Question

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

• A. 10 cm
• B. 20 cm
• C. 40 cm
• D. 15 cm

Answer 1

The correct option is B 20 cm

Let the radius and height of the original cone be R cm and H cm respectively.

Let the radius and height of the smaller cone be r cm and h cm respectively

Volume of initial cone = $\frac{1}{3}\pi R^{2}H=\frac{1}{3}\pi R^2\times 30=10\pi R^2$
Volume of the new cone formed = $\frac{1}{27}\times 10\pi R^2=\frac{1}{3}\pi r^{2}h\Rightarrow h=\frac{10}{9}{\left(\frac{R}{r}\right)}^2$

Triangles $△ACD$ and $△AOB$ are similar (AA similarity)

Therefore, $\frac{R}{r}=\frac{30}{h}$
$\therefore h=\frac{10}{9}\times \frac{30\times 30}{h\times h}$

$h^3=1000$

$h=10cm$
Height from the base $=30-10=20cm$