 # 173 geometry/cover.earth.p

## Description

This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

# 173 geometry/cover.earth.p

A thin membrane covers the surface of the (spherical) earth. One
square meter is added to the area of this membrane to form a larger
sphere. How much is added to the radius and volume of this membrane?

geometry/cover.earth.s

We know that V = (4/3)*pi*r^3 and A = 4*pi*r^2.
We need to find out how much V increases if A increases by 1 m^2.

```  dV / dr = 4 * pi * r^2
dA / dr = 8 * pi * r
dV / dA = (dV / dr) / (dA / dr)
= (4 * pi * r^2) / (8 * pi * r)
= r/2
= 3,250,000 m
```

If the area of the cover is increased by 1 square meter,
then the volume it contains is increased by about 3.25 million cubic meters.

We seem to be getting a lot of mileage out of such a small square of cotton.
However, the new cover would not be very high above the surface of the
planet -- about 6 nanometers (calculate dr/dA).

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