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][3].

<hr />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[![BoxVol][1]][1]
<br />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
<sup>
(Image from [patrickJMT](https://www.youtube.com/watch?v=_oS_LjKse38) YouTube video.)
</sup>
<hr />
<hr />
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]


  [1]: https://i.sstatic.net/c322M.jpg
  [2]: https://i.sstatic.net/t9djz.png
 [3]: https://web.archive.org/web/20171220175135/http://mathforum.org/library/drmath/view/53560.html