Math CentralQuandaries & Queries


Question from Joe, a student:

A snail starts crawling from one end along a uniformly stretched elastic band. It crawls at a rate of 1 foot per minute. The band is initially 10 feet long and is instantaneously and uniformly stretched an additional 10 feet at the end of each minute. The snail maintains his grip on the band during the instant of each stretch. At what points in time is the snail
(a) closest to the far end of the band, and
(b) farthest from the far end of the band?

Clarification : the solution considers that the snail starts crawling at minute 1.

  1. Work out what proportion of the current length the snail covers in each minute. In the first minute he covers 1/10. Then the band stretches to 20 ft and in the next minute she (snails are hermaphroditic) covers 1/20. In the next minute, 1/30, and so on.

    In N minutes, the snail covers 1/10[1 + 1/2 + 1/3 + 1/4 +....] of the band. Does that ever reach 1? Yes, it must. Why?

    Regroup the terms in the square brackets as

    [(1) + (1/2) + (1/3+1/4) + (1/5+...+1/8) + (1/9+...+1/16)+ ...]

    Each group in parentheses adds to more than 1/2 (why? how exactly are the groups defined?. So once you have 20 of them (in fact 19, because the first is double-sized) they will add to more than 10. How many individual terms is this? By the end of this many minutes the snail will have reached the end of the band.

    Now, this is an upper bound; to find the time when the snail gets there more exactly requires some advanced calculus. (It would actually be easier to compute if the band was being stretched continuously.) Can you write a program to find for which N the sum first exceeds 10? I get about 12367 minutes using MAPLE.

  2. Initially the snail loses more distance (measured from the goal) to stretching than it makes up crawling. When he is close to its goal almost all the stretching takes place behind her so he loses little to the stretch, less than she covers crawling. When he is furthest behind, she loses exactly as much to stretching as he covered that minute. What proportion of the way along the band must she be for the portion of the 10ft stretch that occurs in the band ahead to cancel a minute's crawl? How many minutes does it take to get that far?

Good Hunting!

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