Question: 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 is $ \frac{1}{27}th $ of the volume of the given cone, at what height above the base is the section made?
Options:
A) 19 cm
B) 20 cm
C) 12 cm
D) 15 cm
Show Answer
Answer:
Correct Answer: B
Solution:
- Let the height of small cone be h cm.
Given, volume of small cone
$ =\frac{1}{27}\times $ volume of large cone
$ \Rightarrow $ $ V _{s}=\frac{1}{27}V _{B} $
$ \Rightarrow $ $ \frac{V _{s}}{V _{B}}=\frac{1}{27} $
(i)
Then, $ \frac{\frac{1}{3}\pi r^{2}h}{\frac{1}{3}\pi R^{2}\times 30}=\frac{1}{27} $ $ [\because h=30] $
$ \Rightarrow $ $ \frac{h}{30}\times {{( \frac{r}{R} )}^{2}}=\frac{1}{27} $
(ii)
In above figure, $ \Delta ABC\sim \Delta ADE $
$ \Rightarrow $ $ \frac{AB}{BC}=\frac{AD}{DE} $
$ \Rightarrow $ $ \frac{h}{r}=\frac{30}{R} $
$ \Rightarrow $ $ \frac{r}{R}=\frac{h}{30} $
Now, from Eq. (ii)
$ \frac{1}{27}={{( \frac{h}{30} )}^{2}}\times \frac{h}{30} $
$ \Rightarrow $ $ {{(30)}^{3}}=27h^{3} $
$ \Rightarrow $ $ {{(30)}^{3}}={{(3h)}^{3}} $
$ \Rightarrow $ $ h=10 $
Then, $ BD=30-h=30-10=20cm $
BETA
