Question
Fri March 13, 2015 By: R Aditya Narayanan

A solid cone of base radius 10cm is cut into two parts through the midpoint of its height , by a plane parallel to its base. Find the ratio in the volumes of two parts of the cone

Expert Reply
Vimala Ramamurthy
Sat March 14, 2015
G i v e n space t h a t space A E equals E B equals H over 2 comma space w h e r e space H space i s space t h e space h e i g h t space o f space t h e space c o n e space A B C 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 E A D equals angle B A C space space space space space space left square bracket c o m m o n right square bracket T h u s 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 triangle A E D tilde space triangle A B C T h u s comma space t h e space s i d e s space a r e space p r o p o r t i o n a l. rightwards double arrow fraction numerator A E over denominator A B end fraction equals fraction numerator E D over denominator B C end fraction equals r over 10 comma space w h e r e space r space i s space t h g e space r a d i u s space o f space t h e space c o n e comma space w h e n space a space p l a n e space p a r a l l e l space t o space i t s space b a s e space c u t s space t h e space h e i g h t space a t space i t s space m i d p o i n t. rightwards double arrow fraction numerator begin display style H over 2 end style over denominator H end fraction equals r over 10 comma space w h e r e space H space i s space t h e space h e i g h t space o f space t h e space c o n e rightwards double arrow 1 half equals r over 10 rightwards double arrow r equals 5 space c m V o l u m e space o f space t h e space f r u s t u m space o f space t h e space c o n e equals V o l u m e space o f space t h e space c o n e space A B C minus V o l u m e space o f space t h e space u p p e r space p a r t space o f space t h e space c o n e space A E D equals 1 third πR squared straight H minus 1 third straight pi straight R over 4 squared straight H over 2 comma space Here comma space straight R equals 10 space cm space and space straight R over 2 equals 5 space cm rightwards double arrow straight V straight o straight l straight u straight m straight e space straight o straight f space straight t straight h straight e space straight f straight r straight u straight s straight t straight u straight m space straight o straight f space straight t straight h straight e space straight c straight o straight n straight e equals 1 third straight pi 100 cross times straight H minus 1 third straight pi open parentheses 25 close parentheses straight H over 2 equals fraction numerator 175 πH over denominator 6 end fraction Volume space of space the space cone space AED equals 1 third straight pi 5 squared straight H over 2 equals fraction numerator 25 πH over denominator 6 end fraction Ratios space of space their space volumes space equals fraction numerator fraction numerator 25 πH over denominator 6 end fraction over denominator fraction numerator 175 πH over denominator 6 end fraction end fraction equals 1 over 7
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