A Hollow cone is cut by a plane parallel to the base and upper portion is removed. If the curved surface of the remainder is 8/9 of the curved surface of the whole cone; find the ration of the line-segment into which the cone's altitude is divided by the plane.

Explanation

Let OAB be the given hollow cone cut by the plane CD parallel to base AB and let cone OCD eb removed. Then, the remainder is the frustum CABD of the given cone.
Let OE = h units, OF = H units,
OD = l units, OB = L units,
ED = r units, FB = R units,
ED = r units, FB = R units.
In `Delta OED" and " Delta OFB,` we have
`/_ EOD = /_FOB(" common ")`,
`/_OED=/_OFB = 90^(@)`.
`:. Delta OED Delta OFB`
`rArr" "(OE)/(OF)=(OD)/(OB)=(ED)/(FB)rArrh/H=l/L=r/R" ...(i)[by Thales' theorem]"`
Now, (curved surface area of the frustum CABD
`8/9("curved surface area of the cone OAB")`
`rArr" (curved surface are of the OCD)"`
= ( curved surface of the cone OAB )
- (curved surface of the frustum CABD)
= ( curved surface of the cone OAB )
`-8/9("curved surface of the cone OAB")`
`1/9("curved surface of the cone OAB")`
`rArr pirl=1/9piRL`
`rArr" "(r/R)(l/L)=1/9rArr(h/Hxxh/H)=1/9rArrh/H=1/3" [using (i)]"`
`rArr H = 3h." ...(ii)"`
Now, EF=(OF-OE)=(H-h)=(3h-h)=2h
`:." "(OE)/(EF)=h/(2h)=1/2`
Hence, OE : EF = 1 : 2