We are tasked with finding the minimum cost of pills that satisfy certain nutritional requirements using a system of linear inequalities. This can be solved using linear programming.

Variables:

Let:

• $x$ = the number of Pill I taken
• $y$ = the number of Pill II taken

Nutritional Constraints:

• Each Pill I contains 4 units of calcium, 3 units of iron, and 4 units of vitamin A.
• Each Pill II contains 1 unit of calcium, 2 units of iron, and 7 units of vitamin A.

The patient needs a minimum of:

• 200 units of calcium,
• 250 units of iron,
• 420 units of vitamin A.

This gives us the following system of inequalities:

1. Calcium constraint: $4x + y \geq 200$
2. Iron constraint: $3x + 2y \geq 250$
3. Vitamin A constraint: $4x + 7y \geq 420$

Cost Function:

The cost of Pill I is $0.10, and the cost of Pill II is $0.15. Therefore, the total cost is given by: $\text{Cost} = 0.10x + 0.15y$

Objective:

Minimize $0.10x + 0.15y$ subject to the above constraints.

To solve this linear programming problem, I will proceed using graphical methods or by applying a standard algorithm like the Simplex method.The patient should take 70 pills of Pill I and 20 pills of Pill II to meet the nutritional requirements. The minimum cost of the pills will be $10.00.

Would you like more details or explanations on any part of the solution?

Here are 5 related questions for further exploration:

1. How would the solution change if the price of Pill II increased to $0.20?
2. What happens if the patient needs more than 420 units of Vitamin A?
3. Can you solve this problem by hand using graphical methods?
4. How would the cost be affected if the patient needed an additional nutrient, like Vitamin C?
5. How would this model change if the patient had a budget constraint?

Tip: Linear programming can also be used in various fields like logistics, economics, and healthcare to optimize costs and resources effectively.