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(C) Given that the volume of the cylindrical vessel is $V$ and it has no lid on the top.Let $r$ be the radius and $h$ be the height of the cylinder.Th

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$r=\sqrt[3]{\frac{V}{\pi}}, h=\sqrt[3]{\frac{V}{\pi}}$

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(C) Given that the volume of the cylindrical vessel is $V$ and it has no lid on the top.Let $r$ be the radius and $h$ be the height of the cylinder.The volume of the cylinder is given by $V = \pi r^2 h$,which implies $h = \frac{V}{\pi r^2}$.The surface area $S$ of the metal sheet used (including the base but no lid) is given by:$S = 2\pi rh + \pi r^2$Substituting $h = \frac{V}{\pi r^2}$ into the expression for $S$:$S(r) = 2\pi r \left(\frac{V}{\pi r^2}\right) + \pi r^2 = \frac{2V}{r} + \pi r^2$To find the minimum surface area,we differentiate $S(r)$ with respect to $r$ and set it to zero:$S'(r) = -\frac{2V}{r^2} + 2\pi r = 0$$2\pi r = \frac{2V}{r^2} \Rightarrow r^3 = \frac{V}{\pi} \Rightarrow r = \sqrt[3]{\frac{V}{\pi}}$Now,substitute $r$ back into the expression for $h$:$h = \frac{V}{\pi r^2} = \frac{V}{\pi (V/\pi)^{2/3}} = \frac{V}{\pi} \cdot \left(\frac{\pi}{V}\right)^{2/3} = \left(\frac{V}{\pi}\right)^{1/3} = \sqrt[3]{\frac{V}{\pi}}$Thus,the radius and height for minimum surface area are $r = \sqrt[3]{\frac{V}{\pi}}$ and $h = \sqrt[3]{\frac{V}{\pi}}$.

If a cylindrical vessel of given volume $V$ with no lid on the top is to be made from a sheet of metal,then the radius $(r)$ and height $(h)$ of the vessel so that the metal sheet used is minimum,is

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