Question

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? Form an LPP from the above and solve it graphically.

Answer 1

Let x and y hectares of land be allocated to crop A and B respectively. If Z is the profit then

Z = 10500x + 9000y ......(i)

We have to maximize Z subject to the constraints:

x + y ≤ 50 ..........(ii)

20x + 10y ≤ 800 ⇒ 2x + y ≤ 80 …(iii)

x ≥ 0, y ≥ 0 ........(iv)

The graph of system of inequalities (ii) to (iv) are drawn, which gives feasible region OABC with corner points O (0, 0), A (40, 0), B (30, 20) and C (0, 50).

Table for x + y = 50

x	0	50
y	50	0

Table for 2x + y = 80

x	0	40
y	80	0

Feasible region is bounded.

Now,

Hence, the co-operative society of farmers will get the maximum profit of Rs 4,95,000 by allocating 30 hectares for crop A and 20 hectares for crop B.