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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Aicharya · Accepted Answer

We have given, $V=$ volume of cylindrical vessel $r=$ radius and $h=$ height

As we know for cylindrical vessel $ v=\pi r^2 h $

Let $S$ be the area of metal sheet used to form a cylindrical vessel. Then, $ \begin{aligned} & S=2 \pi r h+\pi r^2=2 \pi r \frac{V}{\pi r^2}+\pi r^2 \quad \text { [using Eq. }(i) \text { ] } \\ & \Rightarrow \quad S(r)=\frac{2 V}{r}+\pi r^2 \end{aligned} $ For minimum value of $S(r), S^{\prime}(r)=0$ $ \begin{aligned} & \Rightarrow \quad \frac{-2 V}{r}+2 \pi r=0 \Rightarrow \frac{2 V}{r^2}=2 \pi r \\ & \Rightarrow r^3=\frac{V}{\pi} \Rightarrow r=\sqrt[3]{\frac{V}{\pi}} \end{aligned} $ Now, by using the value of $r$ in Eq. ( $i)$ $ \begin{aligned} V & =\pi\left(\frac{V}{\pi}\right)^{\frac{2}{3}} \cdot h \\ \Rightarrow \quad & h=\frac{V}{\pi} \times \frac{\pi^{2 / 3}}{v^{2 / 3}} \Rightarrow h=\frac{V^{1 / 3}}{\pi^{1 / 3}}=\sqrt[3]{\frac{V}{\pi}} \end{aligned} $ Hence, the required radius is $r=\sqrt[3]{\frac{V}{\pi}}$ and height is $ 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…

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