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 ₹10,500 and ₹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. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?

Let the land allocated for crop A be x hectares and that of crop B be y hectares

and it is given that profit from crops A are Rs.10,500 and from crop B are Rs. 9000

Hence to maximize the profit, the objective function is

max Z = 10500x + 9000y

Now, Subject to constraints,

It is given that maximum area of the land available for two crops is 50 hectares

∴ x + y ≤ 50 …(i)

It is given that the liquid herbicide to be used for crops A and B are at the rate of 20 litres and 10 litres per hectare respectively.

Maximum amount of herbicide to be used is 800 litres

∴ 20x + 10y ≤ 800

⇒ 2x + y ≤ 80 …(ii)

Now let us draw the graph for the lines

AB: x + y = 50

CD: 2x + y = 80

Consider the line x + y = 50

Put x = 0, y = 0 then 0 ≤ 50 is true.

Hence the region x + y ≤ 50 lies below the line.

Consider the line 2x + y = 80

Put x = 0, y = 0 then 0 ≤ 80 is true.

Hence the region 2x + y ≤ 80 lies below the line.

The feasible region OABC is the shaded portion shown in the fig.

The coordinates of B is (30, 20)

The corner points are O(0,0), A(40, 0), B(30, 20), C(0, 50)

Let us obtain the values of the objective function

Z = 10500x + 9000y

At the points (x,y)the value of the objective function subjected to

Z = 10500x + 9000y

At O (0,0)the value of the objective function

Z = 0

At A(40, 0) the value of the objective function

Z = 10500 × 40 + 9000 × 0 = 420000

At B (30, 20) the value of the objective function

Z = 10500 × 30 + 9000 × 20 = 315000 + 180000 = 495000

At Q (0, 50) the value of the objective function

Z = 10500 × 0 + 9000 × 50 = 450000

The maximum profit is Rs.495000 at B(30, 20)

Hence, 30 hectares of land should be allocated for crop A and 20 hectares for crop B.

Yes, the protection of wildlife is utmost necessary to preserve the balance in environment.