This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
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?
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