Question

A cone is cut by a plane parallel to the base and
upper part is removed. If the curved surface area
of upper cone is = times the curved surface of
original cone. Find the ratio of line segment to
which the cone's height is divided by the plane.​

Answer 1

Answer:

If the curved surface area

of upper cone is = times the curved surface of

original cone.

Please mention the value by how many times the CSA exceeds.

Step-by-step explanation:

If your question is this:

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 8/9th of the curved surface of the whole cone, find the ratio of the line segments into which the cone's altitude is divided by the plane.

Then here's your answer..

Let R is the radius, H is the height and L is the slant height of the original cone and

let r is the radius, h is the height and l is the slant height of the smaller cone respectively.

Now in ΔOAB and ΔOCD,

∠OAB = ∠OCD  {each 90}

∠AOB = ∠COD {common}

So, by AA similarity,

ΔOAB ≅ ΔOCD

=> OB/OD = AB/CD = OA/OC

=> l/L = r/R = h/H

Now, curved surface area of the smaller cone = curved surface area of the cone - curved surface area of the frustum

=> curved surface area of the smaller cone = (1 - 8/9) * curved surface area of the cone

=> curved surface area of the smaller cone = (1/9) * curved surface area of the cone

=> curved surface area of the smaller cone/curved surface area of the cone = 1/9

=> πrl/πRL = 1/9

=> rl/RL = 1/9

=> (r/R)*(l/L) = 1/9

=> (h/H)*(h/H) = 1/9    {using equation 1}

=> (h/H)² = 1/9

=> (h/H) = 1/3

=> h = H/3

Now, OA/AC = h/(h - h)

=> OA/AC = (H/3)/(H - H/3)

=> OA/AC = (H/3)/(2H/3)

=> OA/AC = 1/2

=> OA : AC = 1 : 2

$\boxed{\text{1:2}}$

Answer 2

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