What is the Circumference of the Reservoir?

Date: 04/09/2003 at 12:25:11
From: Lisa
Subject: Finding the circumference of a circle

Dear Dr. Math,

I saw this problem in a forum on the Internet; we know what the answer 
(supposedly) is (it was posted as 480 yards), but so far no one has 
explained how they arrived at that answer. That said, it has been a 
long time since I used this type of math and I am stumped. I am 
looking for the formula/equation(s). Can you help?

John and Tina are standing at opposite sides of a round reservoir, on 
the path. At the same time, they start running around the path in 
opposite directions. They meet for the first time after John has 
traveled 100 yards, and they meet again 60 yards before Tina has
completed her first lap. What is the circumference of the circular 
path around the reservoir?

One poster states: John has run half a circumference less 40 yards (as 
he was 100 yards past his start point the first meeting, and 60 yards 
past the opposite point the second meeting). And Tina has run half a 
circumference plus 40 yards. So we know after two meetings, Tina has 
done 80 yards more than John. Assuming constant speed by both, it 
follows that after one meeting, Tina has done 40 yards more than Jon -
and we know at that time John had done 100 yards. So while John had 
done 100, Tina must have done 100 + 40 = 140 yards. Added together we 
get the value for half a circumference, 100 + 140 = 240. So a whole 
circumference is 480 yards.

Maybe so, but John has not run half a circle less 40 yards, he has 
actually run half a circle plus 60; he HAS YET to run half a circle 
less 40. So how do you get an understandable equation out of this? I 
am confused.

Thanks for your consideration,
L. Owens 

Date: 04/09/2003 at 13:34:14
From: Doctor Douglas
Subject: Re: Finding the circumference of a circle

Hi Lisa

Thanks for writing. I agree, this type of word problem can be very 
confusing. But things are helped with a diagram: 
          

        A     .          John starts at J and goes clockwise
     .           .       Tina starts at T and goes counterclockwise
    .             .
    J             T      first meeting is at A  (J to A is 100 yd)
    .             .      second meeting is at B (B to T is 60 yd)
     .           B       
        .     X          J is opposite to T, X is opposite to A    


The poster above reasons as follows:

From meeting 1 to meeting 2, John runs d(AX)-d(BX) = C/2 - 40. This is 
because J is opposite T and A is opposite X, so d(TX) = 100, d(TB)=60, 
and d(BX) = 40. The confusion stems from the poster not saying what 
interval is being considered; the poster is saying that John runs a 
distance C/2 - 40 (and Tina runs C/2+d(BX) = C/2+40) during the time 
between the first and second meeting. Thus in the time interval 
between meetings A and B, Tina runs 80 yd more than John.

Here's how I would proceed (not necessarily using the information
above) with this problem:

  At meeting B, Tina has run a total of C-60 yards from the start.
                John has run a total of C/2+60 yards from the start.

  At meeting A, Tina has run a total of C/2-100 yards from the start.
                John has run a total of 100 yards from the start.

The rates to accomplish this should be consistent. That is, we must
find C such that the ratio of their running rates is constant:

                  start to B          start to A
                ---------------     -------------
  Tina/John:    (C-60)/(C/2+60) ?=? (C/2-100)/100

We are solving for C:  

   100 (C-60) = (C/2-100)(C/2+60)
   400 (C-60) = (C-200)(C+120)
 400C - 24000 = C^2 - 80C - 24000
    400C      = C^2 - 80C
      0       = C^2 - 480C
      0       = C(C-480)
     
So either C=0 (which is nonsense) or C=480 yd. Now let's check this, 
using the expressions for the distance run by each of the individuals:

                       John      Tina
  start to A:           100      140      ratio is 1.4
  start to B:           300      420      ratio is 1.4

So our arithmetic is vindicated. Note that the distances run between 
A and B are 200 yd (John) and 280 yd (Tina). The ratio is still 1.4, 
and we verify that over this interval between A and B, Tina does run 
80 yds more than John.

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

Date: 04/09/2003 at 15:04:35
From: Lisa
Subject: Thank you (Finding the circumference of a circle)

Thank you very much for your quick reply and thorough explanation. I 
think I understand this now; at any rate, it's good exercise for a 
part of my brain that is too seldom used these days!

Lisa