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).