A factory manufactures chairs and tables, each requiring the use of three operations: Cutting, Assembly, and Finishing. The first operation can be used at most 39 hours; the second at most 44 hours; and the third at most 24 hours. A chair requires 1 hour of cutting, 2 hours of assembly, and 1 hour of finishing; a table needs 2 hours of cutting, 1 hour of assembly, and 1 hour of finishing. If the profit is $20 per unit for a chair and $30 for a table, how many units of each should be manufactured to maximize profit?
Amount of chairs: 
Amount of tables: 

This problem is a linear programming problem where we need to maximize the profit based on the constraints for the time available in each operation.

Let's define:
- $ x $ = number of chairs to be manufactured
- $ y $ = number of tables to be manufactured

### Objective Function:
The goal is to maximize the profit $ P $:
\[
P = 20x + 30y
\]
where:
- $ 20x $ is the profit from chairs
- $ 30y $ is the profit from tables

### Constraints:
There are three operations (Cutting, Assembly, and Finishing), each with a limited number of available hours.

1. **Cutting Constraint:**
   A chair requires 1 hour of cutting, and a table requires 2 hours of cutting. The factory has at most 39 hours for cutting:
   \[
   1x + 2y \leq 39
   \]

2. **Assembly Constraint:**
   A chair requires 2 hours of assembly, and a table requires 1 hour of assembly. The factory has at most 44 hours for assembly:
   \[
   2x + 1y \leq 44
   \]

3. **Finishing Constraint:**
   Both chairs and tables require 1 hour of finishing. The factory has at most 24 hours for finishing:
   \[
   1x + 1y \leq 24
   \]

4. **Non-negativity Constraints:**
   The number of chairs and tables cannot be negative:
   \[
   x \geq 0, \quad y \geq 0
   \]

### Solving the Linear Programming Problem:
We now solve the system of inequalities to find the values of $ x $ and $ y $ that maximize the profit.

Let's solve this step by step.The factory should manufacture:
- 9 chairs
- 15 tables

This combination will maximize the profit, yielding a total profit of \$630.

Would you like further details or clarifications? Here are some related questions you might consider:

1. How do we set up linear programming problems in different scenarios?
2. What methods can be used to solve linear programming problems manually?
3. What are some real-world applications of linear programming besides manufacturing?
4. How does changing the profit per chair or table affect the solution?
5. What happens if the available hours for each operation increase?

**Tip:** Always double-check the constraints when setting up a linear programming problem to ensure all real-world limits are respected.

A factory manufactures chairs and tables, each requiring the use of three operations: Cutting, Assembly, and Finishing. The first operation can be used at most 39 hours; the second at most 44 hours; and the third at most 24 hours. A chair requires 1 hour of cutting, 2 hours of assembly, and 1 hour of finishing; a table needs 2 hours of cutting, 1 hour of assembly, and 1 hour of finishing. If the profit is $20 per unit for a chair and $30 for a table, how many units of each should be manufactured to maximize profit?

Solve a linear programming problem to maximize profit in a factory manufacturing chairs and tables. Learn how to set up and solve the system of inequalities for cutting, assembly, and finishing constraints. Ideal for high school students studying optimization.

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