http://www.vagueambition.com/napkin-rings/
Here are Todays Amazing Napkin Rings Deals
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More Great Information on Napkin Rings:
weekly wrap up + geometric inlay (Design*Sponge)
Remember the crazy awesome geometrics happening at Anthro? Here are the final
bits and pieces that are my favorite, ranging from gorgeous hardware knobs to
coasters, trivets and bottle openers/corkscrews (which are all resin inlay,
not paint). Using all of this in one room would drive me crazy, but one of
these pieces would be...
you make a napkin ring by drilling a hole through a solid ball. height of napkin ring is H.?
Find volume of napking ring in terms of H only. you can use any calculus method
At first glance is seems you would need more than just H, one might think that the radius of the ball would be needed too.
However upon further thought, it does seem that H alone might be enough, since as the radius of the ball increases (and therefore that of the hole to maintain a constant H) the thickness of the ring decreases while the circumference increases, whether the two exactly compensate, I don't really know.
Since the problem says "Find volume of napkin ring in terms of H only" and does not say prove that that is possible, one could, at some risk of being labels a smart a** take the simplest case where H = 2r (and the hole has a zero radius) in which case the volume is 4/3π(H/2)^3
I don't know how to prove (or disprove) that this works for any other case, but I suspect it might, at least might give you some ideas for further thought.
Since this seemed like an interesting question, I was hoping some one who actually knew the answer would reply, but it's been a while with no answers.
[EDIT]
Another thought, from the formula for a circle; x² + y² = a²
we know that the radius of the hole will be related to the radius of the ball thus, Rhole^2 + (H/2)^2 = Rball^2
Also, maybe helpful (maybe not) is the relationship of a conicsection to the a sphere and a cylinder, see: http://descartes.cnice.mecd.es/materiales_didacticos/ing_redondos/arquime2.htm
Since the volume of the double cone would be 2*1/3π(H/2)^3 and the volume of the napkin ring would be (π Rball^2 * H - 2/3π(H/2)^3) - πRhole^2 * H, it seems to me there is enough information to answer the question, but how to put it into a coherent statement is escaping me, particularly considering the instructions; "you can use any calculus method" where I am totaly lost.