This is a linear programming problem where we want to minimize John's cost while ensuring he gets enough vitamins, minerals, and calories. Here's a breakdown of the problem and how we can set it up to solve:

Variables:

Let:

• $x$ = number of boxes of Cereal A
• $y$ = number of boxes of Cereal B

Cost function (Objective):

We need to minimize the total cost, which is: $\text{Total cost} = 0.50x + 0.55y$

Constraints:

John needs at least:

• 475 vitamins,
• 550 minerals,
• 1275 calories.

Each box of Cereal A provides 10 vitamins, 5 minerals, and 15 calories, while each box of Cereal B provides 5 vitamins, 10 minerals, and 15 calories.

Thus, the constraints are:

• Vitamins: $10x + 5y \geq 475$
• Minerals: $5x + 10y \geq 550$
• Calories: $15x + 15y \geq 1275$

We also assume non-negative values for $x$ and $y$: $x \geq 0, \quad y \geq 0$

Linear Programming Formulation:

Minimize: $Z = 0.50x + 0.55y$ Subject to: $10x + 5y \geq 475$ $5x + 10y \geq 550$ $15x + 15y \geq 1275$ $x \geq 0, \quad y \geq 0$

This is now a standard linear programming problem that can be solved using methods like the Simplex Method, or graphically if it's manageable.

Would you like to proceed with solving it step-by-step or using software to compute the exact solution?

Here are 5 related questions that can help you explore the concept further:

1. How would changing the cost per box of Cereal A or B affect the minimum cost?
2. What happens if John needs more than 475 units of vitamins? How would that change the solution?
3. Could John meet his nutrient requirements using only one type of cereal? How would that affect the cost?
4. What if John wanted to maximize his calorie intake instead of minimizing cost?
5. How could you solve this problem graphically?

Tip:

When solving linear programming problems graphically, always consider the points of intersection of the constraint lines as potential solutions, as the optimal solution will always lie on one of these intersection points.