A solid metallic sphere of radius 4 cm is melted and recast into 4 identical cubes. What is the side of the cube?

The volume of a solid metallic sphere of radius r is given by:
Volume of sphere=
3
4
​
πr
3

Given that the radius of the sphere r=4 cm:

Volume of sphere=
3
4
​
×π×4
3
=
3
4
​
×π×64=
3
256π
​
 cm
3

Let the side of each identical cube be a. The volume of 4 identical cubes is:

Total volume of 4 cubes=4a
3

Since the sphere is melted and recast into these 4 cubes, their volumes must be equal:

4a
3
=
3
256π
​

a
3
=
3×4
256π
​

a
3
=
3
64π
​

Taking the cube root on both sides:

a=
3

3
64π
​


​
=4
3

3
π
​


​
 cm=
3

3
64π
​


​
 cm
Let's look at the given options format. The option D represents
3

3
16π
​


​
 cm if written as
3
16π
​

1/3
. However, re-evaluating the options provided in the prompt:
Option D says 4\pi33 which translates to 4
3

3
π
​


​
=
3

3
64π
​


​
.
Thus, the side of the cube is 4
3

3
π
​


​
 cm or
3

3
64π
​


​
 cm.