Duration of a Clock's Strike

Date: 01/21/2004 at 19:50:30
From: Stacey
Subject: how long does it take for a clock to strike 10

If it takes seven seconds for a clock to strike 7 o'clock, then how 
long does it take to strike 10 o'clock?  It seems as if the answer is
10, but I have worked a problem just like this before and I can't
remember how to start it.

Date: 01/22/2004 at 09:04:23
From: Doctor Peterson
Subject: Re: how long does it take for a clock to strike 10

Hi, Stacey.

The answer depends on how you think of a strike.  If each "bong" lasts 
until the next one starts, and you are counting how long the sounds 
last, then you are right:

strikes     1     2     3     4     5     6     7
         *=====*=====*=====*=====*=====*=====*=====
         +-----+-----+-----+-----+-----+-----+-----+
seconds  0     1     2     3     4     5     6     7

Each strike takes a whole second, and ten of them will take 10 
seconds.

But if you think only of the actual strikes (when the bell is hit), 
and ignore how long the sound lasts afterward, then the picture is 
different:

strikes  1      2      3      4      5      6      7
         *      *      *      *      *      *      *
         +-----+-----+-----+-----+-----+-----+-----+
seconds  0     1     2     3     4     5     6     7

Now there are 6 spaces BETWEEN strikes, which each last MORE than a 
second.

Can you finish the problem, interpreting it that way?

Of course, in reality each bong would take some time that may be more 
or less than the time between strikes, and we don't have enough 
information to determine the total time we hear the clock.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/