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A Closed Cylinder Has Volume 2156 Cm3. What Will Be the Radius of Its Base So that Its Total Surface Area is Minimum ? - Mathematics

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Question

A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?

Solution

\[\text { Let the height, radius of the base and surface area of the cylinder be h, r and S, respectively . Then }, \]

\[\text { Volume =} \pi r^2 h\]

\[ \Rightarrow 2156 = \pi r^2 h\]

\[ \Rightarrow 2156 = \frac{22}{7} r^2 h\]

\[ \Rightarrow h = \frac{2156 \times 7}{22 r^2}\]

\[ \Rightarrow h = \frac{686}{r^2} . . . \left( 1 \right)\]

\[\text { Surface area } = 2\pi r h + 2\pi r^2 \]

\[ \Rightarrow S = \frac{4312}{r} + \frac{44 r^2}{7} \left[ \text { From eq } . \left( 1 \right) \right]\]

\[ \Rightarrow \frac{dS}{dr} = \frac{4312}{- r^2} + \frac{88r}{7}\]

\[\text { For maximum or minimum values of S, we must have }\]

\[\frac{dS}{dr} = 0\]

\[ \Rightarrow \frac{4312}{- r^2} + \frac{88r}{7} = 0\]

\[ \Rightarrow \frac{4312}{r^2} = \frac{88r}{7}\]

\[ \Rightarrow r^3 = \frac{4312 \times 7}{88}\]

\[ \Rightarrow r^3 = 343\]

\[ \Rightarrow r = 7 cm\]

\[\text { Now }, \]

\[\frac{d^2 s}{d r^2} = \frac{8624}{r^3} + \frac{88}{7}\]

\[ \Rightarrow \frac{d^2 s}{d r^2} = \frac{8624}{343} + \frac{88}{7}\]

\[ \Rightarrow \frac{d^2 s}{d r^2} = \frac{176}{7} > 0\]

\[\text{ So, the surface area is minimum when r = 7 cm }.\]

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APPEARS IN

 RD Sharma Solution for Mathematics for Class 12 (Set of 2 Volume) (2018 (Latest))
Chapter 18: Maxima and Minima
Ex. 18.5 | Q: 26 | Page no. 73
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A Closed Cylinder Has Volume 2156 Cm3. What Will Be the Radius of Its Base So that Its Total Surface Area is Minimum ? Concept: Graph of Maxima and Minima.
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