Question

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

Solution

## 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, $\frac{dT}{dr}=\frac{d}{dr}\left[2\mathrm{\pi }r\left(r+h\right)\right]\phantom{\rule{0ex}{0ex}}⇒\frac{dT}{dr}=\frac{d}{dr}\left(2\mathrm{\pi }{r}^{2}\right)+\frac{d}{dr}\left(2\mathrm{\pi }rh\right)\phantom{\rule{0ex}{0ex}}⇒\frac{dT}{dr}=4\mathrm{\pi }r+2\mathrm{\pi }h\phantom{\rule{0ex}{0ex}}⇒\frac{dT}{dr}=2\mathrm{\pi }\left(r+h\right)$MathematicsRD Sharma XII Vol 1 (2015)Standard XII

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