10th Class
Mathematics
Surface Areas and Volumes
Question Bank
Surface Areas and Volumes
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, \[AC=30-10=20\text{ }cm~\]