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 8 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 4 cents/oz and Brand B costs 5 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.



To solve this optimization problem, we will define variables for the amount of each dog food brand to be used and set up inequalities based on the nutritional requirements, then solve to minimize the cost.

### Step 1: Define the Variables
Let:
- \( x \) = ounces of Brand A to be used per serving.
- \( y \) = ounces of Brand B to be used per serving.

### Step 2: Set Up the Constraints
We are given several constraints:

1. **Total weight constraint**: The total amount of food per serving should be no more than 8 ounces.
   \[
   x + y \leq 8
   \]

2. **Nutrient I constraint**: Each serving should contain at least 29 units of Nutrient I.
   \[
   3x + 5y \geq 29
   \]

3. **Nutrient II constraint**: Each serving should contain at least 20 units of Nutrient II.
   \[
   4x + 2y \geq 20
   \]

4. **Non-negativity constraints**: The amounts of Brand A and Brand B used must be non-negative.
   \[
   x \geq 0 \quad \text{and} \quad y \geq 0
   \]

### Step 3: Set Up the Objective Function
The objective is to minimize the cost. The cost per ounce is:
- 4 cents for Brand A (so the cost contribution is \(4x\)).
- 5 cents for Brand B (so the cost contribution is \(5y\)).

Thus, the objective function is:
\[
\text{Minimize } C = 4x + 5y
\]

### Step 4: Solve the Linear Programming Problem
We have the following linear programming setup:

#### Objective:
Minimize \( C = 4x + 5y \)

#### Subject to:
\[
\begin{cases}
x + y \leq 8 \\
3x + 5y \geq 29 \\
4x + 2y \geq 20 \\
x \geq 0 \\
y \geq 0
\end{cases}
\]

Solving this problem involves graphing the feasible region defined by the constraints and finding the point(s) that minimize the objective function within this region.

Let me calculate the feasible solution and find the optimal values of \( x \) and \( y \).The optimal solution for minimizing the cost while meeting the nutrient requirements is:

- **Brand A**: 3 ounces
- **Brand B**: 4 ounces
- **Minimum Cost**: 32 cents per serving

This combination meets all constraints and provides the necessary nutrients at the lowest cost.

Would you like further details on any of these steps or an explanation on how each constraint is verified in this solution?

Here are 5 follow-up questions that build on this problem:
1. How would the solution change if the cost of Brand B increased to 6 cents per ounce?
2. What if the total weight per serving limit increased from 8 ounces to 10 ounces?
3. How would a requirement increase for Nutrient I to 35 units per serving affect the cost?
4. Could other brands with different nutrient compositions provide a more cost-effective solution?
5. How can you verify that this solution is optimal within the feasible region?

**Tip:** In optimization problems, always verify that the optimal solution meets all constraints by substituting values back into each constraint.

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 8 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 4 cents/oz and Brand B costs 5 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.

Learn how to solve an optimization problem involving dog food nutrition using linear programming. This problem requires setting up constraints to ensure a minimum of 29 units of Nutrient I and 20 units of Nutrient II per 8-ounce serving, with the goal of minimizing cost. Suitable for high school and college students studying optimization.

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