Name: Area of Segment
Uploaded: 2019-07-19
Duration: 7 min 50 s
Description: Do you know the result for the area of a segment? Well, this video talks about the derivation of it. So, let's dive in and master the concept in just a few m...

Question

A circular cone is cut by a plane parallel to the base and the conical portion is removed. If the curved surface area of the frustum is

\frac{15}{16}

of the curved surface area of the whole cone, prove that the height of the frustum is

\frac{3}{4}

of the height of the whole cone.

Accepted Answer

Given:

Let $ABC$ is the cone of height $H$ with radius $R$ and slant $L$

Let $AXY$ is the cone of height $h$ with radius $r$ and slant $l$

Suppose it is cut by a plane parallel to the base.

$\frac{AO}{AP} = \frac{OC}{PY} = \frac{AC}{AY}$

$\frac{H}{h} = \frac{R}{r} = \frac{L}{l}$ ...(i)

From the given, the curved surface area of the frustum is $\frac{15}{16}$ of the curved surface area of the whole cone

Curved surface area of the smaller cone $=$ Curved surface area of cone $-$ Curved surface area of the frustum

Curved surface area of the smaller cone $=$ Curved surface area of the whole cone $- \frac{15}{16}$ Curved surface area of the whole cone.

Curved surface area of the smaller cone $= (1 - \frac{15}{16})$ Curved surface area of the whole cone.

Curved surface area of the smaller cone $= \frac{1}{16}$ Curved surface area of the whole cone.

$∴ \frac{Curved surface area of the smaller cone}{Curved surface area of the whole cone} = \frac{1}{16}$

$\Rightarrow \frac{πrl}{πRL} = \frac{1}{16}$

Using (i)

$\frac{r \times l}{R \times L} = \frac{h^{2}}{H^{2}} = \frac{1}{16}$

$\frac{h}{H} = \frac{1}{4}$

$∴ h = \frac{H}{4}$

$\frac{height of the frustum}{height of the whole cone} = \frac{H - h}{H} = \frac{H - \frac{H}{4}}{H} = \frac{\frac{3 H}{4}}{H} = \frac{3 H}{4} \times \frac{1}{H}$

$\frac{height of the frustum}{height of the whole cone} = \frac{3}{4}$

Hence, proved the height of the frustum is $\frac{3}{4}$ of the height of the whole cone.

A circular cone is cut by a plane parallel to the base and the conical portion is removed. If the curved surface area of the frustum is $\frac{15}{16}$ of the curved surface area of the whole cone, prove that the height of the frustum is $\frac{3}{4}$ of the height of the whole cone.

Important Questions on Mensuration

Learn and Prepare for any exam you want !

A circular cone is cut by a plane parallel to the base and the conical portion is removed. If the curved surface area of the frustum is 1516 of the curved surface area of the whole cone, prove that the height of the frustum is 34 of the height of the whole cone.

Important Questions on Mensuration

Learn and Prepare for any exam you want !

A circular cone is cut by a plane parallel to the base and the conical portion is removed. If the curved surface area of the frustum is $\frac{15}{16}$ of the curved surface area of the whole cone, prove that the height of the frustum is $\frac{3}{4}$ of the height of the whole cone.