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, 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.

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, 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 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 be Rs. 1980