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:

\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.