Question

A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is 8/9 of the curved of the whole cone, find the ratio of the line segments into which the altitude of the cone is divided by the plane.

Answer 1

In Fig. , the smaller cone APQ has been cut off through the plane PQ || BC. Let r and R be the radii of the smaller and larger cone and l and L their slant heights respectively.
Here, in the adjoining figure
OQ = r, MC = R, AQ = l, AC = L.
Now, ΔAOQ ~ ΔAMC
⇒ OQ/MC= AQ/AC
⇒ r/R= l/L

⇒ r=Rl/L         … (i)

Since, curved surface area of the remainder = 8/9 of the curved surface area of the whole cone, therefore, we get,

Hence, the required ratio of their heights = 1:2