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Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm^3, which has the minimum surface area?

Let $r$ $and$ $h$ be the radius and height of the cylinder, respectively. Then,

Volume $(V)$ of the cylinder \[=\text{ }\pi {{r}^{2}}~h\]

\[\to \text{ }100\text{ }=\text{ }\pi {{r}^{2}}~h\]

\[\to h\text{ }=\text{ }100/\text{ }\pi {{r}^{2}}\]

Surface area $(S)$ of the cylinder \[=\text{ }2\text{ }\pi {{r}^{2}}~+\text{ }2\text{ }\pi r\text{ }h\text{ }=\text{ }2\text{ }\pi {{r}^{2}}~+\text{ }2\text{ }\pi r\text{ }\times \text{ }100/\text{ }\pi {{r}^{2}}\]

RD Sharma Solutions for Class 12 Maths Chapter 18 Maxima and Minima Image 19