Question

A cone of radius $8cm$ and height $12cm$ is divided into two parts by a plane through the mid-point of its axis parallel to its base. Find the ratio of the volumes of two parts.

Answer 1

Step 1- Finding the radius of the smaller cone

To find the ratio of the volume of the two parts made by dividing the cone into two parts by a plane through the mid-point of its axis parallel to its base.

From the figure, Here $OM=12cm$

Hence, let us assume the mid-point of cone is P

$$\begin{gathered} OP=PM=6cm \\ From△OPDand△OMN \\ \angle POD=\angle POD\left[Common\right] \\ \angle OPD=\angle OMN[Both90°] \end{gathered}$$

Hence, by the Angle-Angle similarity criterion

So, $△OPD~△OMN$

We know that

Similar triangles have corresponding sides in equal ratio,

$$\begin{gathered} \frac{PD}{MN}=\frac{OP}{OM} \\ \frac{PD}{8}=\frac{6}{12} \\ PD=4cm \end{gathered}$$

So, the radius of the cone OCD is $4cm$

Radius of base of cone ORN is $MN=8cm$

Step 2- Finding the volume of the smaller cone

For the cone, OCD

Base Radius, $r=PD=4cm$

Height,$h=OP=6cm$

$$\begin{gathered} VolumeofconeOCD=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h} \\ \mathrm{Volume}\mathrm{of}\mathrm{cone}\mathrm{OCD}=\frac{1}{3}\mathrm{\pi }\times 4^2\times 6 \\ \mathrm{Volume}\mathrm{of}\mathrm{cone}\mathrm{OCD}=32\mathrm{\pi } \end{gathered}$$

Step 3- Finding the volume of the frustum

Now, for second part of cone i.e. Frustum CDNR

Bottom radius, $r_1=MN=8cm$

Top Radius, $r_2=PD=4cm$

Height,$h=PM=6cm$

$$\begin{gathered} Volumeoffrustum=\frac{1}{3}\mathrm{\pi h}\left({\mathrm{R}}^2+{\mathrm{r}}^2+\mathrm{Rr}\right) \\ \mathrm{Volume}\mathrm{of}\mathrm{frustum}=\frac{1}{3}\mathrm{\pi }\times 6\left(8^2+4^2+8\times 4\right) \\ \mathrm{Volume}\mathrm{of}\mathrm{frustum}=224\mathrm{\pi } \end{gathered}$$

Step 4- Finding the ratio of the two parts

$$\begin{gathered} \mathrm{Ratio}=\frac{\mathrm{Ratio}\mathrm{of}\mathrm{first}\mathrm{part}}{\mathrm{Ratio}\mathrm{of}\mathrm{second}\mathrm{part}} \\ \mathrm{Ratio}=\frac{\mathrm{Volume}\mathrm{of}\mathrm{cone}\mathrm{OCD}}{\mathrm{Volume}\mathrm{of}\mathrm{frustum}\mathrm{CDNR}} \\ \mathrm{Ratio}=\frac{32\mathrm{\pi }}{224\mathrm{\pi }} \\ \mathrm{Ratio}=1:7 \end{gathered}$$

Hence, the ratio of the volume of the two parts made by dividing the cone into two parts by a plane through the mid-point of its axis parallel to its base is $1:7$.