Question

The height of a cone is $20\text{cm}$. A small cone is cut off from the top by a plane parallel to the base. If its volume is $\frac{1}{125}$ of the volume of the original cone, determine at what height above the base the section is made.

Answer 1

Volume of a cone of radius $r$ and height $h$ is $=\frac{\pi r^{2}h}{3}$

$\Rightarrow \frac{1}{3}\pi r^{2}h=\frac{1}{125}\frac{1}{3}\pi R^2\left(20\right)\Rightarrow r^{2}h=R^2\frac{\left(20\right)}{125}-\left(1\right)$

Also $BE||CD$ (given)

In $△ABE$ & $△ACD$

$\angle BAE=\angle CAD$ (common )

$\angle ABE=\angle ACD$ (corresponding angles)

$△ABE\sim △ACD$ by $AA$ criterion

thereby the sides will be in proportion

$\Rightarrow \frac{BE}{CD}=\frac{AB}{AC}\Rightarrow \frac{r}{R}=\frac{h}{20}-\left(3\right)$

Substituting $\left(2\right)$ in $\left(1\right)$ we get

$\left(\frac{r^2}{R^2}\right)h=20\Rightarrow {\left(\frac{h}{20}\right)}^{2}h=\frac{20}{125}\Rightarrow h^3=\frac{20\times 20\times 20}{5\times 5\times 5}$

$\Rightarrow h=\frac{20}{5}=4\text{cm}$

Therefore, the section is made $(20-4)\text{cm}=16\text{cm}$ above the base of original cone.