Question

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

Answer 1

We have,

Height of cone $=30cm.$

Let the small cone is cut of height “h” from the top

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

Then ,

The volume of big cone is $=V_1=\frac{1}{3}\pi {r_1}^{2}h$

$\begin{align}

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

$\Rightarrow V_1=10\pi {r_1}^{2}cuc{m}^3.$

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

But given,

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

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

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

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

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

Now, The figure $\Delta ACD\sim \Delta AOB$ (A.A. similarity )

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

$\Rightarrow h=10cm.$

Thus at a height 20 cm.

The base a small cone is cut.