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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 1 27 of the Volume of the Given Cone, Then the Height Above the - Mathematics

MCQ

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, then the height above the base at which the section has been made, is

Options

  • 10 cm

  • 15 cm

  • 20 cm

  • 25 cm

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Solution

Let VAB be cone of height 30 cm and base radius r1 cm.

Suppose it is cut off by a plane parallel to the base at a height h2 from the base of the cone.

Clearly 

\[∆ VOD~ ∆ VO'B\]

Therefore,

\[\frac{OV}{O'V} = \frac{OD}{O'B}\]

\[ \Rightarrow \frac{h_1}{30} = \frac{r_2}{r_1}\]

But,

\[\text { Volume of cone VCD}  = \frac{1}{27}\text { Volume of cone VAB }\]

\[ \Rightarrow \frac{1}{3}\pi \left( r_2 \right)^2 h_1 = \frac{1}{27}\left( \frac{1}{3}\pi \left( r_1 \right)^2 30 \right)\]

\[ \Rightarrow \left( \frac{r_2}{r_1} \right)^2 h_1 = \frac{10}{9}\]

\[ \Rightarrow \left( \frac{h_1}{30} \right)^2 h_1 = \frac{10}{9}\]

\[ \Rightarrow h_1 = 10\]

Hence,

Required height 

= 30 -10

= 20 cm

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 10 Maths
Chapter 14 Surface Areas and Volumes
Q 11 | Page 88
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