ntt@dciem.UUCP (ntt) (08/16/83)
I assume the item referred to was intended as a joke or puzzle!
Certainly, it is true that for the objects specified we have:
sphere: volume = (4/3)*pi*r^3 surface = 4*pi*r^2
cylinder: volume = (6/3)*pi*r^3 surface = 6*pi*r^2
since the cylinder has h = 2*r.
And it is true that the surface/volume ratios are numerically equal to 3/r.
But the objects being compared are not equal in volume, nor in area (if r>0).
The standard claim about the sphere having the smallest surface/volume
ratio refers to comparisons among objects of equal volume (or of equal
surface). To equate the volumes in this case we must see that r has
different meanings for the sphere and the cylinder, and force
r[sph]/r[cyl] = (3/2)^(1/3)
And then 3/r[sph] is not equal to 3/r[cyl].
Alternatively we can equate surface areas, in which case 1/2 replaces 1/3.
Mark Brader, NTT Systems Inc., Toronto (decvax!dciem!ntt)ntt@dciem.UUCP (ntt) (08/17/83)
Ouch, got my own address wrong. Should be: decvax!utzoo!dciem!ntt