A woodworking shop makes foot-stools, coffee tables, and rocking chairs. Each spends time in a fabrication shop, a gluing shop, and a painting shop. The following table shows the number of hours each item spends in each shop. If there are 46 hours available in the fabrication shop, 41 hours available in the gluing shop and 87 hours available in the painting shop, how many of each item can be made if all the available hours are to be used? (Order your answers from smallest number of rocking chairs to largest number of rocking chairs.)
Fabrication
shop hours 	Gluing
shop hours 	Painting
shop hours
Foot-stool	2	1	3
Coffee table	1	1	2
Rocking chair	4	8	12
Foot-stools  	Coffee tables  	Rocking chairs
Scenario #1 (smallest number of rocking chairs)
Scenario #2
Scenario #3
Scenario #4 (largest number of rocking chairs)

We are given a linear programming problem that requires us to determine how many foot-stools, coffee tables, and rocking chairs can be made, while using up all the available hours in three different shops: fabrication, gluing, and painting.

To summarize:
- **Fabrication hours available:** 46
- **Gluing hours available:** 41
- **Painting hours available:** 87

Each item requires the following hours:

|                | **Fabrication hours** | **Gluing hours** | **Painting hours** |
|----------------|-----------------------|------------------|--------------------|
| **Foot-stool** | 2                     | 1                | 3                  |
| **Coffee table**| 1                     | 1                | 2                  |
| **Rocking chair**| 4                    | 8                | 12                 |

Let \( x_1 \) represent the number of foot-stools, \( x_2 \) represent the number of coffee tables, and \( x_3 \) represent the number of rocking chairs. We need to solve the system of equations to maximize the number of each type of item made, while using all the available hours.

### Constraints:
- Fabrication shop: \( 2x_1 + x_2 + 4x_3 = 46 \)
- Gluing shop: \( x_1 + x_2 + 8x_3 = 41 \)
- Painting shop: \( 3x_1 + 2x_2 + 12x_3 = 87 \)

The goal is to find different possible values for \( x_1 \), \( x_2 \), and \( x_3 \), ordered by increasing values of \( x_3 \), the number of rocking chairs.

### Solving the System:

We can solve this system by trial and error or algebraic manipulation (substitution or elimination) to find integer values that satisfy the constraints. I'll calculate it to find feasible solutions for each scenario and arrange them by the number of rocking chairs from smallest to largest.

Let me perform the calculations.Here are the solutions where all available hours are used, with the number of foot-stools, coffee tables, and rocking chairs ordered from the smallest to the largest number of rocking chairs:

1. **Scenario #1 (0 rocking chairs):**
   - Foot-stools: 5
   - Coffee tables: 36
   - Rocking chairs: 0

2. **Scenario #2 (1 rocking chair):**
   - Foot-stools: 9
   - Coffee tables: 24
   - Rocking chairs: 1

3. **Scenario #3 (2 rocking chairs):**
   - Foot-stools: 13
   - Coffee tables: 12
   - Rocking chairs: 2

4. **Scenario #4 (3 rocking chairs):**
   - Foot-stools: 17
   - Coffee tables: 0
   - Rocking chairs: 3

These are all the valid solutions based on the constraints. Let me know if you want more details on the calculation process!

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Here are some related questions for further practice:

1. How would the solution change if there were fewer hours in the painting shop?
2. What happens if the time spent gluing is doubled for each item?
3. How many items could be produced if the fabrication shop had more hours available?
4. Can you find a general formula for the number of items if the shop capacities change?
5. What if an additional item with different hour requirements were introduced?

**Tip:** In linear programming, always check that the solution makes sense in terms of both the constraints and practicality (e.g., non-negative integer results).

A woodworking shop makes foot-stools, coffee tables, and rocking chairs. Each spends time in a fabrication shop, a gluing shop, and a painting shop. The following table shows the number of hours each item spends in each shop. If there are 46 hours available in the fabrication shop, 41 hours available in the gluing shop and 87 hours available in the painting shop, how many of each item can be made if all the available hours are to be used? (Order your answers from smallest number of rocking chairs to largest number of rocking chairs.)

Learn how to solve a linear programming problem involving foot-stools, coffee tables, and rocking chairs in a woodworking shop. Using available hours for fabrication, gluing, and painting shops, this solution explains how to determine the number of each item that can be made. Ideal for Grades 10-12 students, this exercise covers systems of equations and algebraic techniques.

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