Question

The largest right circular cone is curved out of a cube whose edge is of length $p$ units. Find the volume of the cube.

• A. $\frac{2\pi p^3}{15}$
• B. $\frac{4\pi p^3}{15}$
• C. $\frac{\pi p^3}{15}$
• D. $\frac{\pi p^3}{12}$

Answer 1

Answer: (D)

$\frac{\pi p^3}{12}$

The largest sphere-cone out will have the diameter of its base and its height equal to the side of the cube.

$\Rightarrow$ Radius of the base of the cone $=\frac{p}{2}$

Volume of a cone is determined by $\frac{1}{3}\pi r^{2}h$ where $r$ is the radius of the base of the cone $h$ is the height of the cone.

So,

$V=\frac{1}{3}\times \pi \times {\left(\frac{p}{2}\right)}^2\times p$

$=\frac{\pi p^3}{12}$

Hence, option $D$ is correct.