Question

A birthday conical cap is cut by a plane parallel to its base and the upper part is used as a new cap for a toy. The CSA of this new cap is 1/9 th of the CSA of the whole cone.Find the ratio of the line segments into which the cone's altitude is divided by the plane.

Answer 1

Let the radius, height and slant height of a conical cap be R, H and L units respectively.

And the radius, height and slant height of new cone be r,hr,h and ll respectively.


According to quesiton,
πrl=1/9πRL
=> r/R×l/L=1/9 ..........(i)

Now, in triangle ADE and triangle ABC,

angle A = angle A [Common]

angle ADE = angle ABC [Corresponding angles]

Therefore, ΔADE∼ΔABCΔADE∼ΔABC [AA similarity]

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

Now, in eq. (i), h/H×h/H=1/9

=> h/H=1/3 => H/h=3/1

=> H/h−1=3/1−1 =>
(H−h)/h=(3−2)/1
(H−h)/h=3−2/1

=> ( H−h)/h=2/1

Therefore, the required ratio is 2 : 1.