Set up the constraints and objective function for the following linear programming problem. A discount store is offered two styles of slightly damaged coffee tables. The store has storage space for 70 tables and 105 hours of labor for repairing the defects. Each table of style A requires 1 hour of labor to repair, and each table of style B requires 2 hours. Style A is priced at $58 each, and style B at $44 each. How many of each style should be ordered to maximize gross sales?

To set up the linear programming problem, we need to define the variables, the objective function, and the constraints based on the given information.

### Variables
Let:
- $ x $ be the number of style A coffee tables.
- $ y $ be the number of style B coffee tables.

### Objective Function
We want to maximize the gross sales, which can be represented as:
\[ \text{Maximize } Z = 58x + 44y \]

### Constraints
1. **Storage Space Constraint:**
   The store has storage space for 70 tables in total.
   \[ x + y \leq 70 \]

2. **Labor Hours Constraint:**
   The store has 105 hours of labor available.
   - Each style A table requires 1 hour of labor.
   - Each style B table requires 2 hours of labor.
   \[ 1x + 2y \leq 105 \]

3. **Non-negativity Constraints:**
   The number of tables ordered cannot be negative.
   \[ x \geq 0 \]
   \[ y \geq 0 \]

### Summary
The linear programming problem can be summarized as:

\[
\begin{aligned}
\text{Maximize } & Z = 58x + 44y \\
\text{Subject to:} & \\
& x + y \leq 70 \\
& x + 2y \leq 105 \\
& x \geq 0 \\
& y \geq 0
\end{aligned}
\]

Would you like further details or have any questions?

Here are some related questions:
1. What is the graphical method for solving linear programming problems?
2. How does the simplex method work for solving linear programming problems?
3. What is a feasible region in linear programming?
4. How do you determine the optimal solution in linear programming?
5. What are slack variables in linear programming?
6. How can you check if a linear programming problem is unbounded?
7. What is the dual problem in linear programming?
8. How do constraints affect the feasible region and the optimal solution?

**Tip:** When solving linear programming problems graphically, always identify the feasible region and evaluate the objective function at the vertices of this region to find the optimal solution.

Learn how to set up and solve a linear programming problem to maximize gross sales for a discount store offering two styles of slightly damaged coffee tables. This problem involves defining variables, setting up an objective function, and applying constraints on storage space and labor hours. Suitable for college-level students and professionals.

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