A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further, not more than 800 litres of 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 maximize the total profit?<p></p>

A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further, not more than 800 litres of 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 maximize the total profit?

Solution:

Let x hectares for crop A and y hectares for crop B be allocated.

x + y ≤ 50 hectares
Profit from crop A = 10500 Rs/hectare
Profit from crop B = 9000 Rs/hectare
Herbicide usage:
Crop A: 20 L/hectare
Crop B: 10 L/hectare
Total herbicide usage ≤ 800 litres
We need to maximize profit given by Z = 10500x + 9000y
subject to:
x + y ≤ 50
20x + 10y ≤ 800
x, y ≥ 0
After plotting the graph, the corner points are:

Corner Point	Value of Z
O(0,0)	0
A(40,0)	420000
B(30,20)	315000 + 180000 = 495000
C(0,50)	450000
Maximum profit is at point B (30, 20).
Therefore, 30 hectares for crop A and 20 hectares for crop B.

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