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

Hi friend,

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Let the Height of the larger cone = H

Let the height of the smaller cone = h

Let the radius of the Larger cone = R

Let the radius of the smaller circle = r

h/H = r/R = l/L

Curved Surface area of the frustum = (8/9) Curved surface area of the cone.

pi(R + r)(L – l) = (8/9)piRL

(1 + r/R)(1 – l/L) = (8/9)

(1 + h/H)(1 – h/H ) = (8/9)

On simplifying, we get h2/H2 = 1/9

Therefore h/H = 1/3

Therefore h/(H- h) = 1/2

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Hope this helps u..
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Answer 2

let R, H and L be the radius, height and slant height respectively 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 (angle angle similarity)
curved surface area of the smaller cone
⇒ curved surface area of cone – curved surface area of the frustum
∴  1 : 2 is the answer