Thursday, July 01, 2010

Is this solution cheating?

Consider the following problem:

A hole is drilled through the center of a sphere. The cylinder-with-caps is removed. The length of the removed cylinder (it also has caps on it which do not count for the length) is 6 inches. What is the volume of the remaining solid?

There are two ways to do this problem.
  1. Here is the solution using calculus:
  2. Here is a solution which you may consider cheating. The very asking of the question implies that the answer can be determined from the data given. Hence we can CHOOSE an instance of the problem and KNOW that our solution for this instance is always the solution. We choose to have a cylinder of radius 0 (so its just is a line of length 6). Hence the answer is the Volume of a Sphere that is 6 inches in diameter: (4/3)(π)33=36π.
There are two ways this may be considered is cheating.
  1. Minor one: Deriving the volume of a sphere itself requires calculus so I didn't really get around that issue. However, the Volume of a sphere is well known so I think this is a quibble. (Does anyone know a non-calculus proof for the formula for the the volume of a sphere?)
  2. Major one: We used the fact that the answer can be determined from the data to find the answer. Is this appropriate?
How would you grade this if given as an answer on an exam? Here are some thoughts:
  1. If you put this on an exam what would you do if a student had this solution? Reward them for thinking outside the box or penalize them for not showing they know calculus?
  2. What if it was on a mathematics competition?
  3. Best solution might be to make it a multiple choice question so they do not need to show how they did it. Those that think of the clever solution are rewarded by spending less time on it--- unless it took them a long time to think of the clever solution. Those that do it via calculus also get it right. You might want to make one of the choices Cannot be determined from the data given.


  1. Without any further context, 2) is absolutely a legal answer, since it implies that the students had realized the uniqueness of the answer.

    IMHO, this is a bad question if you really want to test whether the student knows calculus, because it allows one to take the short-cut of answering something like 2). A countermeasure is simply stating that the cilinder has a strictly positive volume.

    Otherwise, it is a good question in a GRE-like test (multiple choices or not), seeking for the attitude of the taker toward quantitative reasoning.

  2. If you do not want that someone comes up with the second answer, you could simply modify the question to: "Proof that the volume of the remaining solid does not depend on the diameter of the sphere and the cylinder (and compute this volume)!"

    "(Does anyone know a non-calculus proof for the formula for the the volume of a sphere?)" How does one define the volume of a sphere without using calculus?

  3. I'd rather ask, does anybody know a non-calculus proof that the volume must be independent of the ball radius?

    Or if not, then a less brute-force proof than simply computing the volume and seeing that the expression doesn't contain r?

  4. The Greeks knew the volume of a sphere, AFAIK using Cavalieri's principle (not by that name). Here's a picture (think cross-sections):

    Actually, by Cavalieri's principle, you can solve the whole problem:

    In particular, you can see it doesn't depend on the original radius without using calculus.

  5. Why don't you just ask for the answer in terms of the radius r of the cylinder-with-caps?

  6. @Tyson, the goal of the exam question is to make a blog entry later.

    I feel for those students.

  7. Unless you've stated not to use the known formula, either answer is correct.

    In fact, the second answer is superior, as the responder is clearly pragmatic. I tend to find pragmatism to be a virtue.

  8. @Anonymous Rex

    I always had the impression that the volume of a sphere inherently required 'calculus' in some form.

    Can Cavalieri's principle be proved without 'calculus'?

    Archimedes first had the volume of the sphere; he used rudimentary infitesimals, arguably 'calculus'.

    Obviously, the defn of 'calculus' is ambiguous here.

  9. You can solve the problem by knowing the formulas for the volume of the cilinder, the sphere and the spherical cap and a bit of trigonometry.

    This just avoids knowing directly the triple integrals.

  10. The second answer is correct, but not worth full credit, because it doesn't come with a proof of correctness. Why doesn't the answer depend on the size of the sphere?

    You wouldn't have posed the question if it didn't have an answer!” is NOT valid reasoning, because, actually, yes, I would! (I've deliberately inserted nonsense question into exams, and then given full credit for the answer “WTF? This question makes no sense!!”)

    Yes, I am mean.

  11. Whether this is "cheating" or not depends on the context. This reminds me of multiple choice questions on high school math contests. In that context, solution 2 is a reward for the kind of ingenuity those contests test. (It is only a matter of checking the formula on a suitable instance.) If someone asked this orally I would also consider it OK. For a long written answer I would not be happy without some argument that it is independent of the radius of the sphere (≥ 3) but I could understand someone complaining if they were marked down because the question doesn't say "as a function of the radius of the sphere".

  12. "You might want to make one of the [multiple] choices Cannot be determined from the data given."

    That is really cruel! It forces the examinee to check that the answer can be determined from the data given, instead of getting to take it for granted.

  13. @Anonymous: Cruel? Do you really mean it?
    If you were an engineer building bridges, would you really say -- hey, I didn't know I have to measure the radius ...

    Answer 2) is certainly ingenious, but with no justification it is just a bold guess.

    The proper attitude towards math questions is IMHO this: if some parameter is not mentioned, choose a suitable one and solve a problem with a parameter. If the question does not make sense, find most "similar" question that does and answer that.
    If you are asked something that is false, find the most similar thing that is true. Anything else is just baby-sitting :-)

  14. Would you give credit for the correct answer?

  15. Lance, what about a summary of this year's job market? Who in theory is going for a postdoc or job this year, and where?

  16. I think we cannot even define the volume of a sphere without talking about limits, and allowing limits the area can be found easily. How would you define the volume of a sphere without limits?

  17. The answer is that the problem is indeterminate. Think about taking a rock core. The boring bit has a finite thickness which was not specified.

    I've noticed that many word problems have this kind of flaw. Always drove me crazy on tests.