A right circular cone is divided by a plane parallel to its base in two equal volumes.  In what ratio the plane divides the axis 

Asked by tultuldutta58 | 7th Jan, 2018, 09:54: PM

Expert Answer:

begin mathsize 16px style CD equals straight r
AB equals straight R
OC equals straight h
OA equals straight H end style
begin mathsize 16px style Let space the space height space and space the space radius space of space whole space cone space be space straight H space space and space straight R space respectively. space
The space cone space is space divided space into space two space parts space by space drawing space straight a space plane space
through space the space mid space point space of space its space height space and space parallel space to space the space base. space space
Let space the space radius space of space the space smaller space cone space be space straight r space cm. space
In space increment OCD space and increment OAB comma space
angle OCD space equals space angle OAB space space left parenthesis 90 degree right parenthesis space
angle COD space equals space angle AOB space space left parenthesis Common right parenthesis space
increment OCD space tilde space increment OAB space space left parenthesis AA space Similarity space criterion right parenthesis space space

straight r over straight R equals straight h over straight H....... left parenthesis 1 right parenthesis
fraction numerator volume space of space small space cone over denominator volume space of space large space cone end fraction equals fraction numerator begin display style fraction numerator πr squared straight h over denominator 3 end fraction end style over denominator fraction numerator πR squared straight H over denominator 3 end fraction end fraction space space equals 1 half
fraction numerator straight r squared straight h over denominator straight R squared straight H end fraction equals 1 half
from space left parenthesis 1 right parenthesis
straight h cubed over straight H cubed equals 1 half
straight h over straight H equals fraction numerator 1 over denominator cube root of 2 end fraction
now space we space need space
fraction numerator straight h over denominator straight H minus straight h end fraction equals fraction numerator 1 over denominator cube root of 2 minus 1 end fraction end style

Answered by Arun | 9th Jan, 2018, 12:35: PM

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