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. (Round your answer to two decimal places.)

Answer: the amount needed is around 25.64m^2

Step-by-step explanation:

Here we have a thickness of 0.07cm and a surface of 54m

So we have the integral of volume between:

54m and 54m + 0.0007m

this is equal to:

v = ∫sin (θ) r^2drdθdφ

Where dr goes as we know, from 54m to 54m+0.0007m, dφ goes from 0 to 2*pi, and θ goes from 0 to pi.

Solving the integral, we have that:

v = 2*pi*∫sin (θ) rdrdθ

v = 4*pi*∫rdr = (4/3) * pi * ((54.0007m) ^3 - 54m^3) = 25.64m^3