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### Given Information
1. We have two brands of dog food: Brand \( A \) and Brand \( B \).
2. Each ounce of Brand \( A \) contains:
   - 3 units of Nutrient I
   - 4 units of Nutrient II
3. Each ounce of Brand \( B \) contains:
   - 5 units of Nutrient I
   - 2 units of Nutrient II
4. Cost:
   - Brand \( A \): 3 cents per ounce
   - Brand \( B \): 4 cents per ounce

### Requirements
1. The serving size should be no larger than 10 ounces.
2. The serving must contain at least:
   - 29 units of Nutrient I
   - 20 units of Nutrient II

### Variables
Let:
- \( x \) be the ounces of Brand \( A \).
- \( y \) be the ounces of Brand \( B \).

### Objective
Minimize the cost, which is given by:
\[
\text{Cost} = 3x + 4y
\]

### Constraints
1. **Total Serving Size**:
   \[
   x + y \leq 10
   \]
2. **Nutrient I Requirement**:
   \[
   3x + 5y \geq 29
   \]
3. **Nutrient II Requirement**:
   \[
   4x + 2y \geq 20
   \]
4. **Non-negativity**:
   \[
   x \geq 0, \quad y \geq 0
   \]

### Solution
This is a linear programming problem. We need to solve it by optimizing the cost function \( 3x + 4y \) subject to the given constraints.

Let me calculate this and find the optimal values for \( x \) and \( y \).The optimal solution to minimize the cost while meeting all the nutritional and size requirements is:

- **Brand \( A \):** 3 ounces
- **Brand \( B \):** 4 ounces

The minimum cost achieved with this combination is **25 cents**.

Would you like a detailed breakdown of the solution process, or do you have any other questions?

Here are some related questions for further exploration:

1. How would the solution change if the cost of Brand \( B \) was 5 cents per ounce instead?
2. What if the nutrient requirements increased to 35 units of Nutrient I and 25 units of Nutrient II?
3. How would the solution change if there was no restriction on the serving size?
4. Can we achieve the requirements if Brand \( A \) were not available at all?
5. How would an additional nutrient requirement affect the solution?

**Tip:** Linear programming is a valuable tool in optimizing resources under constraints and is widely used in areas like diet planning, logistics, and manufacturing.

A veterinarian has been asked to prepare a diet for a group of dogs to be used in a nutrition study at the School of Animal Science. It has been stipulated that each serving should be no larger than 10 oz and must contain at least 29 units of Nutrient I and 20 units of Nutrient II. The vet has decided that the diet may be prepared from two brands of dog food: Brand A and Brand B. Each ounce of Brand A contains 3 units of Nutrient I and 4 units of Nutrient II. Each ounce of Brand B contains 5 units of Nutrient I and 2 units of Nutrient II. Brand A costs 3 cents/oz and Brand B costs 4 cents/oz. Determine how many ounces of each brand of dog food should be used per serving to meet the given requirements at a minimum cost.

Explore a linear programming solution for determining the optimal ounces of Brand A and Brand B dog food to meet nutritional requirements at minimum cost. The problem involves constraints on serving size, nutrient requirements, and cost minimization, ideal for high school students learning linear optimization techniques.

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