Write True or False and justify your answer.
An edge of a cube measures $r \, cm$. If the largest possible right circular cone is cut out of this cube,then the volume of the cone (in $cm^3$) is $\frac{1}{6} \pi r^3$.

(B) False.
For the largest possible right circular cone cut out of a cube of edge $r$,the height of the cone $(h)$ must be equal to the edge of the cube,so $h = r \, cm$.
The diameter of the base of the cone must be equal to the edge of the cube,so the radius of the base $(R)$ is $\frac{r}{2} \, cm$.
The volume of a cone is given by the formula $V = \frac{1}{3} \pi R^2 h$.
Substituting the values: $V = \frac{1}{3} \pi \left(\frac{r}{2}\right)^2 \cdot r = \frac{1}{3} \pi \left(\frac{r^2}{4}\right) \cdot r = \frac{1}{12} \pi r^3 \, cm^3$.
Since $\frac{1}{12} \pi r^3 \neq \frac{1}{6} \pi r^3$,the given statement is False.