Question

A cone is divided into two parts by drawing a plane through a point which divides its height in the ratio $1:2$ starting from the vertex and the plane is parallel to the base. Find the ratio of the volume of the two parts.

• A. $1:26$
• B. $1:13$
• C. $1:12$
• D. $1:2$

Answer 1

The correct option is A $1:26$

Let the plane XY divide the height AD of cone ABC such that $AE:ED=1:2$,
where AED is the axis of the cone. Let $r_2$ and $r_1$ be the radii of the circular section XY and the base BC of the cone respectively and let $h_{1}h$ and $h_1$ be their heights.
$\frac{h_1}{h}=\frac{3}{2}$
$\Rightarrow h_1=\frac{3}{2}h$
$\frac{r_1}{r_2}=\frac{h_1}{h_1-h}=\frac{\frac{3}{2}h}{\frac{1}{2}h}=3$
Volume of the cone AXY=$\frac{1}{3}\pi r_2^2(h_1-h)$
$=\frac{1}{3}\pi r_2^2(\frac{3}{2}h-h)$
$=\frac{1}{6}\pi r_2^{2}h$
VolumeoffrustumXYBC=$\frac{1}{3}\pi h(r_1^2+r_2^2+r_1{r}_2)$
=$\frac{1}{3}\pi h(9r_2^2+r_2^2+3r_2^2)$
=$\frac{1}{3}\pi h\left(13r_2^2\right)$
$\frac{\text{Volume of cone AXY}}{\text{Volume of cone XYBC}}=\frac{\frac{1}{6}\pi r_2^{2}h}{\frac{13}{3}\pi r_2^{2}h}$
$=\frac{1}{26}$
The ratio between the volume of the cone AXY and the remaining portion BCXY is $1:26$