Math Problem Statement
Formulate but do not solve the following exercise as a linear programming problem.
A farmer plans to plant two crops, A and B. The cost of cultivating Crop A is $50/acre, whereas the cost of cultivating Crop B is $70/acre. The farmer has a maximum of $7400 available for land cultivation. Each acre of Crop A requires 20 labor-hours, and each acre of Crop B requires 25 labor-hours. The farmer has a maximum of 2900 labor-hours available. If she expects to make a profit of $160/acre on Crop A and $250/acre on Crop B, how many acres of each crop, x and y, respectively, should she plant to maximize her profit P in dollars? Skip the steps and show the answers
Solution
The linear programming formulation for this problem is as follows:
Decision Variables:
- Let represent the number of acres of Crop A to be planted.
- Let represent the number of acres of Crop B to be planted.
Objective Function:
- Maximize profit , where: (since the profit per acre is $160 for Crop A and $250 for Crop B).
Constraints:
-
Budget constraint (Cultivation costs): (since the farmer has a maximum of $7400 for cultivation costs, with Crop A costing $50/acre and Crop B costing $70/acre).
-
Labor constraint: (since the farmer has a maximum of 2900 labor-hours available, with Crop A requiring 20 labor-hours/acre and Crop B requiring 25 labor-hours/acre).
-
Non-negativity constraints: (since the number of acres planted cannot be negative).
Summary:
- Maximize:
- Subject to:
This is the linear programming formulation of the problem.
Would you like any further details or explanations?
Expanding on this:
- How would changes in profit per acre impact the optimal solution?
- What would happen if the labor-hours constraint increased?
- Could the model be adjusted to account for additional crop types?
- How would the problem change if there were additional budget limitations?
- What methods can be used to solve this linear programming problem efficiently?
Tip: In linear programming, sensitivity analysis can help understand how changes in constraints or the objective function affect the optimal solution.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Optimization
Formulas
Objective Function: P = 160x + 250y
Budget Constraint: 50x + 70y ≤ 7400
Labor Constraint: 20x + 25y ≤ 2900
Non-negativity Constraint: x ≥ 0, y ≥ 0
Theorems
Linear Programming Theorem
Suitable Grade Level
Undergraduate or Advanced High School
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