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09 Sep

Pythagorean Guess

A frequently cited math curiosity relates the relative increase in the rope length between the rope laid on the Earth's equator and that around an average size watermelon. In both case the sought increase in length is due to the uniform expansion of the rope to, say, 1 ft away from the surface. It often comes as a surprise - even for those who remember the C = πD formula for the length of the circumference of the diameter D. In both cases the length of the rope will increase by exactly the same amount - 2π.

I have come across another surprising result (which at first I thought related to the aforementioned one) in a new book by Paul H. Nahin.

A straight, precisely one-mile-long stretch of a continuous railroad track is laid during a cold night, with the two ends firmly fixed in the ground. The next morning the hot sun causes the track to expand by exactly one foot, and so the track buckles up. Now, off the top of your head, is the midpoint of the track raised above the ground by (1) several inches, (2) several feet, or (3) several yards?

Answer

References

1. P. H. Nahin, Number-Crunching, Princeton University Press, 2011

We'll make use of the Pythagorean theorem to obtain a rough estimate. We assume that the buckled track formed an isosceles triangle. This is certainly would not be true in practice as the track would bent smoothly.

Dropping a perpendicular from the apex divides that shape into two right triangles. Now recollect that 1m = 5280ft so that a half mile is 2640ft. This is the length of the horizontal leg of either of the right triangles. Their hypotenuse is longer by .5ft. The Pythagorean theorem gives then an estimate for the short leg - the height of the buckle:

$\sqrt{2640.5^2 - 2640^2} \approx 51.38336\ldots$

Nahin reports that using more sophisticated methods Forman S. Acton gave a by far more accurate estimate of 44.49845. Just about an 11% error compared to the straightforward application of the Pythagorean theorem.

Pat Ballew had a more detailed discussion of this problem.

3 Responses to “Pythagorean Guess”

1. 1
jessica Says:

I couldn't get that part about watermelon and the Earth. What relative increase? We have a rope around both of them and than what?

2. 2
Alexander Bogomolny Says:

Ropes are wrapped around the Earth and a watermelon. Measure their length. Now, shift both 1' away from their current position. The question is, How longer the ropes need to be made to keep the circles closed - when they are 1' away, one from the Earth, the other from the watermelon?

3. 3
jessica Says:

Now I get it 🙂 And they do increase by the same amount. Really not that obvious at all.
Thanks!

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