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The Hare and Tortoise Rematch (Posted on 2009-03-26) Difficulty: 3 of 5
Still smarting from his rather embarrassing loss, Harvey the Hare appealed to Aesop for a rematch against Tommy the Tortoise. After giving it much thought to make any such race as fair as possible, knowing that Harvey can run exactly 50 times as fast as Tommy, Aesop agreed to the new race but only under the following odd and seemingly unfair conditions:

1. This time, each would start/finish the race at the same point, but would run on separate (occasionally intersecting) courses of different lengths. Harvey's course consists of three straight legs, while Tommy's course runs along the external arcs of 3 tangent (non-overlapping) circles having radii greater than 5 and less than 10 feet.
2. The second leg of each course was 2 feet longer than the first leg, while the third leg for each was 2 feet longer than the second.
3. The "infield" areas enclosed by each course in (square) feet were both 4 times the total length of the respective course in feet.
4. But most surprisingly of all, Aesop insisted that Harvey got to run the shorter of the two courses!!!


Note that the graphic is not to scale.

What were the lengths of Harvey and Tommy's respective race courses, and the area enclosed by each, in feet?

Bonus question!

Race day arrived, with throngs of story book characters arriving to cheer along their favourite runner (although Rip van Winkle did sleep through the whole thing, and Sneezy was laid up at home with a cold!)

Mother Goose as the Official Starter popped a balloon and the two were off! Not surprisingly, Harvey was away like a shot and in true fable form, given his superior speed, decided that a nap along the way would once again be in order. After snoozing for exactly 60 minutes, however, the roar of the crowd cheering Tommy along to the finish line woke Harvey up, but despite his best efforts, he lost the race yet again, this time by less than 30 seconds!

To one decimal place, what were Tommy and Harvey's running speeds in feet/minute?

(Hint: Since this is, after all, a fable, you can ignore any acceleration, topography, wind, "cornering" issues, etc. Just assume that when they run, both Tommy and Harvey run flat out at a constant speed.)

See The Solution Submitted by rod hines    
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Comments: ( Back to comment list | You must be logged in to post comments.)
Question re: Complete solution | Comment 9 of 15 |
(In reply to Complete solution by Daniel)

Daniel,

Adding to the comment by brianjn, could you please list the equations you used that the Excel's Solver function solved.

Thanks.

bractals

I was playing around with Geometer's Sketchpad and could get a configuration that satisfied area = four times arc length, but I could never get one that also satisfied the difference of two in the arc lengths.

Edited on March 27, 2009, 12:21 pm
  Posted by Bractals on 2009-03-27 12:15:43

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