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.)
Daniel has essentially solved this problem and correctly identified the necessary dimensions (length and enclosed areas) of the two tracks, and running speeds for Harvey and Tommy. (Although as Daniel infers in his third post, Harvey does not indeed lose by exactly 30 seconds, but actually by less than 30 seconds as was stated. Tommy runs his 102.8511 ft. at 1.7 ft./min in 60.500647 minutes, while Harvey runs his 84 ft. at 85 ft./min. in 0.9882352 minutes, plus the 60 min. nap = 60.9882352, for a difference of 0.487882 min., or 29.255292 seconds.)<p>
As has also been raised, Herron's formula is appropriately used to determine the triangular track area, which can be accomplished with a calculator or any on-line triangle calculator.<p>
In view of the many comments about the difficulty in determining the arc angles and circle radii, I do apologize if I misled anyone about use of a calculator. I simply intended to convey that the geometry itself is fairly straightforward, and that the math can be carried out on a calculator equipped with pi, sqrt, inv and cos functions. That alone, of course, will not be sufficient to accurately calculate specific arc angles or radii, which would require some time consuming trial/error (i.e. iterations) on a calculator that a computer would certainly facilitate. I too am not aware of any simple geometric method that would assist with those two key elements, so again, my apology.<p>
If anyone's interested, I'm happy to post the full formula used in the Solution.
Edited on April 1, 2009, 12:44 pm
Edited on April 1, 2009, 12:45 pm
Complete Solution
