A hemispherical dome of radius 40 feet is to be given 7 coats of paint, each of which is 1/100 inch thick. How do you use linear approximation to estimate the volume of paint needed for the job?

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Answer 1 · 2016-09-25T20:59:50

#approx " 117.3 ft"^3 #

I would go with the (brilliant) result that, for a sphere:

#V = 4/3 pi r^3# and #(dV)/(dr) = 4 pi r^2#

[ie the surface area of a sphere of radius #r# is the derivative wrt #r# of its volume]

To first order we can say that:

#delta V = (dV)/(dr) delta r = 4 pi r^2 delta r#

and so with #delta r = 7 * 1/(12*100)# (7 layers, and adjusting to Imperial ft measurements), we have

#delta V = 4 pi ( 40)^2 * 7/100 = 112/3 pi approx " 117.3 ft"^3 #

Actual increase is

#DeltaV = 4/3 pi ( (40+ 7/1200)^3- 40^3) = " 117.3 ft"^3 #

I did those to 1 dp, so you'll find a slightly bigger discrepancy if you plug the numbers into a calculator