If an open cylinder of given surface area has maximum volume, then its radius is

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Aicharya · Accepted Answer

Cylinder is open, radius of base $=R$, height $=H$ Surface area $A=2 \pi R H+\pi R^2$ $ \begin{aligned} & \Rightarrow H=\frac{A-\pi R^2}{2 \pi R} \ & V=\pi R^2 H=\pi R^2\left(\frac{A-\pi R^2}{2 \pi R}ight)=\frac{R}{2}\left(A-\pi R^2ight) \ & \frac{d v}{d R}=\frac{A}{2}-\frac{3 \pi R^2}{2}=0 \Rightarrow A=3 \pi R^2 \ & \Rightarrow \frac{d^2 V}{d R^2}=-3 \pi R < 0 \ & \Rightarrow 	ext { at } A=3 \pi R^2, 	ext { maximum volume } \ & A=3 \pi R^2=2 \pi R H+\pi R^2 \ & \Rightarrow R=H \end{aligned} $

If an open cylinder of given surface area has maximum volume, then its radius is

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