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Question Number 144200 by bramlexs22 last updated on 23/Jun/21

$$\mathrm{Find},\mathrm{among}\mathrm{all}\mathrm{right}\mathrm{circular}$$$$\mathrm{cylinders}\mathrm{of}\mathrm{fixed}\mathrm{volume}\mathrm{V}$$$$\mathrm{that}\mathrm{one}\mathrm{with}\mathrm{smallest}\mathrm{surface}\mathrm{area}$$$$(\mathrm{counting}\mathrm{the}\mathrm{areas}\mathrm{of}\mathrm{the}\mathrm{faces}$$$$\mathrm{at}\mathrm{top}\mathrm{and}\mathrm{bottom})$$

Answered by MJS_new last updated on 23/Jun/21

$$S=2\pi r^2+2\pi rh$$$$V=\pi r^{2}h\Rightarrow h=\frac{V}{\pi r^2}$$$$\Rightarrow$$$$S=2\pi r^2+\frac{2V}{r}$$$$\frac{dS}{dr}=4\pi r-\frac{2V}{r^2}=0\Rightarrow r=\sqrt[3]{\frac{V}{2\pi }}\wedge h=\sqrt[3]{\frac{4V}{\pi }}$$$$\mathrm{or}\mathrm{simply}h=2r$$

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