The height of a cone is 40 cm. A small cone is cut off at the top by a plane par

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The height of a cone is 40 cm. A small cone is cut off at the top by a plane parallel to the base. If the volume of the small cone be

\frac{1}{64}

of the volume of the given cone, at what height ( in cm) above the base is the section made ?

Options:

A . 20

B . 30

C . 40

D . 50

Answer: Option B
:
B

The Height Of A Cone Is 40 Cm. A Small Cone Is Cut Off At Th...

Let

R

be the radius of the given cone,

r

the radius of the small cone,

h

be the height of the frustum and

h_{1}

be the height of the small cone.
In the figure,

△ O N C \sim △ O M A

∴ \frac{O N}{O M} = \frac{N C}{M A}

[Sides of similar triangles are proportional]

\Rightarrow \frac{h_{1}}{40} = \frac{r}{R}

\Rightarrow h_{1} = (\frac{r}{R}) \times 40

.......(i)
We are given that

\frac{V o l u m e o f s m a l l c o n e}{v o l u m e o f g i v e n c o n e} = \frac{1}{64}

\Rightarrow \frac{\frac{1}{3} π r^{2} \times h_{1}}{\frac{1}{3} π R^{2} \times 40} = \frac{1}{64}

\Rightarrow \frac{r^{2}}{R^{2}} \times \frac{1}{40} \times [(\frac{r}{R}) 40] = \frac{1}{64}

[ By (i)]

\Rightarrow {(\frac{r}{R})}^{3} = \frac{1}{64} = {(\frac{1}{4})}^{3}

\Rightarrow

\frac{r}{R} = \frac{1}{4}

...... (ii)
From (i) and (ii)

h_{1} = \frac{1}{4} \times 40 = 10 c m

∴, h = 40 - h_{1} = (40 - 10) c m

\Rightarrow h = 30 c m

Hence, the section is made at a height of 30 cm above the base of the cone .

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