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The problem appears to be focused on determining the optimal amounts of milk to purchase from two dairies, Dairy I and Dairy II, to get a blend with the maximum amount of butterfat content while staying within budget constraints. Here's a breakdown of the problem setup:

### Problem Setup:
- **Milk from Dairy I**:
  - Price: $2.40 per gallon
  - Butterfat content: 3.7%
  - Maximum supply: 55 gallons

- **Milk from Dairy II**:
  - Price: $0.80 per gallon
  - Butterfat content: 2.9%
  - Maximum supply: 80 gallons

- **Budget constraint**: The company has at most $144 available to purchase milk.
- **Goal**: The company wants to buy enough milk from each supplier to get at most 100 gallons of milk with the highest possible butterfat content.

### Step-by-Step Approach:

1. **Let**:
   - $ x_1 $ be the number of gallons from Dairy I.
   - $ x_2 $ be the number of gallons from Dairy II.

2. **Objective**: Maximize butterfat content.
   - The total butterfat content is $ 0.037 \cdot x_1 + 0.029 \cdot x_2 $.

3. **Constraints**:
   - Total gallons: $ x_1 + x_2 \leq 100 $ gallons.
   - Budget: $ 2.40 \cdot x_1 + 0.80 \cdot x_2 \leq 144 $.
   - Supply limits: $ x_1 \leq 55 $ and $ x_2 \leq 80 $.
   - Non-negativity: $ x_1, x_2 \geq 0 $.

This is a linear programming problem where we maximize the butterfat content subject to the constraints.

### Solving:
You can solve this using graphical methods or simplex methods in linear programming.

Would you like a detailed solution for how to maximize the butterfat or further clarification on the steps? Here are 5 related questions you might consider:

1. How does linear programming help in maximizing a function under constraints?
2. Can the problem be solved graphically or does it require the simplex method?
3. What would happen if the cost or butterfat percentages changed?
4. How would you interpret the shadow price in this context?
5. Could you formulate similar problems involving other ingredients and constraints?

**Tip**: When dealing with linear programming, always ensure to clearly define both your objective function and all constraints to avoid errors.

A dairy company gets milk from two dairies and then blends the milk to get the desired amount of butterfat. Milk from Dairy I costs $2.40 per gallon, and milk from Dairy II costs $0.80 per gallon. At most $144 is available for purchasing milk. Dairy I can supply at most 55 gallons averaging 3.7% butterfat, and Dairy II can supply at most 80 gallons averaging 2.9% butterfat. The company needs to get at most 100 gallons of milk with the maximum amount of butterfat.

Learn how to maximize butterfat content by purchasing milk from two dairies under a $144 budget constraint. This linear programming problem involves finding the optimal mix of milk from Dairy I and Dairy II while staying within the budget and supply limits. Suitable for college-level students studying linear programming and optimization techniques.

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