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Smarties - Part II

Have you every noticed that when you open a box of Smarties, there seems to be a lot of air in there?

What is amount of air in each box? What percentage of each box is air? What size would the box be if there was no air? Why do they not make the box that size? How many Smarties can you *actually* fit in a box? What do you need to figure this out? Clearly we need some boxes of Smarties to begin collecting data.

We already know how many Smarties are in each box. Now we need to know the dimensions of the box and the dimensions of a Smartie. A Smartie is an oblate spheroid:
We can approximate the volume of a Smartie by considering a cylinder with the Smarties' diameter of 12 mm and height 5 mm.

Using this, we can calculate the Smartie volume of each box:

Now we need to find the volume of each box, then the volume of air:

Next, students can calculate the percentage of air in each box:

Whoa! That's a lot of air. Did I go wrong somewhere???

Students can then design their own box that will minimize the amount of air for a particular number of Smarties. Choose a number of Smarties, determine the volume required then choose/calculate the dimensions of a rectangular prism that will have the required volume. The grand finale would be for them to build their box and see if their Smarties fit!
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