Rope Around The Earth Refined 


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2019/01/22  Rope Around the Earth  RefinedAs we previously mentioned, the Rope Around the Earth problem is a lovely one, and really, everyone should know it. But even if you do, there's still a nice added extra. In this post we'll have a look at that.
The Rope Around the EarthSomeone discovers that the 40 million metres of rope either isn't 40 million metres, or that the Great Circle around which they stretched it isn't exactly 40 million metres (there's a surprise!), because when they stretched the rope around the world, the rope was six metres longer than needed. Well, as in other puzzles involving 40 million metres of rope, they didn't want to cut it, because it was expensive, and tying a knot just seems too inelegant, so they fuse the ends together and think. They come to the conclusion that making the circle bigger would fix the problem and take up the slack, so they make plans to prop up the rope, same height everywhere, (never mind that it would sag), and that would take up the excess. So how high do the props need to be? There will be a lot of them, because they will need to be closely spaced, and we don't want to make them the wrong size. So how high will the rope be?
If you've not met this before then I urge you most strongly to have a think about this, even if only for a minute or two. If you have met it before, make sure you can remember the answer, or work it out again. Remember, this version has six metres of extra rope, and is asking how high it needs to be propped. The solutionWhether you find this easy, hard, surprising, or obvious, depends a lot on your background, experience, and personality. I like it a lot, possibly because I met it a long time ago and regard it as an old friend. The simple answer is to say that the extra radius is $6/(2\pi)$, and since $\pi$ is roughly three, the answer is roughly one metre.
But there's an interesting bit of calculus here. Since we know that circumference is $2\pi$ times radius, so $C=2\pi R$, and we can differentiate with respect to $C$ to get
Refining our answerThere's a lovely refinement we can do on this estimated answer. We got the answer of "about a metre" by assuming that $\pi$ is about $3$. But it's not, it's about $3.15$ (wait for it ...)
Let's get a calculator and do the actual divisions. We were dividing by $3.15$ and claiming it's roughly $0.95$, whereas the actual answer is:
If instead we want to divide by $3.14$ we can say that 3.14 is 0.01 less than 3.15, which is about $\frac{1}{3}$%, so we have to increase our answer by about 0.0033, giving an estimate of:
But using a calculator we get:
Well, I liked it. AddendumIf you use $\pi\approx 22/7$ then we want to compute $3$ divided by that, so $21/22$. Multiply numerator and denominator by 5 each to get $(105)/(110)$ and multiply again, this time by 9 to get $(945)/(990)$. Add another 1% to get:
Make the approximation of adding 1 top and bottom, so we have $955.45/1000$, which is 0.95545. Compare with $3/\pi$, which is $3.95493...$ and we're pretty close. We can play the same game with $\pi\approx 355/113$ and compute (or approximate) $339/355$, but I think that's enough for now.
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Quotation from Tim BernersLee 