Further disclaimer: This is my solution. There’s always the chance I could be missing something. If so, I’d love to hear from you!! 🙂
Note first that a 0.5″ diameter gumball would ‘fit’ in a tiny CUBE that is 0.5″ x 0.5 “x 0.5”. So if we tackle the problem with cubes of that size, it’ll not only be easier, it’ll give us a little extra ‘room for error’ on the final solution.
Second, we can see that a cube of side 1 inch will contain EIGHT of the smaller cubes above. (See 1-inch cube figure below.)
Third, since 8 of our ‘gumball cubes’ will fit in one square inch, we only need 5000/8 (or 625) cubic inches of volume.
Finally, of course, there are several ways to get containers with approximately (a little extra room doesn’t hurt!) 625 cubic inches of volume. We could go with a spherical container (which one solution did), but probably it’s more practical (opinion?!) to use a ‘rectangular’ box. Since the volume of a ‘box-type’ figure is length x width x height, we need any three dimensions that will multiply to 625 or slightly more.
For example, a box of dimensions 10 x 10 x 7 (inches) would have volume 700, which is plenty big enough. So is a long, skinny box of 6 x 6 x 18 (748) – though it gives more room than we need (which may or may not be a problem.) If we’re searching for an actual cube, then 9 x 9 x 9 is big enough (729). Interestingly enough, an 8 x 8 x 8 cube only has a volume of 512, but it might (not sure) actually be big enough, as we over-estimated at the start, and the gumballs will ‘settle’ themselves better than cubes will. 🙂
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