Question

The height of a cone is $30cm$. A frustum is out off from this cone by a plane parallel to the base of the cone. If the volume of the frustum is $\frac{1}{27}$ of the volume of the cone, find the height of the frustum.

• A. $20cm$
• B. $12cm$
• C. $15cm$
• D. $18cm$

Answer 1

The correct option is A $20cm$

Lets consider a cone of Radius $R$.

Let a cone of height $h$ is cut off from the top of this cone whose base is parallel to the original cone. The radius of the the cone cut off be $r$.

Here, $H=30cm$

In $△APC$ and $△AQE$, $PC\parallel QE$

Therefore,

$△APC\sim △AQE$

$=>\frac{AP}{AQ}=\frac{PC}{QE}$

$=>\frac{h}{H}=\frac{r}{R}$ (I)

Given, Volume of the cone$ABC$$=\frac{1}{27}$Volume of the cone $ADE$

$=>\frac{\text{Volume of the cone ABC}}{\text{Volume of the cone ADE}}=\frac{1}{27}$

$=>\frac{\frac{1}{3}\pi r^{2}h}{\frac{1}{3}\pi R^{2}h}=\frac{1}{27}$

$=>{\left(\frac{r}{R}\right)}^2\times \frac{h}{H}=\frac{1}{27}$

$=>{\left(\frac{h}{H}\right)}^2\times \frac{h}{H}=\frac{1}{27}$ ...(from (i))

$=>{\left(\frac{h}{H}\right)}^3=\frac{1}{27}$

$=>\frac{h}{H}=\frac{1}{3}$

$=>h=\frac{1}{3}H$

$=>h=\frac{1}{3}\times 30cm$

$=>h=10cm$

Now, $PQ=H-h$

$=30cm-10cm$

$=20cm$

Hence, the section is cut at the height of $20cm$ from the base.