The height of a cone is 40 cm. A small cone is cut off at the top by a plane parallel to the base. If the volume of the small cone be 164 of the volume of the given cone, at what height ( in cm) above the base is the section made ?
30
Let R be the radius of the given cone, r the radius of the small cone, h be the height of the frustum and h1 be the height of the small cone.
In the figure,△ONC∼△OMA
∴ONOM=NCMA [Sides of similar triangles are proportional]
⇒h140=rR
⇒h1=(rR)×40 .......(i)
We are given that Volume of small conevolume of given cone=164
⇒13πr2×h113πR2×40=164
⇒r2R2×140×[(rR)40]=164 [ By (i) ]
⇒(rR)3=164=(14)3
⇒ rR=14 ...... (ii)
From (i) and (ii) h1=14×40=10 cm
∴,h=40−h1=(40−10) cm
⇒h=30 cm
Hence, the section is made at a height of 30 cm above the base of the cone .