Question

A hollow cone is cut by a plane parallel to the base and then the upper portion is removed. so now If the csa of 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.​

Answer 1

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 Let the:-

⭕️ Height of the larger cone = H

⭕️Height of the smaller cone = h

⭕️ radius of the Larger cone = R

⭕️radius of the smaller circle = r

$⇒ h/H = r/R = l/L$

It's given that the :-

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

$⇒ π (R + r) (L – l) = (8/9) × π × R × L$

 $⇒ (1 + r/R) (1 – l/L) = (8/9)$

 $⇒ (1 + h/H) (1 – h/H ) = (8/9)$

⭕️Simplifying, we get h²/H² = 1/9

 $∴ h/H = 1/3$

 $∴ h/(H- h) = 1/2$

Answer 2

Step-by-step explanation:

Assume that the ratio of the altitude of the bigger and the smaller cone be k:1.

Let R and r be the radii of the bigger and the smaller cone respectively.

Let H and h be the height of the bigger and the smaller cone respectively.

Consider the similar triangles △ AGC & △ AFE ,

By the property of similarity, we have,

AF

AG

=

FE

GC

H

h

=

R

r

=

k

1

, where k is some constant.

Curved surface area of bigger cone = πRL, where L is the slant height of the bigger cone.

Curved surface area of smaller cone = πrl, where l is the slant height of the smaller cone.

Again by the property of similarity, we have,

L

l

=

R

r

=

k

1

Given that the ratio of the curved surface area of the frustum of the cone to the whole cone is

9

8

.

The ratio of the curved surface area of the smaller cone to the bigger cone is

9

1

.

πRL

πrl

=

k

2

1

=

9

1

k=3

H

h

=

3

1

Therefore,

H−h

h

=

3−1

1

=

2

1

Hence, the required ratio is 1:2.

solution