The graph shows the feasible region (shaded area) where the constraints are satisfied:

1. **The orange line** represents the constraint \(3T + 6C \leq 120\).
2. **The yellow line** represents the constraint \(5T + 2C \leq 80\).

The shaded region is where both constraints overlap, and it represents all possible combinations of tables (\(T\)) and chairs (\(C\)) that can be produced while satisfying the time limits in both departments.

### Next Steps:
To find the optimal solution, we should:
1. Identify the vertices (corners) of the feasible region.
2. Evaluate the objective function \(P = 800T + 500C\) at these vertices.
3. The vertex that gives the highest value of \(P\) will be the optimal production plan.

Would you like to proceed with calculating the vertices and solving for the optimal solution?

### Related Questions:
1. What are the coordinates of the intersection points of the constraints?
2. How does changing the profit per table or chair impact the feasible region?
3. Can you determine the optimal solution if a new constraint is added?
4. What if there is a requirement to produce at least a certain number of tables and chairs?
5. How would the solution change if the available hours in one of the departments were increased?

### Tip:
In graphical methods, always verify the feasible region by checking the intersection points of the constraints, as these points usually provide the optimal solutions.

Explore how to graph constraints to find the feasible region and optimize production plans. Understand the intersection points of constraints and evaluate the objective function to determine the optimal production plan for tables and chairs.

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