Question

A hollow cone is cut by a plane parallel to the base and upper part is removed to make a Turkish cap. If the curved surface area of the remainder is 24/25 of the curved surface area of the whole cone ,find the ratio of the line segments into which the cones height is divided by the plane from which the cut is made.

Answer 1

Let R, H and L be the radius, height and slant height of the original cone respectively and r, h and l be the radius, height and slant height of the smaller cone respectively.

In ∆OAB and ∆OCD,

∠OAB = ∠OCD (90°)
∠AOB = ∠COD (Common)
∴ ∆ OAB ≅ ∆OCD (AA Similarity)

Curved surface area of the smaller cone
= Curved surface area of cone – Curved surface area of the frustum

Thus, the cones altitude is divided in the ratio 1 : 2.