QUESTION

# The height of a cone is 30 centimeters. A small cone is cut off at the top by a plane parallel to the base volume be $\dfrac{1}{27}$times of the given cone. At what height above cone, at what height above the base is the section?

Hint: We will use the volume of the cone to determine the height of the cone which is required. Volume of cone is given by ${{V}_{1}}=\dfrac{1}{3}\pi {{r}^{2}}h$, where r is the radius of the base and h is the height of the cone.

Given that the cone is of height 30 centimeters. From that cone, the small cone at the top by a plane parallel to the base volume be $\dfrac{1}{27}$ times of the given cone. We have to determine at what height above the cone and base is the section.
We have to find at what height it is cut from its base.
Let h be the height and r be the radius of the smaller cone. And H be the height of the given cone and R be the radius of the given cone.
Given H= 30cm
Volume of the smaller cone is ${{V}_{1}}=\dfrac{1}{3}\pi {{r}^{2}}h$……..(i)
Volume of the given cone is ${{V}_{2}}=\dfrac{1}{3}\pi {{R}^{2}}H$
Substituting the value of H in above expression we get,
$V_2$ = $\dfrac{1}{3}\pi {{R}^{2}}(30)$……….(ii)
Because the triangle or cone are similar in shape then from the property of similar triangles,
\begin{align} & \dfrac{h}{r}=\dfrac{H}{R} \\ & \Rightarrow \dfrac{h}{r}=\dfrac{30}{R} \\ & \Rightarrow r=\dfrac{(h)(R)}{30} \\ \end{align}
Substitute the value of r in equation (i)
\begin{align} & \Rightarrow {{V}_{1}}=\dfrac{1}{3}\pi {{\left( \dfrac{(h)(R)}{30} \right)}^{2}}h \\ & \Rightarrow {{V}_{1}}=\dfrac{1}{3}\pi \dfrac{({{h}^{3}}){{(R)}^{2}}}{{{(30)}^{2}}} \\ \end{align}
We are given, the volume of smaller cone is $\dfrac{1}{27}$ of the given cone, which gives,
$\Rightarrow {{V}_{1}}=\dfrac{1}{27}{{V}_{2}}$
$\Rightarrow \dfrac{1}{3}\pi \dfrac{({{h}^{3}}){{(R)}^{2}}}{{{(30)}^{2}}}=\dfrac{1}{27}\left( \dfrac{1}{3}\pi {{R}^{2}}(30) \right)$
\begin{align} & \Rightarrow {{h}^{3}}=\dfrac{{{(30)}^{3}}}{27} \\ & \Rightarrow {{h}^{3}}=1000 \\ & \Rightarrow h=10cm \\ \end{align}
Therefore, the smaller cone is cut from 30-10 = 20 cm from the base.
Hence, the height which was to be determined is 20cm.

Note: The possibility of error in this question can be not considering different volumes and therefore different height and radius of the cone of both the different cones, which will automatically give incorrect solutions.