question_answer

A farmer has a supply of chemical fertilizers of type A which contains 10% nitrogen and 6% phosphoric acid and type B contains 5% of nitrogen and 10% of phosphoric acid. After soil testing, it is found that at least 7 kg of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertilizers of types A and B cost Rs. 5 and Rs. 8 per kg, respectively. By using LPF; find how many kilograms of each type of fertilizers should be bought to meet the requirement and cost be minimum? Solve the problem graphically. We generally deal with two types of farming (i) Farming using fertilizers (ii) Organic farming Which farming you would perfer and why?

Answer:

Let x kg of type A and y kg of type B fertilizer be mixed by the farmer to meet the requirement with minimum cost. Then, linear programming problem is Minimise\[C=5x+8y\].  Subject to the constraints \[\frac{10}{100}x+\frac{5}{100}y\ge 7\] \[2x+y\ge 140\]                 ?(i)                         \[\frac{6}{100}x+\frac{10}{100}y\ge 7\]                                     \[3x+5y\ge 350\]                          ?(ii)             and                   \[x\ge 0,\,\,y\ge 0\]             Lines                 \[2x+y=140\]                ?(iii)             and                   \[3x+5y=350\]              ?(iv) are drawn on the same graph. The shaded region is the feasible region. The lines meet at P (50, 40). Now, \[C=5x+8y\] At \[\frac{dl}{d\theta }=-\,a\cos ec\theta \cot \theta +b\sec \theta \tan \theta \] At \[(50,\,\,40)\Rightarrow C=5\times 50+8\times 40=Rs.\,570\] At \[B\,(0,\,\,140)\Rightarrow C=0+8\times 140=Rs.\,1120\] \[\therefore \]Cost is minimum at P (50, 40). i.e. 50 kg of type A and 40 kg type B is mixed to meet the requirement. Value Organic farming save the energy and protect environment to slow down the global warming.