How can you forebode that amount of wood in a napkin ring is the same, regardless of the size of the sphere used ?
Stewart, Clegg, Watson. Calculus Early Transcendentals, 2021, 9th edition. page 467. Problem 64.
When I attempted this question, I thought I flubbed, because my answer lacked R and r! Even after seeing the solution, I still can’t intuit why the napkin ring’s volume is independent of R and r. Before attempting any paperwork, how can a student forefeel that the napkin ring’s volume shall be independent of R and r ?
1 answer
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User | Comment | Date |
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Sunny | (no comment) | Jun 20, 2024 at 18:12 |
(a) Guess which ring has more wood in it.
The word "guess" shows that this is not necessarily intended to be guessed correctly. The guess is asked for precisely because the result is unintuitive, and surprising to most people. Writing down your guess emphasises the contrast between what you initially expect, and what you later calculate exactly (in part (b)).
If you wanted to discuss ways of potentially anticipating this kind of unexpected result with students (after they have made their own attempt at guessing), you could talk about methods of getting a quick insight without fully calculating the result.
For example, with the height $h$ held constant, the larger the ring the thinner it must be. This can be seen to some extent from the two rings in the image. This initial intuition can be confirmed by noting that the thickness of the ring can be made arbitrarily close to zero by making the ring larger. This does not yet confirm that the volume will be invariant, but it hints that there is no immediate reason to expect either a small or large ring to have greater volume.
The first half of the question, asking for a guess, appears to be asking for this kind of rough idea, rather than a full calculation, so the result does not need to be conclusive (or even correct). If a student were to give a conclusive proof then I would think they have missed the point of the question.
Intuition from practical objects
Having made the guess, and then been surprised (as intended) by the calculated result, a feeling for why this is surprising can be gained by thinking about how the mathematical result differs from what is possible in reality.
Mathematically, you can calculate the thickness of the ring ($R-r$) in terms of $r$, and see that you can get as close as you like to zero thickness by choosing sufficiently large $r$. However, a ring made of wood cannot be arbitrarily thin due to being made of molecules. Below a certain thickness it will no longer be wood, and long before that it will no longer be strong enough to hold its own shape. In order to get a good intuition for the geometry, it is necessary to let go of the materials based intuitions we have for how thick a ring would need to be to be useful.
Intuition from familiar shapes
The two rings in the image look fairly similar. It's easy to mistakenly expect much larger versions to look bigger but still fairly similar. Personally, my immediate visualisation of a very large ring was not consistent with a cylinder removed from a sphere, but instead a cylinder removed from a torus. This gives a shape that looks more familiar even for arbitrarily large rings, but it gives a misleading intuition about volume. A cylinder removed from a torus would allow the thickness to stay constant as the radius r becomes arbitrarily large, leading to arbitrarily large volume rather than invariant volume.
A guessing question such as this is useful for helping find such misleading tendencies in our intuition.
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