Question

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

Answer 1

R.E.F image

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 be $r_1$

and small cone be $r_2$

$v_1=\frac{1}{3}\pi r^{2}h$

volume of the big cone.

$\Rightarrow v_1=\frac{1}{3}\pi r_1^2\times 30=10\pi r_1^{2}cu.cm$

volume of the small cone,

$\Rightarrow v_2=\frac{1}{3}.\pi r^{2}h$

Given $v_2=\frac{1}{27}.v_1$

$\Rightarrow \frac{v_2}{v_1}=\frac{1}{27}$

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

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

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

From the figure, $△ACD\sim △AOB$ (AA similarity criterion)

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

Hence equation (1), become's

$(\frac{h}{30}{)}^2.\frac{h}{30}=\frac{1}{27}$

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

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

$\therefore$ h = 10 cm

Thus at a height 20 cm above the base a

small cone is cut.