Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site utastro.UUCP
Path: utzoo!linus!decvax!harpo!floyd!cmcl2!philabs!seismo!hao!kpno!ut-sally!utastro!bill
From: bill@utastro.UUCP
Newsgroups: net.math
Subject: Re: Everything you know is wrong!
Message-ID: <687@utastro.UUCP>
Date: Mon, 3-Oct-83 14:38:17 EDT
Article-I.D.: utastro.687
Posted: Mon Oct  3 14:38:17 1983
Date-Received: Sun, 9-Oct-83 18:25:14 EDT
Organization: UTexas Astronomy Dept., Austin, Texas
Lines: 64

med to say) until I found that you get the same 
ratio for a cube:

	Let a cube be circumscribed about the same sphere, so
	that its sides have length 2*r.  Then the ratio of surface 
	area to volume is

	ratio = (6* area of 1 side)/volume

	      = (6 * 4 * r^2)/(8*r^3) = 3/r!!!

When I calculated this I saw the fallacy in Lew's argument.  The
quantity he calculates is not dimensionless, so it is not
a legitimate ratio for comparison.
What we ought to do is to compare the surface areas of a
sphere and a cylinder *whose volumes are the same*.  Then we
can tell which one can be manufactured with the least amount
of "skin".  Thus:

	Let the radius of the sphere be r, and the radius of
	the sphere inscribed within a cylinder of the same
	volume be s.  Then

	Volume of sphere = volume of cylinder

	(4/3)*pi*r^3 = 2*pi*s^3

	s = (2/3)^(1/3)*r

	Now compare the surface areas of the two solids:

	Area of cylinder/area of sphere =

		6*pi*s^2/(4*pi*r^2)

		= (3/2)^(1/3);
	
	So the surface area of the cylinder is about 1.14 times 
	that of a sphere of the same volume.

I think that Lew's "ratio" is very cute, and a little curious.
It's a lovely swindle.  Where did you get it, Lew?

	Bill Jefferys  8-%
	Astronomy Dept, University of Texas, Austin TX 78712   (Snail)
	ihnp4!kpno!utastro!bill   (uucp)
	utastro!bill@utexas-11   (ARPA)