Question

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

Answer 1

Solution:

As, given a plane cuts a shape which is in the form of cone.

The lower part will be in the shape of frustum and the upper part will be in the shape of cone.

→Curved surface of new cone= $\frac{1}{9}\times$  Curved surface of original cone

→→π r l =  $\frac{1}{9}\times$π R L

→→$\frac{r l}{RL}=\frac{1}{9}$-----(1)

As, the two cones will be similar, by looking at the two images of cone which i have drawn below:

In ΔABC and ΔA MN

∠1 = ∠2= 90°

∠3 is common.

ΔABC ~ΔAMN→→[Angle Angle Similarity]

When , triangles are similar their sides are proportional.

$\frac{h}{H}=\frac{r}{R}=\frac{l}{L}$----(2)

Equating (1) and (2)

→→$\frac{l^2}{L^2}=\frac{1}{9}$

→→$\frac{l}{L}=\frac{1}{3}$, as ratio can't be negative.that is $\frac{-1}{3}$------(3)

Equating (2) and (3)

we get , $\frac{h}{H}=\frac{1}{3}$