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? (See Example 1. Let *x* represent the number of tables of style A, *y* the number of tables of style B, and *z* the gross sales in dollars.)

To set up the constraints and objective function for this linear programming problem, we need to identify the decision variables, the constraints based on the problem conditions, and the objective function we aim to maximize.

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

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

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

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

### Objective Function:
We aim to maximize gross sales. The gross sales ($ z $) can be calculated as:
- Each table of style A is priced at $58.
- Each table of style B is priced at $44.

Thus, the objective function to maximize is:
\[
z = 58x + 44y
\]

### Linear Programming Model:
The complete linear programming problem can be formulated as follows:

**Maximize:**
\[
z = 58x + 44y
\]

**Subject to:**
\[
\begin{align*}
x + y & \leq 70 \quad \text{(Storage space constraint)} \\
x + 2y & \leq 105 \quad \text{(Labor hours constraint)} \\
x & \geq 0 \quad \text{(Non-negativity constraint)} \\
y & \geq 0 \quad \text{(Non-negativity constraint)}
\end{align*}
\]

Would you like to see the graphical solution for this problem or have any other questions?

### Follow-up Questions:
1. How do we solve this linear programming problem graphically?
2. What are the corner points of the feasible region, and how are they determined?
3. How can we verify the optimal solution algebraically?
4. What happens if the constraints are changed (e.g., more storage space or labor hours)?
5. How does changing the price of each table affect the optimal solution?
6. Can this problem be solved using the simplex method?
7. What are the applications of linear programming in real-world scenarios similar to this problem?
8. How can we use software tools to solve linear programming problems efficiently?

### Tip:
Linear programming is a powerful tool for optimizing resources under given constraints. Familiarizing yourself with both graphical and simplex methods can significantly enhance your problem-solving skills in optimization tasks.

Learn how to set up constraints and objective functions for a linear programming problem involving a discount store's decision to order damaged coffee tables. Solve using linear programming techniques, including constraints on storage space and labor hours. Suitable for advanced high school students and beyond.

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