Re: [Phys-L] widget rate puzzle ... reasoning, scaling, et cetera

[John Denker <jsd@av8n.com>]

The question is:

>     It takes 5 minutes for 5 machines to make 5 widgets.
>     So, how many minutes does it take for 100 machines
>     to make 100 widgets? 

Previously I mentioned the importance of checking the
work.  Since I am such a nice guy, I will provide an
example of how to do that.

Here's some advice straight out of Pólya:  If you are
stuck on a problem, try solving an analogous problem.
 -- If you have a specific numerical problem, try
  looking for a general algebraic formula;  sometimes
  that's easier, because it exposes the structure of
  the problem.
 -- On the other side of the same coin, if you have
  an algebraic problem, try doing a few numerical
  examples.  That will help you get the lay of the
  land.
 -- Sometimes it pays to do a problem that is neither
  more general nor less general, just a cousin of
  the one you are trying to solve.

Just as importantly, the same advice applies even if
you are not stuck, for instance if you are looking for
ways to check a successful solution and/or ways of
deepening your understanding of a successful solution.

In that spirit, consider the following transmogrification
of the original question:

     It takes 5 minutes for 5 machines to make 5 widgets.
     So, how many machines does it take to make 100 
     widgets in 100 minutes?

If your reasoning methods are any good, you should be
able to answer the second question just as quickly and
just as confidently as the first.

  On the other hand, if you have any doubts at all
  about your answer to the second question, you
  need to spend some quality time with the first
  question to see if it has any of the same failure
  modes.

I'm dropping some pretty broad hints here.  Y'all
should suspect that (as usual) I'm up to something.
The usual jsd puzzle rules apply:  This is not a word
game.  The answer hinges on getting the physics and
the reasoning right.  I expect to have 100% consensus
on the final answer.  Everything I've said is, so 
far as I know, true and helpful, although not always
maximally helpful.  In particular, I haven't told you
the answer.