Question

The radius of a cone is $\sqrt{2}$ times the height of the cone. A cube of maximum possible volume is cut from the same cone. What is the ratio of the volume of the cone to the volume of the cube?
A. $3.18\pi $
B. $2.35\pi $
C. $1.65\pi $
D. can’t be determined

Answer 1

Hint: We first assume the radius and the height of the cone. We use the relation of two similar triangles to find the side of the cube and then find the ratio of the volume of the cone to the volume of the cube. The formula for the volume of a cone is $\dfrac{1}{3}\pi {{r}^{2}}h$ with $r,h$ being radius and height respectively.

Complete step-by-step solution:
Let us assume that the height of the cone is $AB=h$ units. The radius of a cone is $\sqrt{2}$ times the height of the cone.
Therefore, the radius of a cone is $BD=r=\sqrt{2}\times h=h\sqrt{2}$ units.
The formula for the volume of a cone is $\dfrac{1}{3}\pi {{r}^{2}}h$ cubic units.
So, the volume of the cone is $\dfrac{1}{3}\pi {{\left( h\sqrt{2} \right)}^{2}}\left( h \right)=\dfrac{2\pi {{h}^{3}}}{3}$ cubic units.

Now for the cube we assume that the side of the cube is $x$ unit. Therefore, $CE=BF=x$.
In the given figure we have two triangles where we take $\Delta AFE$ and $\Delta ABD$.
It’s given that $\angle AFE=\angle ABD={{90}^{\circ }}$. Also $\angle AEF=\angle ADB$.
We also know that the sum of three angles of a triangle is always equal to ${{180}^{\circ }}$.
We got two angles of each triangle being equal which means the remaining angle of each triangle is also equal.
So, $\angle FAE=\angle BAD$.
We know that if three angles of two triangles are the same then the triangles are similar triangles and the ratio of opposite sides of similar angles are equal.
Therefore, for $\Delta AFE$ and $\Delta ABD$ we get $\dfrac{AF}{AB}=\dfrac{FE}{BD}$.
Therefore, $\dfrac{h-x}{h}=\dfrac{{}^{x}/{}_{2}}{h\sqrt{2}}$. Simplifying we get
$\begin{align}
  & \dfrac{h-x}{h}=\dfrac{{}^{x}/{}_{2}}{h\sqrt{2}} \\
 & \Rightarrow \dfrac{h-x}{1}=\dfrac{x}{2\sqrt{2}} \\
 & \Rightarrow 2\sqrt{2}\left( h-x \right)=x \\
 & \Rightarrow x\left( 2\sqrt{2}+1 \right)=2\sqrt{2}h \\
 & \Rightarrow x=\dfrac{2\sqrt{2}h}{2\sqrt{2}+1} \\
\end{align}$.
The volume of the cube is ${{x}^{3}}={{\left( \dfrac{2\sqrt{2}h}{2\sqrt{2}+1} \right)}^{3}}$ as the volume is equal to the cube value of a side.
The ratio of the volume of the cone to the volume of the cube is
$\dfrac{\dfrac{2\pi {{h}^{3}}}{3}}{{{\left( \dfrac{2\sqrt{2}h}{2\sqrt{2}+1} \right)}^{3}}}=\dfrac{\pi {{\left( 2\sqrt{2}+1 \right)}^{3}}}{24\sqrt{2}}=1.65\pi $.
The correct option is C.

Note: The midpoint of the base of the cone divides the one side of the cube equally. Therefore, BC is the half of the side length of the side of the cube. The length is $\dfrac{x}{2}$. The representation of the 3-D figure of the cube has been done in the form of a square in 2-D dimension. We have ignored the breadth part. In the relationship we only used height and length.