Olympic Puzzle

From think again!

(I’ll be passing this on to my College Algebra students who just completed their Rational Functions chapter.)

What’s the difference?

In the Olympic games in Munich in 1972 the gold medal on 400 m medley went to the Swedish swimmer Gunnar Larsson. Tim McKee got silver. Their respective times were 4 minutes 31.981 seconds and 4 minutes 31.983 seconds.
The gold medal should naturally go to the fastest swimmer. Let me ask two questions.
If Gunnar Larsson’s lane measured exactly 50 m, while Tim McKee’s lane was slighty longer, McKee may have been the fastest of the two. How much longer would the lane have to be for the two swimmers to be equally fast? Assume they swim with constant speed.
Assume sound travels exactly 344.4 m per second in air. How much closer would the start pistol have to be to Larsson for him to hear it 0.002 seconds earlier than McKee?

For the first question, I simply use the fact that we want the rates to be equal and know that rate is distance divided by time. Let $x$ be the extra length of Tim McKee’s lane, thus we have, because of 8 laps, $\displaystyle \frac{8(50)}{271.981 \mbox{sec}} = \frac{8(50+x)}{271.983 \mbox{sec}}$ Thus, $x\approx 3.67 \times 10^{-4} \mbox{m} = 0.367 \mbox{mm}$. Holy Cow, that’s only 367 microns!

Secondly, the question about the gun is simply a question of how far does sound travel in 0.002 sec which is approximately 0.69 m or 69 cm.

That was certainly a close finish. I like one commenters statement, “Perhaps they should move to starting lights instead of starting guns.”