You are given a rectangular sheet of cardboard that measures 11 in. by 8.5 in. (see the diagram below). A small square of the same size is cut from each corner, and each side folded up along the cuts to from a box with no lid.
1. Anya thinks the cut should be 1.5 inches to create the greatest volume, while Terrence thinks it should be 3 inches.
Explain how both students can determine the formula for the volume of the box.
Determine which student's suggestion would create the larger volume.
Explain how there can be two different volumes when each student starts with the same size cardboard.
2. Why is the value of x limited to 0 in. < x < 4.25 in.?
Answers
The length of the square that will be cut out is the height of the box.
1. a
Anya's method: 8.5 -1.5 =7, 11- 1.5 =9.5, the height is 1.5, so the volume is height x length x width which is 1.5 x 9.5 x 7 =99.75 squared inches.
Terrence's method: 8.5-3 = 5.5, 11-3 = 8. Vol= 5.5 x 8 x 3 =132 squared inches. 99.75 < 132 squared inches, Terrence's idea would create larger volume.
1. b
The box's size depends on the length/width/height of the cardboard being cut, which is why different measurements / cutting methods for the same size cardboard can result in different box sizes.
2. The square would be cut from all four corners, therefore the sum of the 2 squares on the cardboard cannot exceed the short side of the cardboard. The shorter side of the cardboard is 8.5 inches, divided by 2 = 4.25 inches, hence the squares cannot be larger than 4.25 inches. Keep in mind that if you cut exactly 4.25 inches you will have a strip of 2.5 inches width that cannot be turned into a box.
If you want to cut 5 inches squares out, depending on how you draw it, it would either overlap or go outside of the paper because 5+5 is ten, surely on the 11 inches side that would still be perfectly fine but for the 8.5 inches side, there isn't any room for the 10 inches.