Question

A solid cone of base radius 10 cm is cut into two parts through the mid-point of its height, by a plane parallel to its base. Find the ratio in the volumes of two parts of the cone.

Answer 1

Let the height of the cone be H.
Now, the cone is divided into two parts by the parallel plane
∴ OC = CAH2
Now, In ∆OCD and OAB
∠OCD = OAB (Corresponding angles)
∠ODC = OBA (Corresponding angles)
By AA-similarity criterion ∆OCD ∼ ∆OAB
$$\begin{gathered} \therefore \frac{\mathrm{CD}}{\mathrm{AB}}=\frac{\mathrm{OC}}{\mathrm{OA}} \\ \Rightarrow \frac{\mathrm{CD}}{10}=\frac{H}{2\times H} \\ \Rightarrow \mathrm{CD}=5\mathrm{cm} \end{gathered}$$
$$\begin{gathered} \frac{\mathrm{Volume}\mathrm{of}\mathrm{first}\mathrm{part}}{\mathrm{Volume}\mathrm{of}\mathrm{second}\mathrm{part}}=\frac{{\displaystyle \frac{1}{3}}\mathrm{\pi }{\left(\mathrm{CD}\right)}^2\left(\mathrm{OC}\right)}{{\displaystyle \frac{1}{3}}\mathrm{\pi CA}\left[{\left(\mathrm{AB}\right)}^2+\left(\mathrm{AB}\right)\left(\mathrm{CD}\right)+{\mathrm{CD}}^2\right]} \\ =\frac{{\displaystyle {\left(5\right)}^2}}{{\displaystyle \left[{\left(10\right)}^2+\left(10\right)\left(5\right)+5^2\right]}} \\ =\frac{25}{100+50+25} \\ =\frac{25}{175} \\ =\frac{1}{7} \end{gathered}$$⇒CDR=H2×H⇒CD=R2
Volume of smaller coneVolume of whole cone=13π(CD)2OC13π(AB)2OA=(R2)H2R2H=18R2HR