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 surface of the whole cone. find the ratio of the line segments into which the altitude of the cone is divided by the plane.

plzzz show step by step solution..

Answer 1

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

So, cones altitude is divided in the ratio 1 : 2