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.)

to recap, so far in my previous post I found the triangle track to have lengths 26,28,30 and this gives it a total length of 84 and area of 336.

I used Excel's Solver function to find the 3 radii of the circles that form the circular arc track, those radii are 6.217946, 6.546281, and 6.873462.  This gives a track length of 102.8511 and area of 411.4051.  So since the triangle track is shorter that means Harry used the triangle track of length 84 ft and Tommy used the arc track of length 411.4051.

Now to find their rates.

If Tommy runs at rate Rt and Harry runs at Rh that means Rh=50Rt

now if harry has to go a distance of Dh=84 and tommy has to go a distance of Dt=411.4051 then their respective times are Th and Tt and these are given by

Th=(Dh/Rh)+60=(Dh/(50Rt))+60

Tt=(Dt/Rt)

and we want Tommy to win by 30 seconds so we have

Th-Tt=0.5 or

(Dh/(50Rt))+60-(Dt/Rt)=0.5

(Dh-50*Dt)/(50*Rt)=-59.5

Rt=(50*Dt-Dh)/(59.5*50)

putting in the known values we get

Rt=1.700355

Rh=85.01777

Tt=60.48803

Th=60.98803

Now this second part solution is based on my assumption that Rod meant for the difference to be exactly 30 seconds because if we only restrict the difference to be between 0 and 30 seconds then the range of possible Rt's with rounding set it to 1.7 and thus Rh=85.0 which with rounding is the same as my solution above anyway, so I guess it kinda does not matter how you look at it when only looking to a single decimal place.

One last note, I did attempt to have Mathematica find a exact solution but as I suspected it was unable to at least after several hours of computation so that is when I decided to go with Excel Solver which I actually find to be more efficient than Mathematica's numerical solver at least when you are not in need of a large number of decimal places.

Posted by Daniel on 2009-03-27 04:22:19