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 be $\frac{1}{27}$ of the volume of the given cone, at what height, above the base is the section cut ?

• A. 20
• B. 24
• C. 25
• D. 34

Answer 1

Answer: (A)

20
Lets consider a cone of Radius .Let a cone of height 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.

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 $ADE$
$=>\frac{VolumeoftheconeABC}{VolumeoftheconeADE}=\frac{1}{27}$
$=>\frac{\frac{1}{3}\pi r^{2}h}{\frac{1}{3}\pi R^{2}h}=\frac{1}{27}$
$=>(\frac{r}{R}{)}^2\times \frac{h}{H}=\frac{1}{27}$
$=>(\frac{h}{H}{)}^2\times \frac{h}{H}=\frac{1}{27}$ (from (i))
$=>(\frac{h}{H}{)}^3=\frac{1}{27}$
$=>\frac{h}{H}=\frac{1}{3}$
$=>h=\frac{1}{3}H$
$=>h=\frac{1}{3}\times 30cm$
$=>h=10cm$
Now,
$PQ=H-h$
$=30cm-10cm$
$=20cm$
Hence, the section is cut at the height of $20cm$ from the base.