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From: David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz>
Newsgroups: rec.puzzles
Subject: Re: How to Make Cisterns
Date: Sun, 4 May 2025 14:52:05 -0000 (UTC)
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On Sun, 4 May 2025 14:15:57 +0000, IlanMayer wrote:

> SPOILER (Tapered case)

Amazing. Well done. I need to study that.

I certainly couldn't have done that unasisted, but arrived numerically at 
the same volume:

SPOILER (tapered case)
SPOILER
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I can't guarantee this solution, but it looks as though it could be right.

a is the length of the upper edge of the cistern.
x is the length from the corner of the original square sheet to the start 
of the cut.
b is the length of the bottom edge of the cistern.
h is the height of the cistern.

The standard square cut-out max-volume solution:

a = 0.667
x = 0.167
b = 0.667
h = 0.167
vol = 0.0741

The tapering fustrum of a pyramid max-volume solution:

a = 0.782
x = 0.109
b = 0.536
h = 0.197
vol = 0.0864

So, about a 17% increase in volume if the sides are allowed to taper.

-- 
David Entwistle