The Popcorn Box I am making popcorn and do not have a container anywhere in my house. I search thoroughly and do not find anything but an eleven by eight and a half-inch piece of cardboard. I have decided to make a box with a bottom, four sides, and no top. I will do this by cutting slits into the cardboard and then folding flaps, from these slits, up and over. This will create a box shape. I want to get the most volume out of this container.
After solving this problem using an equation and my algebraic skills, I have decided to use one and a half-inch long slits. The equation I created to find the box's maximum volume was: Volume = (11 - 2x)(8.5 - 2x) x x = length of cut I started my technical process of guess, check, and revise by taking the most obvious cut depths.
My first was one inch.
(11 - 2)(8.5 - 2) 1 (9)(6.5) 1 58.5 inches cubed I then started with 2 inches.
(11 - 4)(8.5 - 4) 2 (7)(4.5) 2 63 inches cubed I tried many other lengths and decided to go with the length of 1.5625 or one and nine sixteenths of an inch.
(11 - 3.125)(8.5 - 3.125) (7.875)(5.375) 1.5625 66.1376953125 inches cubed One and nine sixteenths is the best possible answer for the length of the slits so that the box will hold the largest amount of popcorn.