28. an open top box is to be formed by cutting out squares fromthe corners of a 50 centimeter x 30 centimeter rectangular sheet ofmaterial. The height of the box must be a whole number ofcentimeters. What size squares should be cut out to obtain the boxwith maximum volume.
a. Understanding the problem: If 6 centimeter x6 centimeter squares are cut from from the corners, the height ofthe box will be 6 centimeters. In this case, what would the widthand length of the box be?
b. Devising a plan: One plan for solving thisproblem is to systematically consider corner squares of increasingsize. What is the largest square with whole-number dimensions thatcan be cut from the corners and still produce a box?
c. Carrying out the plan: Complete thefollowing table and use inductive reasoning to predict the size ofthe corner squares needed to obtain the box of maximum volume.
| Size of squares (centimeters) | Volume of box (cubic centimeters) |
| 2x2 | |
| 4x4 | |
| 6x6 | |
| 8x8 | |
| 10x10 | |
| 12x12 | |
| 14x14 | |
d. Looking back: The preceding table shows thatas the size of the squares at the corners increase, the volume ofthe box increases for awhile and then decreases. Try a few moresizes for the squares, using whole numbers for dimensions to see ifyou can obtain a greater volume for the box.
