In this assignment you will design and construct boxes similar to those that Tobler and Droste use to hold chocolate. Each of their boxes contains 100 grams of chocolate. Because the density of the two chocolates are nearly the same, the boxes have nearly the same volume. However, their lengths (and shapes) are not the same.
Design and construct with poster board two boxes. The first box is 15 cm long and has cross sections that are regular hexagons of perimeter 12 cm. These are the dimensions of the Droste box. The second box is also to be 15 cm long. Its cross sections are equilateral triangles having a perimeter 12 cm. This box will be shaped like the Toblerone box but it will be shorter (and a little wider). You are to construct the boxes by first designing a two-dimensional shape in one piece that you will draw on poster board and then build the box by folding and taping.
You will need to calculate the volume of the two boxes. The volume of a box whose cross sections are all the same size is given by
To help you find the cross sectional area, break up the cross section of the first box into three equilateral triangles and that of the second box into six equilateral triangles. Note that these smaller triangles all have the same size (what is the length of the side of the small triangle?). You can then get the cross sectional area by first getting the area of the small equilateral triangle. This has sides of length 2 cm, and its altitude (which can be found from the Pythagorean theorem) is Ö3 cm.
The surface area of each box is the area of the long sides together with the area of the two ends. The area of each end is then the same as the cross sectional area.
On your writeup, draw the two-dimensional shape you found for each box that you used to construct the box.
Problem 1. What are the volumes and the surface areas of the two boxes you constructed? How much bigger is the volume of the hexagonal box than the other box? How much bigger is the surface area of the hexagonal box than the surface area of the other box? For both of these give the percentage difference that the larger is compared to the smaller. Also, explain how you found your answers.
Problem 2. If you want a box with triangular cross sections (of the same size as that of the box you made) to have volume equal to that of the box you made with hexagonal cross sections, how long must it be?