Question

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

Answer 1

Let VAB be a cone of height $30cm$ and base radius $r$ cm. Suppose it is cut off by a plane parallel to the base at a height $h$ from the base of the cone.

Clearly $△VOA\sim △V{O}^{\prime }{A}^{\prime }$

$\therefore \frac{VO}{V{O}^{\prime }}=\frac{OA}{O^{\prime }{A}^{\prime }}\Rightarrow \frac{30}{h_1}=\frac{r}{r_1}...\left(i\right)$

It is given that
Volume of cone $V{A}^{\prime }{B}^{\prime }=\frac{1}{27}$ volume of cone VAB

$\Rightarrow \frac{1}{3}\pi r_1^2{h}_1=\frac{1}{3}\times \frac{1}{27}\pi r^2\times 30$

$\Rightarrow {\left(\frac{r}{r_1}\right)}^2{h}_1=\frac{10}{9}$

$\Rightarrow {\left(\frac{h_1}{30}\right)}^2{h}_1=\frac{10}{9}$......[From (i)]

$\Rightarrow h_1^3=1000\Rightarrow h_1=10cm$

$\therefore h=30-h_1=(30-10)cm=20cm$

Hence, the section is made at a height of $20cm$ from the base of the cone.