A sphere is tightly enclosed inside a cube of side length a. If the cube is then inscribed inside another sphere, what is the ratio of the inner to outer sphere volumes?
Show answer & explanation
Correct answer: C
Explanation
Given:
A sphere is inscribed inside a cube.
The same cube is inscribed inside a larger sphere.
By the formula:
Volume of the sphere = (4/3)πr³
Let the side of the cube = a
Radius of the inner sphere = a/2
Volume₁ = (4/3)π(a/2)³
= (4/3)π(a³/8)
= πa³/6
Radius of the outer sphere = (diagonal of the cube)/2
= (a√3)/2
Volume₂ = (4/3)π((a√3)/2)³
= (4/3)π(3√3a³/8)
= π√3a³/2
Ratio = Volume₁ : Volume₂ = (πa³/6) : (π√3a³/2) = 1 3√3