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 1/27 of the volume of the given cone, at what height above the base is the section made?

A)  20 cm                        

B)  25 cm            

C)  10 cm                        

D)  15 cm            

Correct Answer: A

Solution :

Let OC = h and AB = r                 Now, \[\Delta OAB\tilde{\ }\Delta OCD\]                         \[\frac{AB}{CD}=\frac{OA}{OC}\] \[\Rightarrow \]            \[\frac{r}{CD}=\frac{30}{h}\] \[\Rightarrow \]            \[CD=\frac{rh}{30}\] Volume of smaller cone \[=\frac{1}{27}\] (Volume of whole cone) \[\frac{1}{3}\pi {{(CD)}^{2}}h=\frac{1}{27}\times \frac{1}{3}\pi {{r}^{2}}\times 30\] \[27\times {{\left( \frac{rh}{30} \right)}^{2}}h={{r}^{2}}\times 30\,\,\Rightarrow {{h}^{3}}=\frac{30\times 30\times 30}{27}\] \[{{h}^{3}}=10\times 10\times 10\,\,\,\Rightarrow \,\,h=10\,cm.\] Required height, \[AC=30-10=20\text{ }cm~\]