Answer
0 chairs, 36 tables
Work Step by Step
Let x be the number of chairs made and y be the number of tables made. Since profits are 35 dollars per table and 20 dollars per chair, we want to maximize f(x) = 35y + 20x.
Since a chair takes 2 hours of carpentry, a table takes 3 hours of carpentry, and we have 108 hours to work with, the total time spent on carpentry on x chairs and y tables is 2x + 3y, and it must not exceed 108. Thus, 2x + 3y $\leq$ 108.
Likewise, a chair takes 1 hour of finishing, a table takes 1/2 hour of finishing, and we have 20 hours to work with, so total finishing time on x chairs and y tables is x + $\frac{1}{2}$y, and it must not exceed 20. Thus, x + $\frac{1}{2}$y $\leq$ 20.
Finally, x $\geq$ 0 and y $\geq$ 0, since we can't have a negative amount of chairs. With these constraints, we can graph:
Since maximum occurs at the corners of the shared areas, we look at the coordinates of the corners, which are (0,0), (20,0), (0,36), and (3,34). Now, we plug in these x and y values in
f(x) = 35y + 20x and find the largest answer.
(0,0): f(x) = 35(0) + 20(0) = 0
(20,0): f(x) = 35(0) + 20(20) = 400
(0,36): f(x) = 35(36) + 20(0) = 1260
(3,34): f(x) = 35(34) + 20(3) = 1190 + 60 = 1250
Since (0,36) gives the highest value, x = 0 and y = 36 gives the highest profits to the company, or producing 0 chairs and 36 tables.
