Question

In fig, as shown a right circular cone of height 30cm. A small cone is cut off from the top by a plane parallel to the base. If the volume of the small cone is $\frac{1}{27}$ of the volume of given cone, fine at what height above the base is the section made.

Answer 1

Given height of the cone = 30 cm

Let the small cone is cut off at a height ‘h’ from the top

Let the radius of big cone be $r_1$ and small cone be $r_2$.

Volume of the big cone, $V_1=\frac{1}{3}\pi r^{2}h$

$\Rightarrow V_1=\frac{1}{3}\pi r_1^2\times 30=10\pi r_1^{2}cucm$

Volume of small cone, $V_2=\frac{1}{3}\pi r_2^{2}h$

Given $V_2=\frac{1}{27}{V}_1$

$\Rightarrow \frac{V_2}{V_1}=\frac{1}{27}$

$\Rightarrow \frac{\frac{1}{3}\pi r_2^{2}h}{10\pi r_1^2}=\frac{1}{27}$

$\Rightarrow \frac{r_2^{2}h}{30r_1^2}=\frac{1}{27}$

$\Rightarrow {\left(\frac{r_2}{r_1}\right)}^2\times \left(\frac{h}{30}\right)=\frac{1}{27}\to \left(1\right)$

From the figure, $DeltaACD\sim \Delta AOB$ [AA similarity criterion]

$\Rightarrow \frac{r_2}{r_1}=\frac{h}{30}$

Hence equation (1), becomes

${\left(\frac{h}{30}\right)}^2\times \left(\frac{h}{30}\right)=\frac{1}{27}$

$\Rightarrow {\left(\frac{h}{30}\right)}^3={\left(\frac{1}{3}\right)}^3$

$\Rightarrow \frac{h}{30}=\frac{1}{3}$

$\therefore h=10cm$

Thus at a heigh 20 cm above the base a small cone is cut.