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Question Number 124924 by bemath last updated on 07/Dec/20

$$Whatshouldthedimensionsbe$$$$ofacylindersothatforagivenvolume$$$$vitstotalsurfaceSminimum?$$

Commented by liberty last updated on 07/Dec/20

$$Denotingbyrtheradiusofthebaseofthe$$$$cylinderandbyhthealtitude,wehave$$$$S=2\pi r^2+2\pi rhandthevolumeisgiven$$$$v=\pi r^{2}h;whenceh=\frac{v}{\pi r^2}$$$$substitutingthisexpressionofhintothe$$$$formulaforS,giveS=2(\pi r^2+\frac{v}{r});where0<r<\mathrm{\infty }$$$$\frac{dS}{dr}=2(2\pi r-\frac{v}{r^2})=0\Rightarrow r_1=\sqrt[3]{\frac{v}{2\pi }}$$$${\left(\frac{d^{2}S}{{dr}^2}\right)}_{r=r_1}=2{(2\pi +\frac{2v}{r^3})}_{r=r_1}>0$$$$Thusatthepointr=r_{1}thefunctionShas$$$$aminimum.Noticingthat\underset{r\to 0}{\lim }S=\mathrm{\infty }and$$$$\underset{r\to \mathrm{\infty }}{\lim }S=\mathrm{\infty },wearriveatconclusionthat$$$$atr=r_1=\sqrt[3]{\frac{v}{2\pi }}thenh=\frac{v}{\pi r^2}=2\sqrt[3]{\frac{v}{2\pi }}=2r$$$$$$

Answered by som(math1967) last updated on 07/Dec/20

$$\mathrm{let}\mathrm{radius}=\mathrm{r}\mathrm{height}=\mathrm{h}$$$$v=\pi {\mathrm{r}}^2\mathrm{h}\Rightarrow \mathrm{h}=\frac{v}{\pi {\mathrm{r}}^2}$$$$\mathrm{S}=2\pi \mathrm{rh}+2\pi {\mathrm{r}}^2$$$$\mathrm{S}=\frac{2v}{\mathrm{r}}+2\pi {\mathrm{r}}^2$$$$\frac{\mathrm{dS}}{\mathrm{dr}}=-\frac{2v}{{\mathrm{r}}^2}+4\pi \mathrm{r}$$$$\frac{{\mathrm{d}}^2\mathrm{S}}{{\mathrm{dr}}^2}=\frac{4v}{{\mathrm{r}}^3}+4\pi$$$$\therefore \frac{{\mathrm{d}}^2\mathrm{S}}{{\mathrm{dr}}^2}>0\therefore \mathrm{S}\mathrm{minimum}\mathrm{for}$$$$\frac{\mathrm{dS}}{\mathrm{dr}}=0$$$$\therefore -\frac{2v}{{\mathrm{r}}^2}+4\pi \mathrm{r}=0$$$$4\pi {\mathrm{r}}^3=2v$$$$2\pi {\mathrm{r}}^3=\pi {\mathrm{r}}^2\mathrm{h}$$$$2\mathrm{r}=\mathrm{h}$$$$\therefore \mathrm{S}\mathrm{is}\mathrm{minimum}\mathrm{when}$$$$\mathrm{Height}\mathrm{is}\mathrm{eqal}\mathrm{to}\mathrm{diameter}$$$$\mathrm{or}\mathrm{r}:\mathrm{h}=1:2$$

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