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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. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?
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Solution
Let the land allocated for crop A be x hectares and crop B be y hectares.
Maximum area of the land available for two crops is 50 hectares.
∴ x + y ≤ 50
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 + 10 y ≤ 800
⇒ 2x + y ≤ 80
The profits from crops A and B per hectare are Rs 10,500 and Rs 9,000 respectively.
Thus, total profit = Rs (10,500x + 9,000y) = Rs 1500 (7x + 6y)
Therefore, the mathematical formulation of the given problem is
Maximize Z = 1500 (7x + 6y) subject to the constraints
x + y ≤ 50 … (1)
2x + y ≤ 80 … (2)
x ≥ 0 … (3)
y ≥ 0 … (4)
The feasible region determined by constraints (1), (2), (3) and (4) is represented by the shaded region in the following graph:
The corner points of the 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 point |
Z = 1500 (7x + 6y) |
|
|
O (0, 0) |
0 |
|
|
A (40, 0) |
420000 |
|
|
B (30, 20) |
495000 |
←Maximum |
|
C (0, 50) |
420000 |
The maximum profit is at point B (30, 20).
Therefore, 30 hectares of land should be allocated for crop A and 20 hectares of land should be allocated for crop B.
The maximum profit is Rs 495000.
Yes, we agree that the protection of wildlife is utmost necessary to preserve the balance in environment.
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