Question

The height of a cone is 20 cm. A small cone is cut off from the top by a plane parallel to the base. If its volume be 1/125 of the volume of the orginal cone, determine at what height above the base the section in made.

Answer 1

Solution:-
Given : Height of the cone = 20 cm
Let the small cone is cut off at a height 'h' from the top.
Let the radius of the big cone be 'R' and of small cone be 'r'
Let the volume of the big cone be V1 and of small cone be $V_2$
Volume of the big cone = $=\frac{1}{3}\pi R^2\times 20=\frac{20\pi R^2}{3}c{m}^3$

Volume of small cone $=\frac{1}{3}\pi r^{2}h$
⇒ $V_2=\frac{1}{125}$th of the volume of big cone.
⇒

$\frac{V_2}{V_1}=\frac{1}{125}$

$=>\frac{\frac{1}{3}\pi r^{2}h}{\frac{20\pi R^2}{3}}=\frac{1}{125}$

$=>\frac{r^{2}h}{20R^2}=\frac{1}{125}$

$=>\frac{r^2}{R^2}\times \frac{h}{20}=\frac{1}{125}$ ...(1)
⇒ From the figure. Δ ACD ~ Δ AOB (By AA similarity criterion)
⇒$\frac{r}{R}=\frac{h}{20}$
Putting this value of $\frac{r}{R}=\frac{h}{20}$ in equation (1), we get.

$(\frac{h}{20}{)}^2\times \frac{h}{20}=\frac{1}{125}$

$(\frac{h}{20}{)}^3=\frac{1}{125}$

$\frac{h}{20}=\frac{1}{5}$

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

So, the height above the base where the section is made is 20 - 4 = 16 cm
Answer.