There are 2 fertilizers F1 and F2 · F1 consists of 10% nitrogen & 6 % phosphoric acid & F2 consists of 5 % nitrogen & 10 % phosphoric acid After testing the soil conditions, a farmer finds that she needs at least 14 kg of Nitrogen & 14 kg of phosphoric acid for her crop. If F1 costs 6 rs. per kg & F2 costs 5 rs. kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

Class: Grade 10

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1000

700

500

1500

answer is A.

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Detailed Solution

Let the farmer buy x kg of fertilizer F1
4kg of fertilizer F2
F1 consists of 10 % N2 & F2 consists of 5 % N2. However, the farmer requires at least 14 kg of nitrogen
∴ 10 % of x + 5 % of
F1 Consists of 6% Phosphoric acid & F2 consists of 10 % phosphoric acid. Farmers requires at least 14 kg of Phosphoric acid.
∴ 6 % of x + 10 % of
Total cost of fertilizers z = 6 x + 5 y.
The mathematical formulation of the given problem is
Minimize z = 6 x + 5 y      …(i)
subject to the constraints.
The values of z at these points are or follows.
Corner point   7 = 6 x + 5 y
B(100,80)   1000
C(0,280)          1400
It can be seen that feasible region has no common point with 6 x + 5 y < 1000
∴ 100 kg of fertilizer F1 80 kg of fertilizer
F2 should be used to minimize the cost
∴ minimum cost = 1000

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