Question

an hollow cone is cut by a plane paralel to the base and the upper portion is removed. curved surface area of the remainder is 8/9 of the curved surface of the whole cone.find the ratio of the line segment into which the cones altitude is divided by the plane

Answer 1

let for the whole cone,
radius = R
height = H
slant height = L

for the removed of the cone,(which is also a cone)
radius = r
height = h
slant height = l

Since they are from same cone, $\frac{r}{R} = \frac{l}{L} = \frac{h}{H}$

CSA of remainder = 8/9 of original
so CSA of removed portion = 1- 8/9 = 1/9 of original

CSA of original cone = πRL
CSA of removed cone = πrl

$\frac{ \pi rl}{ \pi RL} = \frac{1}{9} \\ \\ \frac{r^2}{R^2}= \frac{1}{9} \\ \\ \frac{r}{R} = \frac{1}{3} \\ \\ \frac{h}{H} = \frac{1}{3}$

So the cones altitude is divided in the ratio $\frac{1}{3}$.