A cone and hemisphere have equal bases and equal volumes then find the ratio of their height

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G i v e n space t h a t space b o t h space t h e space c o n e space a n d space t h e space h e m i s p h e r e s space h a v e space e q u a l space b a s e s space a n d space h e n c e b o t h space h a v e space t h e space s a m e space r a d i u s. L e t space apostrophe r apostrophe space b e space t h e space r i a d u s space o f space b o t h space t h e space c o n e space a n d space t h e space h e m i s p h e r e. V o l u m e space o f space t h e space c o n e equals 1 third πr squared straight h Volume space of space the space hemisphere equals 2 over 3 πr cubed Given space that space both space the space volumes space are space equal. rightwards double arrow 1 third πr squared straight h equals 2 over 3 πr cubed rightwards double arrow straight h equals 2 straight r... left parenthesis 1 right parenthesis Height space of space the space hemisphere equals straight r Thus comma space from space equation space left parenthesis 1 right parenthesis comma space we space have comma straight h over straight r equals 2 over 1 Hence space the space ratio space of space the space heights space of space cone space and space hemisphere space is space 2 colon 1

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