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Chuckles · 10-04-2005, 11:10 PM

Distance walked by Tortoise Total length of the band
1. 3
2. 6
3. 9
4. 12

General _expression for step x:
x 3(x + 1)
Ratio of distance walked to the total length of the band:
x/3(x + 1)

If the band continues stretching in this (incredible) manner, the tenacious tortoise will never reach the end of the band and will never get any nice flower. Instead, the furthest the tortoise will ever get as the total distance of the band approaches infinity, as a ratio of distance walked to the total length of the band, is calculated as follows:
Limit as xinfinity of x/3(x + 1)
Multiply each term by 1/x in numerator and denominator
= Lim x infinity x/x / (3x/x + 3/x)
= Lim xinfinity 1 / (3 + 3/x)
= 1/3
So the poor old tortoise will never cover more than 1/3 of the distance to Achilles and the flower. This assumes that the band stays fixed to the ground at the point where the tortoise starts.

10-04-2005, 11:10 PM	#9 (permalink)
Chuckles Insane Location: California	Distance walked by Tortoise Total length of the band 1. 3 2. 6 3. 9 4. 12 General _expression for step x: x 3(x + 1) Ratio of distance walked to the total length of the band: x/3(x + 1) If the band continues stretching in this (incredible) manner, the tenacious tortoise will never reach the end of the band and will never get any nice flower. Instead, the furthest the tortoise will ever get as the total distance of the band approaches infinity, as a ratio of distance walked to the total length of the band, is calculated as follows: Limit as xinfinity of x/3(x + 1) Multiply each term by 1/x in numerator and denominator = Lim x infinity x/x / (3x/x + 3/x) = Lim xinfinity 1 / (3 + 3/x) = 1/3 So the poor old tortoise will never cover more than 1/3 of the distance to Achilles and the flower. This assumes that the band stays fixed to the ground at the point where the tortoise starts.