*h*

^{3}/3, where

*h*is the half-height of the resulting solid; this is the volume of a sphere of radius

*h*. In fact, the corresponding cross-sectional areas of a "napkin ring" of half-height

*h*and a sphere of radius

*h*are the same, as one can see without calculus; by Cavalieri's principle the volumes are equal.

This is a problem that is usually assigned in our calculus courses when we get to solids of revolution. The way that it's phrased in Stewart's calculus text (which I don't have at hand right now) is in two parts. The first part refers to two such napkin rings -- call them A and B -- and says that napkin ring A comes from a larger-radius sphere than B but also has a larger-radius hole; the student is asked to guess whether A or B has greater volume.

At this point the students invariably ask "what if my guess is wrong?" I laugh and tell them that it doesn't matter; the point is that they should try to think about it for a moment before they plunge into the calculations.

The second part, if I remember correctly (my officemate has the book, so I'll check this tomorrow) asks the students to express the volume as a function of the two radii and the height; often, even if they manage to get the correct answer (which is a bit tricky) they do not seem

*surprised*by this fact, which I was when I first saw it. Then again, it is often difficult to judge whether students are surprised by some statement made in class or on the homework; it's not the sort of thing your average college student is going to let show! (Penn students taking the calculus courses are mostly pre-professional, either in the engineering or business schools; I wonder if the future mathematicians or even the future "pure" scientists would react differently.)

(Incidentally, it is always amusing to look at reviews of textbooks at amazon.com; I often wonder how many of them are written by students who are disgruntled about their class and want somewhere to take it out.)

## 7 comments:

Argh, I don't have time to be playing with calculus right now! Shame on you and your interesting math concepts!

I believe I was surprised when I first encountered this.

In an attempt to develop an intuition about the problem, I think it's interesting to imagine a sphere of clay with a tiny hollow pin through a diameter (the radius of the pin is almost zero). Next suppose that the pin has some clever special construction so that it can expand from within to a cylinder of arbitrary radius. Imagine further that the clay adheres to the pin/cylinder at both points where the pin penetrates it.

It's not too hard to imagine that as the clay "napkin ring" stretched and deformed into a "napkin ring" around the expanding cylinder, it would resemble the residue of a sphere that had stared with whatever radius the napkin ring ends up having. And of course the clay would keep its volume, due to conservation of ass and a reasonable supposition that the density shouldn't change.

Of course these kinds of appeals to imagination can be used nearly equally easily to convince your of something that's either false or true, but at least I hope it shows that it really doesn't totally defy intuition how the napkin ring problem works out, after a little meditation on it.

Oh, I learned a similar thing from Martin Gardner's "aha!" books. The area between two concentric circles depends only on the length of the chord in the big circle that's just tangent to the small circle.

i *just* finished teaching volumes in my calculus class. guess what problem we're going to be working on tomorrow?

thanks!

My students also struggled with this problem earlier in the semester. I haven't watched it recently (so I'm not sure how coherent it is), but I recorded this video live in class to help the online students with the problem.

This was a question in my calculus exam in the last year of high-school. Nobody had been taught the necessary pre-requisites, and I don't think anyone got it right!

This problem is somewhat simpler if you just compute the volume of a h-cyclinder in a r-sphere. The answer is a function of h, but not of r.

Ref:

-- L. A. Graham, Ingenious Mathematical problems and methods, 34 p.23

-- T. M. Apostol, A Century of Calculus II, p.321

-- C. W. Trigg, Mathematical Quickies, 217 p.59

-- David Wells, The Penguin Book of Curious and Interesting Puzzles, 323 p.109

-- Martin Gardner, Hexaflexagons and other..., 7 p.113

-- Howard Eves, Great Moments in Mathematics Before 1650, p.210

-- G. Polya, Mathematics and Plausible Reasoning (Vol.I), p.191, 11.5 p.200

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