Question

A division of the Winston Furniture Company manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $40, and the profit for each chair is $30. In a certain week, the company has 3200 board feet of wood available and 520 labor-hours available. How many tables and chairs should Winston manufacture to maximize its profit? (Let x represent the number of tables Winton manufactures and let y represent the number of chairs they manufacture.)

(x, y) =

What is the maximum profit? $

Answer 1

Let x and y represent the number of tables and chairs manufacture by Winton respectively.

Now x tables will require 40x board feet of wood and 3x labor-hours and y chairs will require 16y board feet of wood and 4y labor-hours. Now, since, the company has 3200 board feet of wood available and 520 labor-hours available, hence 40x+16y ≤ 3200 or, 5x+2y ≤ 400…(1) and 3x+4y ≤ 520…(2)

Further, the profit for each table is $40, and the profit for each chair is $30 so that Winton’s profit for the week is P(x,y) = 40x+30y.

A graph of the lines y = -(5/2)x +200 ( in red) and y = -(3/4)x+130 ( in blue) is attached. The feasible region is on and below both these lines. The 2 lines intersect at the point (40,100) so both the conditions (1) and (2) are satisfied when x = 40 and y = 100.

Thus, the maximum profit will be $ (40*40+30*100) = $ 4600.