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

Find the minimum surface area of a solid circular cylinder , if its volume is 16π cm^3 (leave your answer in terms of π)

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

Answered by ajfour last updated on 26/Oct/17

S=2πrh+2πr^2 V=πr^2 h ⇒ (dh/dr)=−((2V)/(πr^3 )) ...(i) (dS/dr)=2πh+2πr(dh/dr)+4πr =0 using (i), this ⇒ 2πh+2πr(−((2V)/(πr^3 )))+4πr=0 ⇒ h=((2V)/(πr))−2r = ((2(πr^2 h))/(πr^2 ))−2r ⇒ h=2h−2r or h=2r Now V=πr^2 h =πr^2 (2r)=16π cm^3 ⇒ r^3 =8 cm^3 or r=2cm and h=2r =4cm so S=2πrh+2πr^2 =2π(3r^2 ) =2π×12cm^2 S=24π cm^2 .

$${S}=\mathrm{2}\pi{rh}+\mathrm{2}\pi{r}^{\mathrm{2}} \\ $$$${V}=\pi{r}^{\mathrm{2}} {h}\:\:\:\Rightarrow\:\:\frac{{dh}}{{dr}}=−\frac{\mathrm{2}{V}}{\pi{r}^{\mathrm{3}} }\:\:\:...\left({i}\right) \\ $$$$\frac{{dS}}{{dr}}=\mathrm{2}\pi{h}+\mathrm{2}\pi{r}\frac{{dh}}{{dr}}+\mathrm{4}\pi{r}\:=\mathrm{0} \\ $$$${using}\:\left({i}\right),\:{this}\:\Rightarrow \\ $$$$\:\:\:\mathrm{2}\pi{h}+\mathrm{2}\pi{r}\left(−\frac{\mathrm{2}{V}}{\pi{r}^{\mathrm{3}} }\right)+\mathrm{4}\pi{r}=\mathrm{0} \\ $$$$\Rightarrow\:\:\:{h}=\frac{\mathrm{2}{V}}{\pi{r}}−\mathrm{2}{r}\:=\:\frac{\mathrm{2}\left(\pi{r}^{\mathrm{2}} {h}\right)}{\pi{r}^{\mathrm{2}} }−\mathrm{2}{r} \\ $$$$\:\Rightarrow\:\:\:{h}=\mathrm{2}{h}−\mathrm{2}{r} \\ $$$${or}\:{h}=\mathrm{2}{r} \\ $$$${Now}\:{V}=\pi{r}^{\mathrm{2}} {h}\:=\pi{r}^{\mathrm{2}} \left(\mathrm{2}{r}\right)=\mathrm{16}\pi\:{cm}^{\mathrm{3}} \\ $$$$\Rightarrow\:\:{r}^{\mathrm{3}} =\mathrm{8}\:{cm}^{\mathrm{3}} \:\:\:{or}\:\:{r}=\mathrm{2}{cm} \\ $$$${and}\:{h}=\mathrm{2}{r}\:=\mathrm{4}{cm} \\ $$$${so}\:\:{S}=\mathrm{2}\pi{rh}+\mathrm{2}\pi{r}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\pi\left(\mathrm{3}{r}^{\mathrm{2}} \right)\:=\mathrm{2}\pi×\mathrm{12}{cm}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\boldsymbol{{S}}=\mathrm{24}\pi\:{cm}^{\mathrm{2}} \:. \\ $$

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

God bless you sir.

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

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