Answer:
r : s is √3 : 2
Explanation:
The given parameter are;
A cube is inscribed in a sphere
The side length of the cube = s
The radius of the sphere = r
The ratio r : e = Required
It is noted that for a cube inscribed in a sphere, we have;
The diameter of the sphere = The diagonal of the cube
The diameter of the sphere, D = 2 × The radius = 2·r
The square of the diagonal of the cube, d² = s² + s² + s² = 3·s²
∴ The diagonal of the cube, d = (√3)·s
From the relationship between the cube and the sphere in which it is inscribed, (The diameter of the sphere = The diagonal of the cube), we have;
2·r = (√3)·s
∴ r/s = (√3)/2
r : s = √3 : 2.