Question

A right circular cone of maximum volume is cut from a cube of side $a$. The ratio of the volumes of the cone and remaining portion of the cube is

• A. π : 6 – π
• B. 2π : 12 – π
• C. 3π : 12 – π
• D. π : 12 – π

Answer 1

The correct option is D π : 12 – π

A right circular cone of maximum volume can be fit in a cube of side $a$, if base of the cone touches the face of the cube and top of the cone touches the upper face of the cube.

Thus, radius of the cone $\left(r\right)$ is $\frac{a}{2}$ and height of the cone $\left(h\right)$ is $a$.
$\therefore \text{Volume of cone}=\frac{1}{3}\pi r^{2}h$
$=\frac{1}{3}\pi \times {\left(\frac{a}{2}\right)}^2\times a$
$=\frac{\pi a^3}{12}$
∴ Ratio of volumes of the cone to the remaining portion of the cube
$=\frac{\pi a^3}{12}:(a^3-\frac{\pi a^3}{12})$
$=\frac{\pi a^3}{12}:\frac{a^3(12-\pi )}{12}$
$=\pi :(12-\pi )$
Hence, the correct answer is option (d).