Climbing Out of a Well

Date: 04/02/2008 at 11:14:56
From: Michelle
Subject: tricky

A well is 100 feet deep. A muscle bound worm climbs 5 feet up towards
the top of the well each day. At night he falls back 2 feet. What day
will he reach the top of the well?  Draw a picture.

Date: 04/02/2008 at 11:26:07
From: Doctor Ian
Subject: Re: tricky

Hi Michelle,

Here's how I'd start.  It's easier to draw left to right, so I'll make
right mean 'up'.  On the first day, he climbs 5 feet:

   0              5
   |--|--|--|--|-->

That night, he falls back 2 feet:

   0        3     5

   |--|--|--|--|-->

            <--|--|
            
So when he wakes up the next morning, he's 3 feet above the bottom of
the well.  The next day and night, he does it all again:

   0        3     5  6     8

   |--|--|--|--|-->

            <--|--|
 
            |--|--|--|--|-->

                     <--|--|

So on the third day, he starts out 6 feet above the bottom of the well. 

Note that you can think of this as a series of smaller problems:

   At the start he needs to climb out of a 100-foot well.
   After one day he needs to climb out of a 97-foot well.
   After two days he needs to climb out of a 94-foot well.
   After three days he needs to climb out of a 91-foot well.
   After four days he needs to climb out of a 88-foot well.
    .
    .

Eventually, you get down to needing to climb out of a well that is 5
feet or less deep.  In THAT well, he gets out during the day, and
doesn't slide down at night.

Does that make sense?  Let me know if you need more help.

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