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 its volume be $\frac{1}{27}$ of the volume of the given cone at what height above the base the section has been made ?

• A. 10 cm
• B. 15 cm
• C. 20 cm
• D. 25 cm

Answer 1

The correct option is C 20 cm

Here , $H=30cm$

In $△APC$ and $△AQE,PC\parallel QE$

$\therefore △APC\sim △AQE$

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

$\Rightarrow \frac{h}{H}=\frac{r}{R}$ ....(i)

Given that volume of cone ABC $=\frac{1}{27}$ volume of cone ADE

$\Rightarrow \frac{VolumeofconeABC}{VolumeofconeADE}=\frac{1}{27}$

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

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

$\Rightarrow {\left(\frac{h}{H}\right)}^2\times \frac{h}{H}=\frac{1}{27}$ using 1

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

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

$\Rightarrow h=\frac{1}{3}H=\frac{1}{3}\times 30cm=10cm$

Now $PQ=H-h=30cm-10cm=20cm$

Thus , the section of the cone is made at a height of $20cm$ above the base.