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.