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Application Of Derivatives

Question
CBSEENMA12035551

Show that the height of the cylinder, open at the top, of given surface area and greatest volume is equal to the radius of its base.

Solution
Let r be the radius of base of circular cylinder and h be its height. Let V be the volume and S be total surface area.
therefore space space straight S space equals space πr squared plus 2 πrh space rightwards double arrow space space space straight h space equals space fraction numerator straight S minus πr squared over denominator 2 πr end fraction                        ...(1)
       straight V equals space πr squared straight h space equals space πr squared space open parentheses fraction numerator straight S minus πr squared over denominator 2 πr end fraction close parentheses                                  open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets
therefore space space space straight V space equals space 1 half left parenthesis Sr minus πr cubed right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space dV over dr equals space 1 half left parenthesis straight S minus 3 πr squared right parenthesis
space space space space space dV over dr space equals 0 space space space space space rightwards double arrow space space space space 1 half left parenthesis straight S space minus space 3 πr squared right parenthesis space equals space 0 space space space space rightwards double arrow space space space straight r space equals space square root of fraction numerator straight S over denominator 3 straight pi end fraction end root
space space space space space fraction numerator straight d squared straight V over denominator dr squared end fraction space equals space 1 half left parenthesis 0 minus 6 πr right parenthesis space space equals space minus 3 πr
For space space space space straight r space equals space square root of fraction numerator straight S over denominator 3 straight pi end fraction end root comma space fraction numerator straight d squared straight v over denominator dr squared end fraction space equals space minus 3 straight pi square root of fraction numerator straight S over denominator 3 straight pi end fraction end root space equals space minus square root of 3 πS end root less than 0
therefore space space space space space straight V space is space greatest space when space straight r space equals space square root of fraction numerator straight S over denominator 3 straight pi end fraction end root     
From (1),  straight h equals fraction numerator straight S minus straight pi. space begin display style fraction numerator straight S over denominator 3 straight pi end fraction end style over denominator 2 straight pi. space square root of begin display style fraction numerator straight S over denominator 3 straight pi end fraction end style end root end fraction space equals space fraction numerator begin display style fraction numerator 2 straight S over denominator 3 end fraction end style over denominator 2 square root of begin display style πS over 3 end style end root end fraction space equals fraction numerator 2 straight S over denominator 3 end fraction cross times fraction numerator square root of 3 over denominator square root of πS end fraction space equals space square root of fraction numerator straight S over denominator 3 straight pi end fraction end root
therefore height = radius of base         

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