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 = \[2\pi r\left( r + h \right)\]

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

Now,

\[\frac{dT}{dr} = \frac{d}{dr}\left[ 2\pi r\left( r + h \right) \right]\]

\[ \Rightarrow \frac{dT}{dr} = \frac{d}{dr}\left( 2\pi r^2 \right) + \frac{d}{dr}\left( 2\pi r h \right)\]

\[ \Rightarrow \frac{dT}{dr} = 4\pi r + 2\pi h\]

\[ \Rightarrow \frac{dT}{dr} = 2\pi\left(2r + h \right)\]