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A farmer is going to divide her 60 acre farm between two crops. Seed for crop A costs $30 per acre. Seed for crop B costs $15 per acre. The farmer can spend at most $1200 on seed. If crop B brings in a profit of $50 per acre, and crop A brings in a profit of $60 per acre, how many acres of each crop should the farmer plant to maximize her profit? acres of crop A acres of crop B

A farmer is going to divide her 60 acre farm between two crops. Seed for crop A costs $30 per acre. Seed for crop B costs $15 per acre. The farmer can spend at most $1200 on seed. If crop B brings in a profit of $50 per acre, and crop A brings in a profit of $60 per acre, how many acres of each crop should the farmer plant to maximize her profit? acres of crop A acres of crop B

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Crop Allocation Problem **Scenario:** A farmer has a 60-acre farm and plans to divide it between two crops, A and B. The costs and profits associated with each crop are as follows: - **Crop A:** - Seed cost: $30 per acre - Profit: $60 per acre - **Crop B:** - Seed cost: $15 per acre - Profit: $50 per acre The farmer has a budget constraint of $1200 for seed costs. **Task:** Determine the optimal number of acres to plant for crops A and B to maximize the farmer's profit. **Solution Steps:** 1. **Define Variables:** - Let `x` be the acres of crop A. - Let `y` be the acres of crop B. 2. **Constraints:** - Land constraint: \( x + y = 60 \) - Budget constraint: \( 30x + 15y \leq 1200 \) 3. **Profit Equation:** - Total profit: \( 60x + 50y \) 4. **Objective:** - Maximize the profit equation \( 60x + 50y \) while satisfying the constraints. **Interactive Input Fields:** - `_______` acres of crop A - `_______` acres of crop B **Explanation of Graphs/Diagrams:** There are no graphs or diagrams associated with this problem. The solution involves formulating and solving a linear programming problem to determine the optimal values for `x` and `y`.
Transcribed Image Text:### Crop Allocation Problem **Scenario:** A farmer has a 60-acre farm and plans to divide it between two crops, A and B. The costs and profits associated with each crop are as follows: - **Crop A:** - Seed cost: $30 per acre - Profit: $60 per acre - **Crop B:** - Seed cost: $15 per acre - Profit: $50 per acre The farmer has a budget constraint of $1200 for seed costs. **Task:** Determine the optimal number of acres to plant for crops A and B to maximize the farmer's profit. **Solution Steps:** 1. **Define Variables:** - Let `x` be the acres of crop A. - Let `y` be the acres of crop B. 2. **Constraints:** - Land constraint: \( x + y = 60 \) - Budget constraint: \( 30x + 15y \leq 1200 \) 3. **Profit Equation:** - Total profit: \( 60x + 50y \) 4. **Objective:** - Maximize the profit equation \( 60x + 50y \) while satisfying the constraints. **Interactive Input Fields:** - `_______` acres of crop A - `_______` acres of crop B **Explanation of Graphs/Diagrams:** There are no graphs or diagrams associated with this problem. The solution involves formulating and solving a linear programming problem to determine the optimal values for `x` and `y`.
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