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Question Number 23133 by tawa tawa last updated on 26/Oct/17

$$\mathrm{Find}\mathrm{the}\mathrm{minimum}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{a}\mathrm{solid}\mathrm{circular}\mathrm{cylinder},\mathrm{if}\mathrm{its}\mathrm{volume}\mathrm{is}$$$$16\pi {\mathrm{cm}}^3\left(\mathrm{leave}\mathrm{your}\mathrm{answer}\mathrm{in}\mathrm{terms}\mathrm{of}\pi \right)$$

Answered by ajfour last updated on 26/Oct/17

$$S=2\pi rh+2\pi r^2$$$$V=\pi r^{2}h\Rightarrow \frac{dh}{dr}=-\frac{2V}{\pi r^3}...\left(i\right)$$$$\frac{dS}{dr}=2\pi h+2\pi r\frac{dh}{dr}+4\pi r=0$$$$using\left(i\right),this\Rightarrow$$$$2\pi h+2\pi r(-\frac{2V}{\pi r^3})+4\pi r=0$$$$\Rightarrow h=\frac{2V}{\pi r}-2r=\frac{2\left(\pi r^{2}h\right)}{\pi r^2}-2r$$$$\Rightarrow h=2h-2r$$$$orh=2r$$$$NowV=\pi r^{2}h=\pi r^2\left(2r\right)=16\pi {cm}^3$$$$\Rightarrow r^3=8{cm}^{3}orr=2cm$$$$andh=2r=4cm$$$$soS=2\pi rh+2\pi r^2$$$$=2\pi \left(3r^2\right)=2\pi \times 12{cm}^2$$$$\mathit{S}=24\pi {cm}^2.$$

Commented by tawa tawa last updated on 26/Oct/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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