Imagine that the Earth has a band of copper wrapped tightly around its equator. As a prank, you add one kilometer to the length of the copper band so that it now sits above the ground by an equal distance at all locations.
How high is the copper band above the ground? Could a flea squeeze underneath it? What about a man? Or how about a mansion?
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2pi(R+x)=2piR+1000===> x=500/pi=159.15 so it's about a mansion :)
2 π ( r + Δ r ) − 2 π r = 1 0 0 0
Δ r ≈ 1 6 0 m
The key is that the circumference varies linearly with radius by a factor of 2 π which is a bit larger than 6. So when the circumference changes by 1 km, the radius changes by 6 1 km which is room enough for a mansion. So there you have it.
Very well put!
Setting up the radius as a function of the circumference, the formula is r = 2 π C
If C is the fixed circumference of the Earth, 2 π C + 1 gives us the new radius of the band, in kilometers, after adding one kilometer. To find the height at any point around the equator, we simply subtract these two radii and that difference is the clearance under the band.
2 π C + 1 − 2 π C = 2 π 1
The two fractions, already with a common denominator, subtract easily and the Earth's circumference subtracts out, leaving 2 π 1 as the clearance under the band (again, in kilometers)
All that remains is some approximating and real-world reasoning to find the answer. The answer comes out to around 160m, or 1.75 American football fields, which certainly gives enough room for all the objects listed.
Though Earth is confusing and simple math can't be used to go from radius to circumference, it is easy to tell that the band would be almost a tenth of a mile above the surface of the planet meaning a mansion would easily fit underneath it.
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Let the radius of the initial band be r . This means the circumference of the band is C = 2 π r . By extending the band by 1 kilometer, the new circumference becomes C = 2 π r + 1 . Divide this by 2 π to determine the new radius r new = 2 π 2 π r + 1 = r + 2 π 1 . The new radius is now 2 π 1 ≈ 0 . 1 5 9 km longer than before, meaning the band sits 159 meters above the ground, so a mansion can definitely fit underneath it.