Question

Find the volume of the largest right circular cone that can be cut of a cube whose edge is $9cm$.

Answer 1

Hint: To find the volume of largest cone that can be cut out of a cube whose edge is $9cm$, take the maximum height of cone as the length of edge of cube, i.e., $9cm$ and the radius of cone as half of the length of the edge of the cube. Use the formula for calculating the volume of the cone which is $\dfrac{1}{3}\pi {{r}^{2}}h$.

We have to find the volume of the largest cone that can be cut out of a cube whose edge is $9cm$. To do so, we will find the maximum possible value of length and radius of the cone.
Let’s denote the height of the cone by $h$ and radius of the cone by $r$.
We observe that the maximum height of the cone can be equal to the length of the edge of the cube. Thus, we have $h=9cm$.
Similarly, the maximum possible radius of the cone is half of the length of edge of the cube as we have to fit the entire cone inside the cube. Thus, we have $r=\dfrac{9}{2}=4.5cm$.
We will now evaluate the volume of the cone.
We know that the volume of cone is $\dfrac{1}{3}\pi {{r}^{2}}h$, where $r$ denotes the radius of the cone and $h$ denotes the height of the cone.
Substituting $r=4.5cm,h=9cm$ in the above equation, we have the volume of cone $=\dfrac{1}{3}\pi {{r}^{2}}h=\dfrac{1}{3}\left( 3.14 \right){{\left( 4.5 \right)}^{2}}\left( 9 \right)$.
Simplifying the above expression, we have the volume of cone $=190.75c{{m}^{3}}$.
Hence, the volume of the largest right circular cone that can be fit in a cube of edge $9cm$ is $190.75c{{m}^{3}}$.

Note: Be careful about the units while calculating the volume of cones, otherwise we will get an incorrect answer. A right circular cone is a cone where the axis of the cone is the line meeting the vertex to the midpoint of the circular base.