Question

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to its base. If its volume is $\frac{1}{27}$ of the volume of the given cone, at what height above the base is the section made?

Answer 1

Given,
Height of original cone, $H=30\text{cm}$
Let, Radius of original cone $=R$
Height of new cone $=h$
Radius of new cone $=r$

Volume of original cone
$=\frac{1}{3}\pi R^{2}H=\frac{1}{3}\pi R^2\times 30=10\pi R^{2}c{m}^3$

Volume of the new cone formed
$=\frac{1}{3}\pi r^{2}h$

Given, volume of the new cone is $\frac{1}{27}$ times the volume of the original cone.
$⟹\frac{1}{3}\pi r^{2}h=\frac{1}{27}\times 10\pi R^2⟹h=\frac{30}{27}\frac{R^2}{r^2}⟹h=\frac{10}{9}{\left(\frac{R}{r}\right)}^2$

Also, from similarity of triangles, $\frac{R}{r}=\frac{30}{h}$

$\therefore h=\frac{10}{9}\times (\frac{30}{h}{)}^{2}h\times h^2=10\times \frac{900}{9}{h}^3=10\times 100h=\sqrt[3]{310\times 10\times 10}h=10\text{cm}$

The height at which the section is made is 20 cm above the base.