CBSE Class 10 Answered
The height of cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27th of the volume of cone, at what height above the base is the section made?
Asked by Ritwika Sharma | 12 Mar, 2013, 08:48: PM
Expert Answer
Answer : 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 1/27th 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
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) = 27 (since volume of small cone is 1/27 of big cone)
=> R2 H = 27r2 h
=> 30 R2 = 27r2h { since the H = 30 cm given }
=> h = (30R2) / ( 27 r2 )
=> h = (30 /27)(R /r)2 .................................................(2)
From equating (1) and (2)
=> 30 (r /R) = (30 /27) (R/r)2
=>(r /R)3 = 1 / 27
=> r / R = 1/3........................................................(3)
substituting eq (3) in eq (1), we get
=> 30 (r /R) = (30 /27) (R/r)2
=>(r /R)3 = 1 / 27
=> r / R = 1/3........................................................(3)
substituting eq (3) in eq (1), we get
=> h = 30 (r /R)
=> h = 30 (1/3) = 10 cm
The section is made above the base is 30cm - 10cm = 20 cm Answer
=> h = 30 (1/3) = 10 cm
The section is made above the base is 30cm - 10cm = 20 cm Answer
Answered by | 13 Mar, 2013, 12:00: AM
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