Question

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

Answer 1

$△ABC\sim △ADE$ [ By AA similarity]

$\therefore \frac{AB}{AD}=\frac{BC}{DE}$

$\Rightarrow \frac{h}{20}=\frac{r_1}{r_2}$

$V_{ABC}=\frac{1}{125}{V}_{ADE}$

$\Rightarrow \frac{1}{3}\pi r_1^{2}h=\frac{1}{3}\pi r_2^2\times 20\times \frac{1}{125}$

$\Rightarrow {\left(\frac{r_1}{r_2}\right)}^2=\frac{20}{125h}$

$\Rightarrow {\left(\frac{h}{20}\right)}^2=\frac{20}{125h}$

$\Rightarrow h^3=\frac{20\times 20\times 20}{125}=64$

$\Rightarrow h=\sqrt[3]{364}=4cm$

Therefore section is made $(20-4=16cm)$ above the base.