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The problem you shared involves allocating land to two crops, A and B, to maximize profit while considering constraints related to land usage and herbicide limitations. This is a classic Linear Programming Problem (LPP).

### Let’s define the variables:
- Let \( x \) represent the number of hectares allocated to crop A.
- Let \( y \) represent the number of hectares allocated to crop B.

### Objective function:
The profits from crop A and B per hectare are ₹10,500 and ₹9,000, respectively. The objective is to maximize total profit \( Z \), which is given by:
\[
Z = 10500x + 9000y
\]
### Constraints:
1. **Land Constraint**: The total available land is 50 hectares.
   \[
   x + y \leq 50
   \]
2. **Herbicide Constraint**: Crop A requires 20 liters of herbicide per hectare, and crop B requires 10 liters per hectare. The total herbicide used cannot exceed 800 liters.
   \[
   20x + 10y \leq 800
   \]
3. **Non-negativity Constraint**: The number of hectares for both crops must be non-negative.
   \[
   x \geq 0, \quad y \geq 0
   \]

### Formulated LPP:
Maximize \( Z = 10500x + 9000y \)

Subject to:
\[
x + y \leq 50
\]
\[
20x + 10y \leq 800
\]
\[
x \geq 0, \quad y \geq 0
\]

### Solving the LPP graphically:
1. Plot the constraints \( x + y = 50 \) and \( 20x + 10y = 800 \) on a graph.
2. Identify the feasible region that satisfies all constraints.
3. Calculate the values of the objective function at the corner points of the feasible region.
4. The maximum value of \( Z \) will occur at one of the corner points.

Do you want me to graph this for you or explain the graphical method in detail?

### Relative questions:
1. What are the implications of increasing herbicide limits on the solution?
2. How would the solution change if profit rates for crops A and B were reversed?
3. How would the objective function change if a third crop were introduced?
4. How would this LPP be affected by different land constraints (e.g., more or less land)?
5. What if the herbicide required for crop B increased by 50%?

### Tip:
In Linear Programming, graphical methods are ideal for two-variable problems, but simplex methods are better suited for larger, more complex problems.

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 ₹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 in the pond which collects drainage from this land. How much land should be allocated to each crop so as to maximize the profit? Form an LPP and solve it graphically.

Solve a Linear Programming Problem to maximize profit for two crops, A and B, given a total land of 50 hectares and a herbicide constraint of 800 litres. Learn how to formulate and solve the problem graphically with an objective function and constraints.

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