You own a hamburger franchise and are planning to shut down operations for the day, but you are left with 11 buns, 13 defrosted beef Patties, and 10 opened cheese slices. Rather than throw them out, you decide to use them to make burgers that you will sell at a discount. Plain burgers each require 1 beef patty and 1 bun, double cheeseburger each require 2 beef Patties, 1 bun, and 2 slices of cheese, while regular cheeseburgers each require 1 beef patty, 1 bun, and 1 slice of cheese. How many of each should you make?

This is a classic optimization problem where we need to determine how to use all available resources (buns, beef patties, and cheese slices) to make the maximum number of burgers. The three types of burgers are:

1. **Plain burger:** requires 1 bun and 1 beef patty.
2. **Regular cheeseburger:** requires 1 bun, 1 beef patty, and 1 slice of cheese.
3. **Double cheeseburger:** requires 1 bun, 2 beef patties, and 2 slices of cheese.

You have the following resources:
- 11 buns
- 13 beef patties
- 10 cheese slices

Let:
- \( x_1 \) be the number of **plain burgers** made.
- \( x_2 \) be the number of **regular cheeseburgers** made.
- \( x_3 \) be the number of **double cheeseburgers** made.

### Constraints:
- Total buns used: \( x_1 + x_2 + x_3 \leq 11 \)
- Total beef patties used: \( x_1 + x_2 + 2x_3 \leq 13 \)
- Total cheese slices used: \( x_2 + 2x_3 \leq 10 \)

### Objective:
Maximize the number of burgers made, i.e., \( x_1 + x_2 + x_3 \), subject to the constraints.

I'll solve this step-by-step to find the values of \( x_1 \), \( x_2 \), and \( x_3 \).The optimal solution is to make:

- 11 plain burgers,
- 0 regular cheeseburgers,
- 0 double cheeseburgers.

This means you can use all the buns and beef patties without exceeding your available resources.

Do you have any questions or would you like further details?

### Related Questions:
1. What if you wanted to prioritize making cheeseburgers?
2. How would the solution change if you had more cheese slices?
3. What happens if you have fewer buns, say 9 buns?
4. How would the total number of burgers change if a double cheeseburger used 3 patties instead of 2?
5. Can we distribute the resources differently to balance out the burger types?

**Tip:** Linear programming is a powerful technique for maximizing or minimizing a result under certain constraints.

Solve an optimization problem involving limited resources: 11 buns, 13 beef patties, and 10 cheese slices. Learn how to maximize the number of plain burgers, regular cheeseburgers, and double cheeseburgers using linear programming concepts and inequalities. Suitable for students in Grades 10-12.

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