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Suppose we get a rope that's about 40,000 kilometers long, and tie it snug around the Earth's equator (imagine a perfectly spherical and solid earth). Now let's add just 10 meters of slack to the rope and distribute it around the planet so that the rope is equidistant from the surface at all points (vertically).

How much space is between the rope and the Earth's surface? Before solving numerically, take a ballpark guess - can you fit a mountain under the rope? Walk underneath it? Can you even slide a sheet of paper between the rope and the Earth's surface?

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    $\begingroup$ I had a profesor who once posed this question. It is an interesting, counter-intuitive display of geometry, but I'm not quite sure it classifies as a puzzle. $\endgroup$ – Fimpellizieri Mar 7 '16 at 9:11
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    $\begingroup$ @Fimpellizieri I reckon if it's interesting and counter-intuitive, it counts as a puzzle on this site by definition. $\endgroup$ – Lawrence Mar 7 '16 at 23:57
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    $\begingroup$ @Lawrence It's interesting because it is counter-intuitive. Aside from that, it is very basic math and the solution requires no investment from the reader. $\endgroup$ – Fimpellizieri Mar 8 '16 at 2:29
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Circumference is ~ 3 x Diameter. 10 meters in circumference means that the diameter is ~3 meters larger. So the rope should be roughly 1.5 meters above the ground.

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This is off-topic, really, because there's no puzzle around it. I think there easily could be, though.

Perreal is very close with their estimation. The complete math is below with all measurements in meters.

+---------------+--------------+--------------+   
|               |     40k      |     +10      |   
+---------------+--------------+--------------+   
| Circumference |  40,000,000  |  40,000,010  |   
| Diameter      |  12,732,395  |  12,732,399  |   
| Radius        |   6,366,198  |   6,366,199  |   
+---------------+--------------+--------------+   
| Difference    |         1.59 meters         |   
+---------------+--------------+--------------+   
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