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Question Number 18556 by aplus last updated on 24/Jul/17

Answered by Tinkutara last updated on 25/Jul/17

V = πr^2 h  Total surface area, A = 2πr^2  + 2πrh  = 2πr^2  + 2((πr^2 h)/r) = 2πr^2  + ((2V)/r)  Now V = (π/4) m^3   (dA/dr) = 4πr − (π/(2r^2 )) = 0  8r^3  = 1 ⇒ r = (1/2) m

$$\mathrm{V}\:=\:\pi{r}^{\mathrm{2}} {h} \\ $$$$\mathrm{Total}\:\mathrm{surface}\:\mathrm{area},\:{A}\:=\:\mathrm{2}\pi{r}^{\mathrm{2}} \:+\:\mathrm{2}\pi{rh} \\ $$$$=\:\mathrm{2}\pi{r}^{\mathrm{2}} \:+\:\mathrm{2}\frac{\pi{r}^{\mathrm{2}} {h}}{{r}}\:=\:\mathrm{2}\pi{r}^{\mathrm{2}} \:+\:\frac{\mathrm{2}{V}}{{r}} \\ $$$$\mathrm{Now}\:{V}\:=\:\frac{\pi}{\mathrm{4}}\:\mathrm{m}^{\mathrm{3}} \\ $$$$\frac{{dA}}{{dr}}\:=\:\mathrm{4}\pi{r}\:−\:\frac{\pi}{\mathrm{2}{r}^{\mathrm{2}} }\:=\:\mathrm{0} \\ $$$$\mathrm{8}{r}^{\mathrm{3}} \:=\:\mathrm{1}\:\Rightarrow\:{r}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{m} \\ $$

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