Question
Fri March 13, 2015 By: Rida Mukadam

the height of a cone is 60 cm a small cone is cut off at the top by a plane parallel to the base and its volume is 1/64 th the volume of original cone. find the height from the base at which the section is made .

Expert Reply
Vimala Ramamurthy
Sat March 14, 2015
C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
G i v e n space t h a t space t h e space h e i g h t space o f space t h e space c o n e space i s space 60 space c m. A B equals 60 c m N o w space c o n s i d e r space t h e space t r i a n g l e s space triangle A E D space a n d space triangle A B C angle A E D equals angle A B C equals 90 degree angle A D E equals angle A C B T h e r e f o r e comma space b y space A n g l e minus A n g l e space s i m i l a r i t y comma space w e space h a v e space triangle A E D tilde triangle A B C. H e n c e space t h e space c o r r e s p o n d i n g space s i d e s space a r e space p r o p o r t i o n a l. L e t space E D equals r space a n d space A E equals h T h u s comma space w e space h a v e comma fraction numerator A E over denominator A D end fraction equals fraction numerator E D over denominator B C end fraction rightwards double arrow h over 60 equals r over R rightwards double arrow open parentheses r over R close parentheses squared equals open parentheses h over 60 close parentheses squared rightwards double arrow r squared over R squared equals h squared over 60 squared... left parenthesis 1 right parenthesis T h e space r a t i o space o f space t h e space v o l u m e s space o f space t h e space s m a l l e r space c o n e space a n d space t h e space b i g g e r space c o n e space equals 1 over 64 V o l u m e space o f space t h e space s m a l l e r space c o n e equals 1 third πr squared straight h Volume space of space the space larger space cone equals 1 third πR squared cross times 60 Ratio equals fraction numerator 1 third πr squared straight h over denominator 1 third πR squared cross times 60 end fraction rightwards double arrow 1 over 64 equals fraction numerator 1 third πr squared straight h over denominator 1 third πR squared cross times 60 end fraction rightwards double arrow 1 over 64 equals fraction numerator straight r squared straight h over denominator straight R squared cross times 60 end fraction rightwards double arrow 1 over 64 equals fraction numerator straight h squared cross times straight h over denominator 60 squared cross times 60 end fraction.... left square bracket from space equation space left parenthesis 1 right parenthesis right square bracket rightwards double arrow 1 over 64 equals straight h cubed over 60 cubed rightwards double arrow straight h cubed equals 60 cubed over 64 rightwards double arrow straight h cubed equals 60 cubed over 4 cubed rightwards double arrow straight h cubed equals open parentheses 60 over 4 close parentheses cubed rightwards double arrow straight h equals 60 over 4  rightwards double arrow straight h equals 15 space cm Thus space theeight space of space the space smaller space cone space straight s space 15 space cm. Thus comma space the space section space is space made space at space straight a space height space of space 60 minus 15 space equals 45 space cm space from space the space base.
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