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 xx represent the number of acres of Crop A to be planted.
  • Let yy represent the number of acres of Crop B to be planted.

Objective Function:

  • Maximize profit PP, where: P=160x+250yP = 160x + 250y (since the profit per acre is $160 for Crop A and $250 for Crop B).

Constraints:

  1. Budget constraint (Cultivation costs): 50x+70y740050x + 70y \leq 7400 (since the farmer has a maximum of $7400 for cultivation costs, with Crop A costing $50/acre and Crop B costing $70/acre).

  2. Labor constraint: 20x+25y290020x + 25y \leq 2900 (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).

  3. Non-negativity constraints: x0,y0x \geq 0, \quad y \geq 0 (since the number of acres planted cannot be negative).

Summary:

  • Maximize: P=160x+250yP = 160x + 250y
  • Subject to: 50x+70y740050x + 70y \leq 7400 20x+25y290020x + 25y \leq 2900 x0,y0x \geq 0, \quad y \geq 0

This is the linear programming formulation of the problem.

Would you like any further details or explanations?

Expanding on this:

  1. How would changes in profit per acre impact the optimal solution?
  2. What would happen if the labor-hours constraint increased?
  3. Could the model be adjusted to account for additional crop types?
  4. How would the problem change if there were additional budget limitations?
  5. 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