ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
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### Coffee Blending Problem

Deluxe coffee is to be mixed with regular coffee to make at least 48 pounds of a blended coffee. The mixture must contain at least 9 pounds of deluxe coffee. Deluxe coffee costs $5 per pound and regular coffee $4 per pound. How many pounds of each kind of coffee should be used to minimize costs?

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**Question:**  
To attain the minimum cost, use [ ] pounds of deluxe coffee and [ ] pounds of regular coffee.

### Explanation

This problem involves finding the optimal blend of two types of coffee to minimize costs while satisfying specified constraints. In mathematical terms, it's an optimization problem with constraints. 

1. **Constraints:**
   - Total weight: at least 48 pounds
   - Minimum deluxe coffee: at least 9 pounds

2. **Costs:**
   - Deluxe coffee: $5 per pound
   - Regular coffee: $4 per pound

### Solution Approach

To solve such a problem, we use linear programming techniques or a systematic approach to test possible combinations that satisfy the constraints.

1. Let the amount of deluxe coffee be \( D \) pounds.
2. Let the amount of regular coffee be \( R \) pounds.

**Constraints:**
- \( D + R \geq 48 \)
- \( D \geq 9 \)

**Objective:**
Minimize cost, \( C \):
\[ C = 5D + 4R \]

By substituting potential values and ensuring all conditions are met, we can find the values of \( D \) and \( R \) that result in the minimum cost. The solution should be presented by filling the blanks in the question:

**To attain the minimum cost, use [ ] pounds of deluxe coffee and [ ] pounds of regular coffee.** 

This transcription and explanation would help students understand the problem, constraints, and the approach to solve the optimization problem.
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Transcribed Image Text:### Coffee Blending Problem Deluxe coffee is to be mixed with regular coffee to make at least 48 pounds of a blended coffee. The mixture must contain at least 9 pounds of deluxe coffee. Deluxe coffee costs $5 per pound and regular coffee $4 per pound. How many pounds of each kind of coffee should be used to minimize costs? --- **Question:** To attain the minimum cost, use [ ] pounds of deluxe coffee and [ ] pounds of regular coffee. ### Explanation This problem involves finding the optimal blend of two types of coffee to minimize costs while satisfying specified constraints. In mathematical terms, it's an optimization problem with constraints. 1. **Constraints:** - Total weight: at least 48 pounds - Minimum deluxe coffee: at least 9 pounds 2. **Costs:** - Deluxe coffee: $5 per pound - Regular coffee: $4 per pound ### Solution Approach To solve such a problem, we use linear programming techniques or a systematic approach to test possible combinations that satisfy the constraints. 1. Let the amount of deluxe coffee be \( D \) pounds. 2. Let the amount of regular coffee be \( R \) pounds. **Constraints:** - \( D + R \geq 48 \) - \( D \geq 9 \) **Objective:** Minimize cost, \( C \): \[ C = 5D + 4R \] By substituting potential values and ensuring all conditions are met, we can find the values of \( D \) and \( R \) that result in the minimum cost. The solution should be presented by filling the blanks in the question: **To attain the minimum cost, use [ ] pounds of deluxe coffee and [ ] pounds of regular coffee.** This transcription and explanation would help students understand the problem, constraints, and the approach to solve the optimization problem.
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