Wrapping The Earth 


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The "Wrapping The Earth" ProblemOur friend once again has too much to drink, and once again indulges in a little late night internet shopping. This time, however, it's fueled by curiousity, having discovered that there is a material that is infinitely stretchy, but which always retains its area. So while you can stretch it one direction, it will keep its area by shrinking in another direction.
(We are again in puzzle world, so we're assuming the numbers are good enough for our purposes  let's pretend they are exact.) It takes some time for the order to arrive  next day delivery wasn't available from this vendor  but when it does arrive our friend goes to it and heroically wraps the entire Earth, only to discover that there is slightly too much material. Just one square kilometre. One million square metres. Never mind, we can simply prop it up the same height everywhere, as we did with the rope. How high must it be? It's a simple sum ...This one's not too hard, and in truth, not too surprising. We can do it by algebra, or by calculus, both working out quite simply. We'll do it via algebra, and let the calculus wait to the next section. We know that the area of a sphere is given by the formula $A=4\pi R^2$. Letting $E$ represent our extra area, and $h$ being the extra height we need, we have:
Reflections ...There are a few interesting differences here between this and the original "Rope around the Earth" problem. For a start, this answer, unlike the original, does depend on the size of the Earth. But that is clear from the calculus approach. In the original we have:
In contrast, with this problem we are going with the area, and we have:
The Other Wrapping the Earth Problem ...But this was just the warm up. Next time we'll take the step of not propping up the sheet everywhere, but just in one place with a single tent pole ... How high will that have to be? Next time.
AcknowledgementsI believe this problem was first suggested by the Rudi Mathematici, but I'm still chasing the exact source, and will update this post when I have more information: Update: As far as we can tell, it did originate with the Rudi Mathematici as an offhand remark one evening, but this is relying on memory, which always has the potential to be faulty. If anyone knows of an earlier reference, do let me know. This post has been significantly improved by several small changes made by a handful of people, none of whom wish to be credited. As always, my thanks to them, and, obviously, any remaining errors are mine.
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