question_answer

The height of a cone is \[30\text{ }cm.\] From its topside a small cone is cut by a plane parallel to its base. If volume of smaller cone is \[\frac{1}{27}\] of the given cone, then at what height it is cut from its base?

Answer:

Volume of original cone OAB

\[=\frac{1}{3}\pi {{R}^{2}}H\,\,c{{m}^{3}}\]

\[=\frac{1}{3}\pi \times {{R}^{2}}\times 30\,\,c{{m}^{3}}\]

\[=10\pi \,{{R}^{2}}\,c{{m}^{3}}\]

Volume of small cone

\[=\frac{1}{3}\pi {{r}^{2}}h\]

\[\text{=}\frac{\text{1}}{\text{27}}\text{ }\!\!\times\!\!\text{ volume}\,\,\text{of}\,\,\text{cone}\,\,\text{OAB}\,\,\text{(given)}\]

\[\frac{\text{1}}{\text{27}}\text{ }\!\!\times\!\!\text{ 10}\pi {{\text{R}}^{2}}=\frac{1}{3}\pi {{r}^{2}}h\]

\[h=\frac{\frac{1}{27}\times 10\pi {{R}^{2}}}{\frac{1}{3}\pi {{r}^{2}}}=\frac{10}{9}{{\left( \frac{R}{r} \right)}^{2}}\]

From similar \[\Delta \,\,OPD\] and \[\Delta \,\,OQB\]

\[\frac{QB}{PD}=\frac{OQ}{OP}=\frac{30}{h}\]

So                    \[\frac{R}{r}=\frac{30}{h}\]

\[h=\frac{10}{9}{{\left( \frac{30}{h} \right)}^{2}}=\frac{9000}{9{{h}^{2}}}\]

\[\Rightarrow \]               \[{{h}^{3}}=1000\] or \[h=10\,\,cm\]

So height from base \[=30-10=20\,\,cm\].