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.