Math Problem Statement

solve

Your small farm encompasses 130 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: $5 for tomatoes, $4 for lettuce, and $2 for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of $2,000 for each acre of tomatoes, $1,400 for each acre of lettuce, and $400 for each acre of carrots. You would like to spend at least $520 on fertilizer and your farm laborers can supply up to 650 hours per week. How many acres of each crop should you plant to maximize total profits?

Solution

To solve this problem, we need to maximize profit while considering constraints on the number of acres used, labor hours, and fertilizer costs. Let's define the variables and then set up the equations.

Variables:

  • Let xx be the number of acres for tomatoes.
  • Let yy be the number of acres for lettuce.
  • Let zz be the number of acres for carrots.

Given Information:

  1. Total acres constraint: x+y+z=130x + y + z = 130

  2. Labor constraint:

    • Tomatoes require 4 hours per acre.
    • Lettuce requires 2 hours per acre.
    • Carrots require 2 hours per acre.

    Total labor constraint is given by: 4x+2y+2z≤6504x + 2y + 2z \leq 650

  3. Fertilizer cost constraint:

    • $5 per acre for tomatoes.
    • $4 per acre for lettuce.
    • $2 per acre for carrots.

    Total fertilizer cost should be at least $520: 5x+4y+2z≥5205x + 4y + 2z \geq 520

  4. Profit function:

    • $2,000 per acre for tomatoes.
    • $1,400 per acre for lettuce.
    • $400 per acre for carrots.

    Therefore, the total profit PP is: P=2000x+1400y+400zP = 2000x + 1400y + 400z

Objective:

Maximize the profit P=2000x+1400y+400zP = 2000x + 1400y + 400z subject to the constraints:

x + y + z &= 130, \\ 4x + 2y + 2z &\leq 650, \\ 5x + 4y + 2z &\geq 520, \\ x, y, z &\geq 0. \end{aligned}$$ ### Solution Approach: This problem can be formulated as a linear programming (LP) problem, where we aim to maximize $$P$$ given the constraints. Let's solve this LP problem using a standard optimization technique like the Simplex method. ### Steps to Solve: 1. Define the LP model in terms of decision variables $$x$$, $$y$$, and $$z$$. 2. Express the objective function and constraints. 3. Use an LP solver to find the optimal values of $$x$$, $$y$$, and $$z$$. Let's calculate this using Python to find the optimal solution.The optimal solution to maximize profit is: - **Tomatoes**: 130 acres - **Lettuce**: 0 acres - **Carrots**: 0 acres This results in a **maximum profit** of $260,000. ### Explanation: - All the available 130 acres should be dedicated to growing tomatoes, which yields the highest profit per acre. - This allocation satisfies all constraints, including the labor hours and fertilizer cost constraints. Would you like more details or have any other questions? Here are five related questions to explore further: 1. How would the optimal solution change if the labor constraints were more restrictive? 2. What if the profit per acre of lettuce increased to $1,600? 3. How would adding a minimum planting requirement for each crop affect the solution? 4. What is the impact on profit if fertilizer costs increase by 10% across all crops? 5. How sensitive is the profit to changes in the available total acreage? **Tip**: In optimization problems, always focus on identifying and prioritizing the most profitable variables within the constraints provided.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Optimization

Formulas

Total acres constraint: x + y + z = 130
Labor constraint: 4x + 2y + 2z ≤ 650
Fertilizer cost constraint: 5x + 4y + 2z ≥ 520
Profit function: P = 2000x + 1400y + 400z

Theorems

Linear programming simplex method

Suitable Grade Level

Grade 11-12

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