So my team and i were asked this question a few years ago on a small Math-A-Thon on my hometown. It went something like this:
"We need to transport a neon tube (or any tube, who cares) of 92cm height, as thin as a hair (which means, as close as 92*0*0 as possible, and such, its width and depth are insignificant) to another country, inside of a box. But there's a problem: any box we want to get through Customs must can't be over 55cm*55cm*55cm. The inside of the box is what counts so its thickness is insignificant (either that or it's a magical box with 0 thickness)
1: Can it be done?
2a: If it can, what are the height, width and depth of the smallest (least volume) box that can contain our tube and can go through Customs?
2b: Same as 2a, but only allowing integer values for width, height and depth. (This was the original 2nd question, but i'm interested in both).
My answer back then:
Q1:
Yes. because if you put the tube diagonally you can fit a $ \sqrt{3 *55^2} = 95.26$ cm, and therefore a 92cm one fits perfectly.
Q2b:
Even through "intuitively" i could tell that the idea was to keep 2 values at 55 and lower the third one as much as possible, since "the volume'd decrease faster than the diagonal" i had no way to prove it. So we ended up getting the int. values by trial and error, which are 55cm, 55cm, 50cm, on any order. As for 2a, if that logic's correct, the answer'd be 55*55*49.13247..
My question to you guys is, is my idea to get answer 2 (bold text) correct? And if so, please provide proof that works for both 2a and 2b.