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 its volume be `1/(27)` of the volume of the given cone, at what height above the base is the section mode?

Explanation

Let the smaller cone have radius = r cm and height = h cm
And, let the radius of the given original cone by R cm
Then, `(r)/(R) = (h)/(30) rArr r = (hR)/(30) ....(i) " " [ :' triangleOCD ~ triangleOAB]`
volume of smaller cone `=(1)/(27) xx`(volume of larger cone)
`:. (1)/(3) pi r^(2) h = (1)/(27) xx ((1)/(3) pi R^(2) xx 30)`
`rArr r^(2) h = (10)/(9) R^(2)`
`rArr ((hR)/(30))^(2) xx h = (10)/(9) R^(2)` [using (i)]
`rArr h^(3) = ((10)/(9) xx 900) = 1000 = (10)^(3) rArr h = 10 cm`
`:. AC = OA - OC = (30 - 10) cm =20 cm`
Hence, the section has been made at a height of 20 cm from the base.