Question

(Allocation problem) A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be

Answer 1

Let `x` hectare of land be allocated for crop A and `y` hectare of land be allocated from crop `B`.
Given that
Maximize profit `z=10500x+9000y`
Subject to `xge0`
`yge0`
`20x+10yle800`
`implies2x+yle80`
and `x+yle50`
Draw the graph of equations `x=0, y=0, x+y=5` and `2x+y=80`.
Now obtain the feasible region for the inequations `xge0, yge0, 2x+yle80` and `x+yle50` and shade it. The convex region is OABC whose vertices are `A(40,0),B(30,20),C(0,50)` and `O(0,0)`. We will find the value of `z` at these vertices.

Therefore the maximum profit is Rs. 495000 for which land allocated for crop `A=30` hectares and for crop `B=20` hectares.