Question

If a cube of maximum possible volume is cut off from a solid sphere of diameter d, then the volume of the remaining (waste) material of the sphere would be equal to

Answer 1

Solution:-
Let the side of the given cube be 'a'
Therefore,
Diagonal of the cube = √3a²
= a√3
Now, according to the question,
a√3 = d
a = d/√3
Volume of the sphere = 4/3 × π × (d/2)³
= 4/3 × π × d³/8
= (1πd³)/6
Volume of the cube = a³ = (d/√3)³
= d³/3√3
Volume of the remaining waste material = (1πd³)/6 - (d³/3√3)
= d³(1π/6 - 1/3√3)
Answer.

Answer 2

Volume of remaining wast material= V = 4/3 π(d/2)³- (d/√3)³
let the side of cube be I, then diagonal of cube will be

= √I²+I²+I²

d=√3.I
I= d/√3

so V= d³/3(π/2 - 1/√3)