Question

The height of a cone is $40$ cm. A small cone is cut off at the top of a plane parallel to its base. If its volume is $\frac{1}{64}$ of the volume of the given cone, at what height above the base is the section cut?

• A. $20$cm
• B. $30$cm
• C. $40$cm
• D. $50$cm

Answer 1

Answer: (B)

$30$cm
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 cone cut off be.

Here,
$H=40cm$
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$ABC$=$\frac{1}{64}$Volume of the cone $ADE$
$=>\frac{\text{Volume of the cone ABC}}{\text{Volume of the cone ADE}}=\frac{1}{64}$

$=>\frac{\frac{1}{3}\pi r^{2}h}{\frac{1}{3}\pi R^{2}H}=\frac{1}{64}$

$=>(\frac{r}{R}{)}^2\times \frac{h}{H}=\frac{1}{64}$

$=>(\frac{h}{H}{)}^2\times \frac{h}{H}=\frac{1}{64}$ (from (i))

$=>(\frac{h}{H}{)}^3=\frac{1}{64}$

$=>\frac{h}{H}=\frac{1}{4}$

$=>h=\frac{1}{4}H$

$=>h=\frac{1}{4}\times 40cm$
$=>h=10cm$

Now,
$PQ=H-h$
$=40cm-10cm$
$=30cm$

Hence, the section is cut at the height of $30cm$ from the base.