Question

If a cone of radius 10 cm is divided into two parts by drawing a plane through the mid-point of its axis, parallel to its base. Compare the volumes of the two parts.

Answer 1

Consider a cone of radius R and height H.

Let a cone of height `H/2` is cutout from this cone whose base is parallel to the original cone. Now is

`Delta AQD and Delta APC, QD || PC`

`Delta AQD ∼ Delta APC`

`(QD)/(PC) = (AQ)/(AP)`

`QD/2 = H/(2/H)`

`QD = R/2`

Volume of cone ABC `=1/3 pi R^2 H`

Volume of cone ADE

`=1/3 pi (R/2)^2 (H/2)`

`=1/8[1/3 piR^2H]`

`=1/8`volume of cone ABC

Now volume of remaining frustum of cons EDCB

= volume of cone ABC − volume of cone AED

\[= \frac{7}{8}(\text { volume of cone ABC})\]

Comparing the volume of cone AED and frustum EDBC we get, the ratio 1 : 7.