$\"\"$

Volume of a sphere is $\"\"$ .

If $\"\"$ then the volume is :

$\"\"$

Therefore, the volume is $\"\"$ .

$\"\"$

From the problem $\"\"$ :

The volume of the resulting ring is $\"\"$ .

Set the equation equal to the half of the volume of the sphere with redius being equal to $\"\"$ .

$\"\"$

The value of $\"\"$ is $\"\"$ .