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distance problem

Posted on 2004-09-17
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Last Modified: 2008-03-06
This a problem that was found on a contest from 1996.

Two swimmers start out from opposite ends (red side and blue side) of a pool.  They cross each other for the first time at line PO which is 32 yards from the red side.  They reach the opposite ends, turn around and cross again at line RS which is 18 yards from the blue side.  Assuming that thye both travel at constant speeds, how many yards is it from the red side to the blue side?
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Question by:djiangr
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rjkimble earned 50 total points
ID: 12089885
Assuming the swimmers swim at a constant speed, we can conclude that the ratio of the distances traveled is constant. If D is the distance in question, then at the first crossing, the  red swimmer has swum 32 yards and the blue swimmer has swum D - 32 yards. At the second crossing, the red swimmer has swum D + 18 yards, and the blue swimmer has swum 2*D - 18 yards. Therefore:

32 / (D-32) = (D+18) / (2*D-18)

A little arithmetic follows:

32 * (2*D-18) = (D-32)*(D+18)

64*D - 576 = D^2 - 14*D - 576

D^2 - 78*D = 0

D = 78 yards

To verify, according to my computations, at the first crossing, the red swimmer has traveled 32 yards and the blue swimmer has traveled 46, a ratio of 16/23. At the second crossing, the red swimmer has traveled  96 yards and the blue swimmer has traveled 138, also a ratio of 16/23. The second set of distances is three times the first set.

The answer seems to be consistent with the conditions of the problem, so I'll guess I got it right. Here's hoping I haven't screwed up the arithmetic somewhere. :-)
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by:wytcom
ID: 12089957
Let L = length of pool (the answer you seek)
t1 = time for both swimmers to cross at PO
t2 = time for both swimmers to cross at RS
vr = speed of swimmer starting at red side
vb = speed of swimmer starting at blue side

Write equations for distance = speed * time

Distance travelled by swimmer from red side to crossing at PO:
  32 = vr t1
Distance travelled by swimmer from blue side to crossing at PO
  L - 32 = vb t1

Distance travelled by swimmer from red side from crossing to crossing at RS
L + 18 = vr t2  

Distance travelled by swimmer from blue side to crossing at RS
L + L - 18  = vb t2

The first pair of equations give vb/vr = (L - 32)/32
Second pair gives vb/vr = (2L - 18)/(L + 18)

So (L - 32)/32 = (2L - 18)/(L + 18)

Solve for L to get the answer...
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by:wytcom
ID: 12089971
I was writing my post while rjkimble submitted his.  I agree with his analysis.
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by:rjkimble
ID: 12090522
Well, this is pretty simple stuff! :-)

It's a Saturday night and I'm bored....
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by:Talmash
ID: 12095120
shorter way :

t1 : 1st meeting , red swimmer passed 32 , both swimmers passed d
t2 : 2nd meeting , red swimmer passed d+18 , both swimmers passed 3*d (!!!)

since sum of velocitied not changed as well as each swimmer's speed =>
t2 = 3*t1

d+18 = 3*32 (the way red swimmer passed  until t2 , is 3 times the way he passed until t1)

d+18 = 96 => d=78 yards

about yesterday night , I had funeral :-(

tal
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by:rjkimble
ID: 12096759
tal: sorry to hear about the funeral.

Very nice observation -- even brilliant. At first I didn't see your point, but after thinking about it, it's a very nice observation. If I could give the points to you, I would. Some solutions are better than others, and in this case your solution is better.
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by:Talmash
ID: 12099979
rj ,

not for the points .

my wife is teaching math at high school and I wrote this question down for her as a very good one ,

I recommended her to give points for a good solution :

1) solve the question using square equation - 10 points ,
2) not using square equations - 15 points ...

"c u" later in other Questions , BTW your profile is realy something .

tal
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