Math question, please help?

The good old magician's "Cube in the Cylinder Trick", eh? ;-)

Ok, seriously:

The fraction or ration of the Cyliner's volume, that the cube takes up, is just:

Volume-of-cube, divided by, the volume of cylinder.

So, first figure the volume of cube, which is the length of its side, cubed: and the cube's edge or side, why they gave it away and simply told us: it's r² (if I'm reading your question aright), so we have to cube that for the volume, and the cube or 3rd power, of the square or 2nd power of r, is the 3-times-2 or sixth power of r -- let's write that as: r^6.

We need to divide that by the volume of the cylinder, so let's find that, it's simple enough, just the area of that circular base, times the height. Circle is πr², and height they gave away as (2r+4), and so this cylinder's volume would be, um, let's see: πr²(2r+4).

We're ready now: divide the cube's volume, by the cylinder's -- and that's:

(r^6) / πr²(2r+4), and two of the r's are cancelling, leaving it simpler, as:

r^4 / π(2r+4), and I guess that's about as simple as I can get it, unless I made a mistake, which you're most welcome to check, of course.

You can see that this answer relies on the cube being entirely inside the cylinder; it doesn't matter where inside, as they're asking us only for the fraction of the two volumes, and we don't need the position to find those. And that's a Very Good Thing! :-)