Analysis of the Furniture Problem

(for evaluators only)

The following analysis represents 3 approaches and things to look for in team responses using these approaches. It explains the theory behind the solution. The graph in figure 1 shows how to determine whether a solution is valid. The scoring guide s hows how to determine a score.

- The number of hours that all the carpenters work per week is 40 x 100 = 4000 hours.
- The number of hours that all the finishers work per week is 40 x 32 = 1280 hours

If we let

C = the number of chairs produced in a week, and

T = the number of tables produced in a week,

then we come up with the following equation for the number of carpenter hours:

4 C + 8 T = 4000 hours *(Equation 1)*

and the following equation for the number of finisher hours:

2 C + T = 1280 hours. (*Equation 2)*

These two equations graph as shown in Figure 1. The solid, heavy line represents the limit of what the carpenters and the finishers can do together. The best solution will be on this line. The equation for the amount of profit the company earns is

P = 30 C + 40 T (Equation 3)

Computing the profit for several points on this line produces this table:

Number of chairs (C) |
Number of tables (T) |
Profit ($) |

0 |
500 |
$20,000 |

640 |
0 |
$19,200 |

520 |
240 |
$25,200 |

So, the maximum profit is the point where the two graphs cross, at 520 chairs and 240 tables.