 10x + 8y

⇒ z = 10(30) + 8(210)

⇒ z = 300 + 1680

⇒ z = 1980

Case 3: C(240, 0)

z = 10x + 8y + 370

⇒ z = 10(240) + 8(0)

⇒ z = 2400 + 0

⇒ z = 2400

The value of z is minimum in the second case at point B(30, 210)

Hence, the quantity of fertilizer A and B be used as **30 Kg and 210 Kg** respectively, and the total cost is **Rs. 1980**

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