Suppose that a furniture manufacturer makes chairs, sofas, and tables. Consider the following chart of labor hours required and available. $\begin{array}{lcccc} & \text { Chair } & \text { Sofa } & \text { Table } & \text { Daily labor available (labor-hours) } \\ \text { Carpentry } & 6 & 3 & 8 & 752 \\ \text { Finishing } & 1 & 1 & 2 & 148 \\ \text { Upholstery } & 4 & 6 & 0 & 416\end{array}$ The profit per chair is $\$ 70$, per sofa $\$ 50$, and per table $\$ 100$. How many pieces of each type of furniture should be manufactured each day to maximize the profit? chairs, $\square$ sofas, and $\square$ tables should be manufactured each day to maximize the profit.

Step 1 ：Let's denote the number of chairs, sofas, and tables to be manufactured each day as x, y, and z respectively.

Step 2 ：The profit function to be maximized is \(P = 70x + 50y + 100z\).

Step 3 ：The constraints are given by the labor hours:

Step 4 ：Carpentry: \(6x + 3y + 8z \leq 752\)

Step 5 ：Finishing: \(x + y + 2z \leq 148\)

Step 6 ：Upholstery: \(4x + 6y \leq 416\)

Step 7 ：We also have the non-negativity constraints: \(x, y, z \geq 0\).

Step 8 ：We can solve this problem using a linear programming solver.

Step 9 ：The output from the linear programming solver gives us the optimal number of chairs, sofas, and tables to be manufactured each day to maximize the profit.

Step 10 ：The optimal solution is to manufacture 86 chairs, 12 sofas, and 25 tables each day.

Step 11 ：Final Answer: \(\boxed{86}\) chairs, \(\boxed{12}\) sofas, and \(\boxed{25}\) tables should be manufactured each day to maximize the profit.