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28. an open top box is to be formed by cutting out squares from the corners...

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Basic Math

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.

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