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A closed cylinder of given volume will have least surface area when the ratio of its height and base radius is

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a

2 : 1

b

1 : 2

c

2 : 3

d

3 : 2

answer is A.

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Detailed Solution

$\begin{aligned} v = π r^{2} h \\ S = 2 π r^{2} + 2 π rh \\ = 2 π (r^{2} + \frac{v}{π r}) \\ \frac{d (S)}{dr} = (2 r - \frac{v}{π r^{2}}) 2 π \\ r^{3} = \frac{v}{2 π} \\ r = {(\frac{v}{2 π})}^{\frac{1}{3}} \\ h = (2)^{\frac{2}{3}} {(\frac{v}{π})}^{\frac{1}{3}} \\ h : r = 2 : 1 \end{aligned}$

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