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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