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.)

let the first leg be x and the second be x+2 and then the third is x+4.  Using Heron's formula it is a simple matter to solve for when the area is 4 times the perimeter and that is when x=26 so the lengths of each leg are 26,28, and 30 and this is actually a right triangle.

Gotta go run errands now, I plan on solving the rest when I get back :-)

 

Posted by Daniel on 2009-03-26 12:30:37