question_answer

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 \[\frac{1}{27}\]th of the volume of the given cone, at what height above the base is the section made? [SSC CGL Tier II, 2014]

A) 19 cm

B) 20 cm

C) 12 cm

D) 15 cm

Correct Answer: B

Solution :

[b] According to the question, From the diagram \[\Delta AQ'\,E\]and \[\Delta AOC\] are similar. Hence,\[\frac{AQ'}{AQ'}=\frac{OC}{Q'E},\frac{30}{h}=\frac{R}{r}\] Let, volume of small cone v, and volume of main cone \[{{\text{v}}_{2}}.\] Then,    \[{{V}_{1}}=\frac{1}{2}{{V}_{2}}\] \[\frac{1}{3\,}\,{{\pi }^{2}}h=\frac{1}{27}\times \frac{1}{3}\,\pi {{R}^{2}}\times 30\] \[\Rightarrow \]   \[{{\left( \frac{r}{r} \right)}^{2}}\times h=\frac{10}{9}\] \[\Rightarrow \]   \[{{\left( \frac{h}{30} \right)}^{2}}\times h=\frac{10}{9}\] \[\Rightarrow \]   \[{{h}^{3}}=\frac{10\times 30\times 30}{9}\] \[\Rightarrow \]   \[{{h}^{3}}=1000\]\[\Rightarrow \]\[h=10\,cm\] Small cone is cut off at hight H. Hence, \[H=30'-h=30-10=20\,cm\]