differentials with approximation?

Question

Use differentials to estimate the amount of paint needed to apply a coat of paint 0.07 cm thick to a hemispherical dome with diameter 54 m.

don't know what formula to use...

Answer 1

Use the volume of a hemisphere formula.

V(r) = (2*pi/3)*r^3

The volume of a thin layer of thickness dr on a hemisphere of radius r can be expressed as how much the volume of the hemisphere changes as you increase the radius from r to dr:

dV = V(r+dr) - V(r)

for small nonzero dr:

dV/dr ~= V'(r)
dV ~= V'(r)*dr

Don't forget to use radius (not diameter) and to put r and dr in the same units so you get an answer in cubic somethings...

PS: I get about 33.55 cubit cubits, but that's probably not the units you needed.
:*) <<-- smiling, with busted nose