Math Problem Statement
Solution
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 represent the number of hectares allocated to crop A.
- Let 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 , which is given by:
Constraints:
- Land Constraint: The total available land is 50 hectares.
- 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.
- Non-negativity Constraint: The number of hectares for both crops must be non-negative.
Formulated LPP:
Maximize
Subject to:
Solving the LPP graphically:
- Plot the constraints and on a graph.
- Identify the feasible region that satisfies all constraints.
- Calculate the values of the objective function at the corner points of the feasible region.
- The maximum value of 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:
- What are the implications of increasing herbicide limits on the solution?
- How would the solution change if profit rates for crops A and B were reversed?
- How would the objective function change if a third crop were introduced?
- How would this LPP be affected by different land constraints (e.g., more or less land)?
- 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.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Optimization
Graphical Method
Formulas
Objective function: Z = 10500x + 9000y
Land Constraint: x + y ≤ 50
Herbicide Constraint: 20x + 10y ≤ 800
Non-Negativity Constraint: x ≥ 0, y ≥ 0
Theorems
Graphical Method for Linear Programming Problems
Suitable Grade Level
Grade 11-12
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