An animal feed to be mixed from soybean meal and oats must contain at least 120 lb of protein, 27 lb of fat, and 10 lb of mineral ash. Each sack of soybeans costs $15 and contains 50 lb of protein, 9 lb of fat, and 5 lb of mineral ash. Each sack of oats costs $5 and contains 15 lb of protein, 5 lb of fat, and 1 lb of mineral ash. How many sacks of each should be used to satisfy the minimum requirements at minimum cost? What is the minimum cost? C
An animal feed to be mixed from soybean meal and oats must contain at least 120 lb of protein, 27 lb of fat, and 10 lb of mineral ash. Each sack of soybeans costs $15 and contains 50 lb of protein, 9 lb of fat, and 5 lb of mineral ash. Each sack of oats costs $5 and contains 15 lb of protein, 5 lb of fat, and 1 lb of mineral ash. How many sacks of each should be used to satisfy the minimum requirements at minimum cost? What is the minimum cost? C
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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![**Optimization Problem: Animal Feed Mixing**
**Problem Statement:**
An animal feed to be mixed from soybean meal and oats must contain at least:
- 120 lb of protein
- 27 lb of fat
- 10 lb of mineral ash
Each sack of soybeans costs $15 and contains:
- 50 lb of protein
- 9 lb of fat
- 5 lb of mineral ash
Each sack of oats costs $5 and contains:
- 15 lb of protein
- 5 lb of fat
- 1 lb of mineral ash
**Question:**
How many sacks of each should be used to satisfy the minimum requirements at minimum cost?
**Objective:**
Minimize the total cost while meeting the nutritional requirements.
**Formulate the problem:**
Let \( x \) be the number of sacks of soybean meal and \( y \) be the number of sacks of oats.
**Constraints:**
1. \( 50x + 15y \ge 120 \) (Protein)
2. \( 9x + 5y \ge 27 \) (Fat)
3. \( 5x + 1y \ge 10 \) (Mineral ash)
**Cost Function:**
\[ \text{Minimize } 15x + 5y \]
**Solution:**
To find the minimum cost \( \$ \)
**Instructions:**
1. **Solve the System of Inequalities:**
Use graphical or algebraic methods to find the feasible region defined by the constraints.
2. **Find the Optimal Solution:**
Evaluate the cost function at the vertices of the feasible region to find the minimum cost.
**Interactive Tools and Options:**
1. **Help Me Solve This:**
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2. **View an Example:**
Refer to a similar example problem to understand the solution process.
3. **Get More Help:**
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Transcribed Image Text:**Optimization Problem: Animal Feed Mixing**
**Problem Statement:**
An animal feed to be mixed from soybean meal and oats must contain at least:
- 120 lb of protein
- 27 lb of fat
- 10 lb of mineral ash
Each sack of soybeans costs $15 and contains:
- 50 lb of protein
- 9 lb of fat
- 5 lb of mineral ash
Each sack of oats costs $5 and contains:
- 15 lb of protein
- 5 lb of fat
- 1 lb of mineral ash
**Question:**
How many sacks of each should be used to satisfy the minimum requirements at minimum cost?
**Objective:**
Minimize the total cost while meeting the nutritional requirements.
**Formulate the problem:**
Let \( x \) be the number of sacks of soybean meal and \( y \) be the number of sacks of oats.
**Constraints:**
1. \( 50x + 15y \ge 120 \) (Protein)
2. \( 9x + 5y \ge 27 \) (Fat)
3. \( 5x + 1y \ge 10 \) (Mineral ash)
**Cost Function:**
\[ \text{Minimize } 15x + 5y \]
**Solution:**
To find the minimum cost \( \$ \)
**Instructions:**
1. **Solve the System of Inequalities:**
Use graphical or algebraic methods to find the feasible region defined by the constraints.
2. **Find the Optimal Solution:**
Evaluate the cost function at the vertices of the feasible region to find the minimum cost.
**Interactive Tools and Options:**
1. **Help Me Solve This:**
Get a step-by-step guide to solving this problem.
2. **View an Example:**
Refer to a similar example problem to understand the solution process.
3. **Get More Help:**
Access additional resources and explanations on optimization problems.
**Additional Features:**
- **Clear all:** Reset the problem to start over.
- **Skill builder:** Engage in practice activities to strengthen your understanding of optimization.
- **Check answer:** Validate your solution and ensure accuracy.
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