## Tuesday, December 27, 2011

### PACING YOURSELF - The Answer Key!

In our last titillating post, we briefly discussed the ways in which different runners pace themselves to arrive at the finish line of a 5 Kilometer race. We choose three mathematicians to compete, each one using his own approach to pacing the his way to the finish line. The award for the fastest and first to arrive at the finish (amid cheers, popping champagne corks, autograph requests and other of the accoutrements accompanying a heroic feat) was a free XXL Tee- Shirt (with the event sponsor's name, Tanks-A-Lot Cesspool Cleaning Service, "We Keep You Running". Here we go....
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Since we are Braintenance, we will choose a five kilometer race, in which three mathematicians are competing. Each has a distinct running pattern, strategy or (ulp!) formula for spanning the distance.

The question to answer regarding each runner is simply this: "How long will it take him (in minutes) to reach the finish line to receive his souvenir Tee-shirt?" Of course, these Ts are only available in one size...XXL.

Note: When we speak of percentage increases or decreases, each such increase or decrease is based upon the initial pace. This is not one of Douglas E. Castle's infamous "compound interest meets calculus" problems.

Here's the lineup of our contestants:

Racer 1:  This fellow runs (on average) one kilometer every 18 minutes.

He runs at a steady pace of one kilometer (British? Kilometre!)  every 18 minutes. The calculation is simple 18 minutes x 5K = 90 minutes.

Racer 2:  This fellow starts the race at a rate of one kilometer every 12 minutes, but his pace declines by 10% per kilometer.

This fellow starts out like a soldier with dysentery, but slows down at a constant rate:

1st Kilometer =12 minutes.
2nd Kilometer =13.2 minutes
3rd Kilometer =14.4 minutes
4th Kilometer =15.6 minutes
5th Kilometer =16.8 minutes

If we add the time it took him to conquer each kilometer, his total time was 72 minutes.

Racer 3:  This fellow starts the race at a rate of one kilometer every 30 minutes (remember the tortoise and the hare?), but his pace increases by 15% per kilometer.

Here's this fellow's pacing pattern:

1st Kilometer = 30 minutes
2nd Kilometer = 25.5 minutes
3rd Kilometer = 21 minutes
4th Kilometer = 16.5 minutes
5th Kilometer = 12 minutes

If we add up this fellow's time (he's apparently a 'late bloomer'), we arrive at a total time of 105 minutes. It is interesting to note that he ran his last kilometer at the same pace at which the second runner ran his first.

And the winner was runner #2 (ironic, considering the name of the sponsor).

Now that we've learned (unknowingly!) about rates of decay, rates of acceleration and how slow most mathematicians tend to be as runners, we can leave the realm of racism, and graduate to more exciting cerebral challenges.

Douglas E. Castle for BRAINTENANCE