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.
Brand A    		
 oz
Brand B    		
 oz


What is the minimum cost per serving?
 cents

To determine the optimal amount of each brand of dog food to use, let:

- \( x \) = ounces of Brand A
- \( y \) = ounces of Brand B

The objective is to minimize the cost per serving, subject to the constraints given.

### Objective Function
The cost function to minimize is:
\[
\text{Cost} = 4x + 5y
\]

### Constraints
1. Total weight constraint:
   \[
   x + y \leq 8
   \]

2. Nutrient I requirement:
   \[
   3x + 5y \geq 29
   \]

3. Nutrient II requirement:
   \[
   4x + 2y \geq 20
   \]

4. Non-negativity constraints:
   \[
   x \geq 0, \quad y \geq 0
   \]

### Solution
To solve this, we can use the method of linear programming. Let's analyze the feasible region defined by the constraints and then determine the optimal solution. Here’s how we proceed:

1. **Solve each inequality** to find points of intersection.
2. **Evaluate the cost function** at each corner point of the feasible region.
3. Choose the point that gives the lowest cost while meeting all constraints.

Let's calculate each step.

#### Step 1: Intersection Points
To find the feasible region, we need to solve the constraints for equality:
- From \( x + y = 8 \)
- From \( 3x + 5y = 29 \)
- From \( 4x + 2y = 20 \)

Solving these equations and evaluating each intersection point within the feasible region will give us possible solutions. Then, we’ll substitute each feasible point into the cost function to find the minimum.

Let’s compute this system.The optimal solution is to use:

- **3 ounces of Brand A**
- **4 ounces of Brand B**

This combination meets all nutrient and weight requirements at a minimum cost of **32 cents per serving**.

Would you like a breakdown of how each constraint was met or more details on the solution steps?

Here are five questions to deepen understanding:
1. How would the solution change if the nutrient requirements increased?
2. What happens to the cost if Brand A becomes more expensive?
3. How can we adjust if the serving size limit changes to 10 oz?
4. What if a third brand with different nutrient values is introduced?
5. How would nutrient content requirements affect diet flexibility?

**Tip:** Linear programming is widely applicable in resource allocation and budgeting, especially for constraints like nutrients in food distribution.

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. Brand A oz Brand B oz What is the minimum cost per serving? cents

Learn how to solve the problem of preparing an optimal dog food diet using linear programming. This example demonstrates how to minimize the cost of feeding dogs with two different brands of dog food, subject to nutrient and weight constraints. The optimal solution is to use 3 ounces of Brand A and 4 ounces of Brand B, with a minimum cost of 32 cents per serving. Suitable for high school students studying linear programming and optimization.

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