Expert: Scott A Wilson - 2/5/2011

Question
Hi, I'm having trouble with this question...can you please help me?

"You may have heard the fable about the tortoise and the hare.  Suppose the tortoise and the hare are running a 2000 m race.  The tortoise proceeds very slowly, never changing its speed.  The hare runs very quickly at the start.  The tortoise travels 20 m every minute.  The hare travels 1000 m in the 1st minute but, in each minute after that, travels only half of the remaining distance (in the 2nd minute, the hare travels half of the remaining 1000 m, and so on)."

QUESTION:  Analyze and compare the two series.  Include information such as S(n) for both the tortoise and the hare, and your conclusion as to who will win the race.  Provide support for your analysis.

***Any help you can give me on this question is greatly appreciated!

Answer
I put the problem in Excel.  The 1st column is minutes passed,
the 2nd column is how far the tortoise went,
and the 3rd column is how far the hare went.

By math, the hare will never reach the finish line,
therefore the tortoise should win.

When rounding, however, the hare will have gone 2000 m in 13 minutes to the nearest meter,

After 12 minutes, the tortoise has gone 230 m, but the hare has gone 2000 to the nearest meter,
but that is only 1999.5 when a decimal is added.  That is equivalant to a little over 19 inches.

After 16 minutes, the hare has gone 2000.0 meters, but that is only 1999.97 meters.
That is the equivalant of a little under 1 1/4 inches left.

After 18 minutes, the hare has gone 2000.00 meters, but that is only 1999.992.
That is slightly under 3/8 of an inch.

After 20 minutes, the hare has gone 2000.000 meters, but that is only 1999.995 meters.
That is a little under 5/64 of an inch to go.

After 26 minutes, the hare has gone 2000.0000 meters, but adding one more decimal accuracty again, it is really 1999.99997 meters.  In English again, that is a little over 1/1024 of an inch to go, and 1/1024 = 1/2^10.

I'm not sure how well it aligns by column, but here is the data:
minute   tort'se   hare   m left   cm left   in left
min   tort   hare   m left   cm left   in left
1   20   1000   1000   100000   39370.0787
2   40   1500   500   50000   19685.0394
3   60   1750   250   25000   9842.5197
4   80   1875   125   12500   4921.2598
5   100   1938   62.5   6250   2460.6299
6   120   1969   31.25   3125   1230.3150
7   140   1984   15.625   1562.5   615.1575
8   160   1992   7.8125   781.25   307.5787
9   180   1996   3.90625   390.625   153.7894
10   200   1998   1.95313   195.313   76.8947
11   220   1999   0.97656   97.6563   38.4473
12   240   2000   0.48828   48.8281   19.2237
13   260   1999.8   0.24414   24.4141   9.6118
14   280   1999.9   0.12207   12.2070   4.8059
15   300   1999.9   0.06104   6.1035   2.4030
16   320   2000.0   0.03052   3.0518   1.2015
17   340   1999.98   0.01526   1.5259   0.6007
18   360   1999.99   0.00763   0.7629   0.3004
19   380   2000.00   0.00381   0.3815   0.1502
20   400   2000.00   0.00191   0.1907   0.0751
21   420   2000.00   0.00095   0.0954   0.0375
22   440   2000.00   0.00048   0.0477   0.0188
23   460   2000.00   0.00024   0.0238   0.0094
24   480   2000.00   0.00012   0.0119   0.0047
25   500   2000.00   0.00006   0.0060   0.0023
26   520   2000.00   0.00003   0.0030   0.0012