Single Variable Calculus, Chapter 3, 3.9, Section 3.9, Problem 34

Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05cm thick to hemispherical dome with diamater 50m.

Recall that the volume of the hemisphere is half of the volume of the sphere so,

$\displaystyle V = \frac{\frac{4}{3}\pi r^3}{2} = \frac{2}{3} \pi r^3$

If we take the derivative with respect to the radius, we have


$
\begin{equation}
\begin{aligned}
\frac{dV}{dr} &= \frac{2\pi}{3} \frac{d}{dr} (r^3)\\
\\
\frac{dV}{dr} &= \frac{2\pi}{\cancel{3}}\left( \cancel{3}r^2 \right)\\
\\
\frac{dV}{dr} &=2 \pi r^2\\
\\
dV &= 2 \pi r^2 dr &&; \text{ recall that } d = 2r ; r = \frac{d}{2}
\end{aligned}
\end{equation}
$

Hence,


$
\begin{equation}
\begin{aligned}
dV & = 2\pi \left( \frac{50}{2} \right)^2 \left( 0.05\text{cm} \times \frac{1m}{100\text{cm}}\right)\\
\\
dV & = 1.96 m^3
\end{aligned}
\end{equation}
$

Therefore, the amount of paint needed is $dV = 1.96m^3$