The question describes a linear programming scenario for a furniture manufacturer with constraints on assembly, finishing, and inspection times. Here's a breakdown of the problem:

Key Information:

1. Products and their requirements:

   • A table requires 3 hours of assembly, 2 hours of finishing, and 1 hour of inspection.
   • A chair requires 2 hours of assembly, 2 hours of finishing, and 1 hour of inspection.

2. Time availability per week:

   • Assembly: 200 hours
   • Finishing: 180 hours
   • Inspection: 40 hours

3. Variables:

   • $T$: Number of tables produced per week.
   • $C$: Number of chairs produced per week.

4. Profit:

   • Table: $120 profit.
   • Chair: $80 profit.

Goal:

Identify the necessary constraint for production based on the provided options.

Solution:

We need to check the constraints of resources against their usage:

1. Assembly time constraint:

   • Each table uses $3T$ hours, and each chair uses $2C$ hours.
   • Total assembly hours $\leq 200$: $3T + 2C \leq 200$

2. Finishing time constraint:

   • Each table uses $2T$ hours, and each chair uses $2C$ hours.
   • Total finishing hours $\leq 180$: $2T + 2C \leq 180$

3. Inspection time constraint:

   • Each table uses $1T$ hour, and each chair uses $1C$ hour.
   • Total inspection hours $\leq 40$: $T + C \leq 40$

Matching Constraints to the Options:

From the question, one of the provided options is: $T + C \leq 200$

While this matches the general idea of total production constraints, it does not correctly reflect the stricter inspection time constraint of $T + C \leq 40$. Thus, the correct constraint should be $T + C \leq 40$, not $T + C \leq 200$.

If you'd like to dive deeper or ask further questions about constraints, let me know!

Expanded Questions:

1. What are the implications of violating any of the constraints in a linear programming problem?
2. Can you explain how to graph these constraints on a coordinate plane?
3. How would the profit equation $120T + 80C$ interact with these constraints?
4. How do slack variables fit into this scenario for unused resources?
5. Could increasing available hours for one resource (e.g., assembly) improve production efficiency?

Tip:

Always ensure constraints are checked against their respective resource limitations to avoid oversimplification!