The Radioactive Cube Investigation

This is a problem that was presented originally by my department chair 21 years ago. I have used it often ever since, but not since I began blogging. It is a great problem at any level in honors or Pre-Calculus. Though some can get it rather quickly, there are definitely some who need it to be broken down. Often, the answer I get without much thinking is "27 cubic units." It is much less. Hopefully, before you read on, you will try the problem. It is great in that it uses a lot of Geometry and nothing that students are likely to be working on at the moment.

I am lucky to have a new course with only seven students. However, four of these students got the answer to this problem rather quickly. The remaining three had the weekend to think about it, but got nowhere. Insert Unifix Cubes. I've had these for a million years.

Students who needed to taped them together like this:

And saw that there were 7 cubes in all that were radioactive...the first cube itself, and the other 6 that radiated from the 6 faces of said cube. So this was 7 cubic units so far.

Then, they cut paper and taped it around the cubes. Students saw that this created 3 4-quarter cylinders, so this added 3 cylinders with radius and height both equal to 1, so π cubic units. These quarter-cylinders came from the edges of the inside cube that radiated radioactivity as a radius of 1 between any two cube faces.

They then realized that they were missing the triangular pieces, shown above. This was from the corner of the middle cube, and it created eight 1/8 spheres, so 1 full sphere with radius 1. Or, 4/3π. 

So the total answer is 7+3π+4/3π, or approximately 20.6 cubic units of radiation.

Some visually talented students drew it like this:

The 7 cubes

The cylinders and spheres built in

Some final results

Students were very proud of themselves either way!