There are many volume-of-a-box questions. I like this one, simpler than what the OP cites:
Given a rectangle, cut out squares from the corners so you can fold it up to a box, without a top, of maximal volume.
The rectangle might be specialized to a square, as below. See also The Math Forum.
[![BoxVol][1]][1]
(Image from [patrickJMT](https://www.youtube.com/watch?v=_oS_LjKse38) YouTube video.)
In response to [@RoryDaulton](https://matheducators.stackexchange.com/questions/14692/list-of-realistic-extremum-problems#comment37319_14692), here is the box problem to which the OP @Student points: [![BoxFlaps][2]][2]
