Question

If a cone is cut into two parts by a horizontal plane passing through the mid-point of its axis, then the ratio of the volumes of the upper part of the cone and the entire cone is:

• A. $1:2$
• B. $1:4$
• C. $1:6$
• D. $1:8$

Answer 1

The correct option is D

$1:8$

Let the radius and height of the given cone = R and H.
Let the radius and height of the upper part of the cone = $r$ and $h$.
Volume of the given cone$=\frac{1}{3}\pi R^{2}H$
From the figure:
$△AQD$ ~$△APC$
$\Rightarrow \frac{AQ}{AP}=\frac{QD}{PC}=\frac{AD}{AC}$
$\Rightarrow \frac{h}{H}=\frac{r}{R}$
$\Rightarrow \frac{\frac{H}{2}}{H}=\frac{r}{R}[\because h=\frac{H}{2}]$
$\Rightarrow \frac{r}{R}=\frac{1}{2}$
Ratio of the volume between the upper part of the cone and the entire cone is

$\frac{1}{3}\pi r^{2}h:\frac{1}{3}\pi R^{2}H=\frac{r^{2}h}{R^{2}H}=(\frac{1}{2}{)}^2\times \frac{\frac{H}{2}}{H}=\frac{1}{8}$
Hence the ratio is $1:8$.