Question

Find the rate of change of the total surface area of a cylinder of radius r and height h, when the radius varies.

Answer 1

Let T be the total surface area of a cylinder. Then,

T = $2\mathrm{\pi }r\left(r+h\right)$

Since the radius varies, we differentiate the total surface area w.r.t. radius r.

Now,

$$\begin{gathered} \frac{dT}{dr}=\frac{d}{dr}\left[2\mathrm{\pi }r\left(r+h\right)\right] \\ \Rightarrow \frac{dT}{dr}=\frac{d}{dr}\left(2\mathrm{\pi }{r}^2\right)+\frac{d}{dr}\left(2\mathrm{\pi }rh\right) \\ \Rightarrow \frac{dT}{dr}=4\mathrm{\pi }r+2\mathrm{\pi }h \\ \Rightarrow \frac{dT}{dr}=2\mathrm{\pi }\left(r+h\right) \end{gathered}$$