Question

A cube whose lateral surface area is 400 cm² is inscribed inside a sphere symmetrically. Find out the volume of the sphere in cm³.

1. 200√3π 
2. 350√3π 
3. 450√3π 
4. 500√3π 

Answer 1

Correct Answer - Option 4 : 500√3π 

Given :

The lateral surface area of the cube is 400 cm²

Cube is inscribed inside a sphere symmetrically. 

Formula used :

The lateral surface area of the cube = 4a²    (where a is the side of the cube)

Longest diagonal of a cube = a√3 (where a is the length of the side of the cube)

Volume of the sphere = (4/3)π r³      (r is the radius of the sphere)

Calculations :

Let the side of the cube be ‘x’ cm

According to the question

4x² = 400    (Lateral surface area of the cube)

⇒ x = 10

Length of the longest diagonal of cube = x√3 = 10√3 cm

The diameter of the sphere = 10√3 cm (Diameter of the sphere is equal to the longest diagonal of the cube inscribed inside it)

Radius of the sphere = 5√3    (Radius = (1/2) of diameter)

Volume of the sphere = (4/3)π (5√3)³

⇒ 500√3π cm³   

∴ The volume of the sphere is 500√3π cm3