How do I do this problem?

Question

A furniture manufacturer makes wooden tables and chairs. The production process involves two basic types of labor: carpentry and finishing. A table requires 2 hours of carpentry and 1 hour of finishing, and a chair requires 3 hours of carpentry and 1/2 hour of finishing. The profit is $35 per table and $20 per chair. The manufacturer's employees can supply a maximum of 108 hours of carpentry work and 20 hours of finishing work per day. How many tables and chairs should be made each day to maximize profit?

Answer 1

This is a Linear Programming type of problem. Let t = # tables, let c = # of chairs.

 

Profit = 35t + 20c is to be maximized.

 

These inequalities apply:

Carpentry:  2t + 3c ≤ 108

Finishing: t + ½•c ≤ 20

 

These inequalities create a trapezoidal FEASIBLE REGION in the first quadrant, using horizontal axis t and vertical axis c.  The vertices (corners) are the Origin, (20,0), (0,36) and the intersection point I of the boundary lines.

 

Solve this system of simultaneous EQUATIONS to find I:

2t + 3c = 108

t + c/2 = 20.    Point I is (3,34).

 

We know that the optimum value of Profit is at a vertex of the feasible region.  So just compute profit for each of the four listed corner points, and choose the maximum Profit value obtained by those four calculations.