The ratio h : 2 r for which S to be minimum will be equal to - Vidyadip

Question

The ratio $h$ : $2 r$ for which $S$ to be minimum will be equal to

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Answer

$\because V=\frac{1}{2} \pi r^2 h$
and $S$ will be minimum, when $(\pi+2) V =\pi^2 r^3$
$\Rightarrow V=\frac{\pi^2 r^3}{\pi+2}$
From (i) and (ii), we get
$\Rightarrow \frac{1}{2} \pi r^2 h=\frac{\pi^2 r^3}{\pi+2} \Rightarrow \pi r^2 h(\pi+2)=2 \pi^2 r^3$
$\Rightarrow h(\pi+2)=2 \pi r \Rightarrow \frac{h}{2 r}=\frac{\pi}{\pi+2}$
Thus, required ratio i.e., $h: 2 r$ is $\pi: \pi+2$.

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