question_answer

A cooperative society of farmers has 50 hec of land to grow two crops A and B. The profit from crops A and B per hectare are estimated as Rs.10500 and Rs. 9000, respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 L/hec and 10 L/hec, respectively. Further not more than 800 L, herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land keeping in mind that the protection of fish and other wildlife is more important than earning profit. How much land should be allocated to each crop, so as to maximise the total profit formulate the above as an LPP and solve it graphically. Do you agree with the message that the protection of wild life is at most necessary to preserve the balance in environment

Answer:

Let the land allocated for crop A be x hectare and crop B be y hectare. Then, maximum area of the land available for two crops is 50 hectare. \[x+y\le 50\] Liquid herbicide to be used for crops A and B are at the rate of 20 L and 10 L per hectare, respectively. Maximum amount of herbicide to be used is 800 L. \[20x+10y\le 800\] and                   \[2x+y\le 80\] The profits from crops A and B per hectare are Rs. 10500 and Rs. 9000 respectively. Thus, total profit \[=(10500x+9000y)=Rs.\,1500\,\,(7x+6y)\] The mathematical formulation of the given problem is Maximise\[Z=1500\,\,(7x+6y)\] Subject to the constraints \[x+y\le 50\]                     ?(i) \[2x+y\le 80\]                  ?(ii)             and                   \[x,\,\,y\ge 0\]                 ?(iii) The corner points of feasible region are O (0, 0), A (40, 0), B (30, 20) and C (0, 50). The values of Z at these corner points are calculated as

Corner	\[Z=1500\,\,(7x+6y)\]
\[O\,(0,\,\,0)\]	\[Z=0\]
\[A\,(40,\,\,0)\]	\[Z=1500\,(7\times 40+6\times 0)=420000\]
\[B\,(30,\,\,20)\]	\[Z=1500\,(7\times 30+6\times 20)\] \[=49500\,\,(maximum)\]
\[C\,(0,\,\,50)\]	\[Z=1500\,(7\times 0+6\times 50)=420000\]

The maximum profit is at point B (30, 20) \[\therefore \]30 hec of land should be allocated for crop A and 20 hec of land should be allocated for crop B. Value Yes, because excess use of herbicide can make drainage water poisonous and shows it harms the life of water living creature and wild life.