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Question

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base if it's volume be half of the volume of the given cone.at what height above the base is the section  made


Solution

Given: The height of cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base and its volume be 1/2th of the volume of cone
To find : height above the base is the section made
 
let the height and radius of original(big) cone be H and R

let the height and radius of cut off (small) cone be h and r

from similar triangles, we know 

H / R = h / r

h = H r / R

since H = 30

h = 30 (r /R) .......................................................(1)

Volume of big cone, V = (1/3) pi R2 H

volume of small cone,v = (1/3) pi r2 h
now dividing , we get 
V / v = (R2 H )/ (r2 h) = 2 (since volume of small cone is 1/2 of big cone)

=> R2 H = 2r2 h 

=> 30 R2 = 2r2h                     { since the H = 30 cm given }

=> h = (30R2) / ( 2 r2 )

=> h = (30 /2)(R /r)2 .................................................(2)
 
 
From equating (1) and (2)
=> 30 (r /R) = (30 /2) (R/r)2
=>(r /R)3 = 1 / 2
=> r / R = (1/2)^1/3.......................................................(3)

substituting eq (3) in eq (1), we get
=> h = 30 (r /R)
=> h = 30 (1/2)^1/3 = 23.8cm

The section is made above the base is 30cm - 23.8cm = 6.2cm Answer  

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