Question

There are two types of fertilisers '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 fertiliser should be used so that nutrient requiremnets are met at a minimum cost

Answer 1

The given information can tabulated as follows:

Fertilizer	Nitrogen	Phosphoric Acid	Cost/kg (in ₹)
A	12%	5%	10
B	4%	5%	8

Let the requirement of fertilizer A by the farmer be x kg and that of B be y kg.

It is given that farmer requires atleast 12 kg of nitrogen and 12 kg of phosphoric acid for his crops.

The inequations thus formed based on the given information are as follows:

$$\begin{gathered} \frac{12}{100}x+\frac{4}{100}y\ge 12 \\ \Rightarrow 12x+4y\ge 1200 \\ \Rightarrow 3x+y\ge 300.....\left(1\right) \end{gathered}$$
Also,

$$\begin{gathered} \frac{5x}{100}+\frac{5y}{100}\ge 12 \\ \Rightarrow 5x+5y\ge 1200 \\ \Rightarrow x+y\ge 240.....\left(2\right) \end{gathered}$$
Total cost of the fertilizer Z = ₹ (10x + 8y)

Therefore, the mathematical formulation of the given linear programming problem can be stated as:

Minimize Z = 10x + 8y

Subject to the constraints

3x + y ≥ 300 .....(1)

x + y ≥ 240 .....(2)

x ≥ 0, y ≥ 0 .....(3)

The feasible region determined by constraints (1) to (3) is graphically represented as:



Here, it is seen that the feasible region is unbounded. The values of Z at the corner points of the feasible region are represented in tabular form as:

Corner Point	Z = 10x + 8y
A(0, 300)	Z = 10 × 0 + 8 × 300 = 2400
B(30, 210)	Z = 10 × 30 + 8 × 210 = 1980
C(240, 0)	Z = 10 × 240 + 8 × 0 = 2400

The open half plane determined by 10x + 8y < 1980 has no point in common with the feasible region. So, the minimum value of Z is 1980.

The minimum value of Z is 1980, which is obtained at x = 30 and y = 210.

Thus, the minimum requirement of fertilizer of type A will be 30 kg and that of type B will be 210 kg.

Also, the total minimum cost of the fertilisers is ₹ 1980.