# A right circular cone has a base radius of $6cm$ and a height of $8cm$. A cube-shaped box is inside the cone so that one face of the box ...

A right circular cone has a base radius of $6$ cm and a height of $8$ cm. A cube-shaped box is inside the cone so that one face of the box is contained in the base of the cone and the four upper corners are in contact with the inner curved surface of the cone.

Furthermore, there is a spherical ball that rests on top of the box,touching the inner curved surface of the cone.

Find the ratio of the volume of the ball to that of the box.

I tried to find the length of the edge of the box.But i could not figure it out as it is $D$.

Hint no. 1

If $x$ is the side of the cube, then the distance from the axis of the cone to a point of contact of a vertex of the cube with the curved surface of the cone is $\frac{x}{\sqrt{2}}$. Use similar triangles to calculate $x$.

Hint no. 2

If $\theta$ is the semi-vertical angle of the cone, $r$ is the radius of the sphere and $h=8-x$ then $$\sin\theta=\frac{r}{h-r}$$

• How is $\sinθ=\frac{r}{h-r}$? Feb 27, 2018 at 13:31
• Yes i have tried drawing it. Feb 27, 2018 at 13:38
• Are talking about the solid angle $θ$ of the cone? Feb 27, 2018 at 13:40
• But how do you find the radius of the sphere? Feb 27, 2018 at 16:25
• Thank you for your precious hints... Feb 28, 2018 at 7:32