Re: Maximize Cistern Volume -- (cut out 4 squares (at Corners) and discard them)

On 05/03/2025 11:34 AM, HenHanna wrote:

On 03/05/2025  David Entwistle wrote:

Adapted from "Amusements in Mathematics" by Henry Ernest Dudeney.
>
      Given a large sheet of zinc, measuring (before cutting) one metre
square,
you are asked to cut out square pieces (all of the same size) from the
four corners of this sheet. You are then to fold up the sides of the
resulting shape, solder the edges and make a cistern. The cistern is
open
at the top.
>
     The puzzle is :   what size should the cut out pieces be, such that
the
cistern will hold the greatest possible quantity of water?

>
>
Thanks!   (my slight editing may have introduced English Usage  errors)
>
>
___________________
>
>
>
        Ok... i remember High-School  Calculus now.
>
                   V    =   x * (1-2x)^2
>
                   V'   =   12 x^2  - 8 x  + 1
>
I let the derivative be 0 and solve it ,  and i get     x = 1/2,   1/6
>
            at  x=0  the slope is 1
   whereas  at  x=1/2,  the slope is Zero!!!
>
_______________
>
at  x=1/2,  the slope is Zero!!!
>
         It's not obvious why,   Can someone explain this?
>
>
at  x=0  the slope is 1  --------- I can visualize this.
                         It's like a Bean-Sprout,  coming out of the
Earth.
>
>
at  x=1/2,  the slope is Zero!!!  ---- I can't visualize this one.

The cube has the most volume among regular tesseracts,
sort of due the isoperimetric theorem and regular spheres
being incident each face of a reugular tesseract,
and the isoperimetric theorem.
So, if you're making one cistern, wasting the cuts,
whether the units are the width or height, sort of
result pyramidal, while, if you simply make as many
cisterns as possible from one sheet of zinc, they're
a bunch of cubes.
Since water pressure increases in height under gravity,
the shorter cistern would be stronger to hold water
than the same volumn taller cistern.
It's sort of like soup cans vis-a-vis fish tins, or the idea
that cans, to get the most volume of a cylinder, have
at least two solutions.
Then how this cistern will be moved or what hole
it will be put in, and at what height or under what
pressures, also come into play.
You might frame it several ways, starting with
wasting none of the material, starting with
wasting all of the material, then that in the
middle it's cubes.
Usually framed as whether food cans are tall or wide,
while apples and oranges are spherical.