Melting a cube of wax

Let $s$ be the length of a side of the cube, and $r$ be the radius of the spherical mold.

The statement that the cube of wax fills half the spherical mold tells us the relationship between the volumes:

$V_{cube} = V_{sphere}/2$

Therefore, using the formulas for those volumes,

$s^3 = \frac{\frac 4 3 \pi r^3}{2}$

Solving for s,

$s = (\frac 2 3 \pi)^{1/3} r$

The question of whether or not the cube would fit inside the sphere can be answered by checking if the cube's largest dimension is smaller than the diameter of the sphere. Through application of the Pythagorean theorem, we find that the longest diagonal distance between corners in the cube is $\sqrt{3} s$ . Therefore we have to check if it's true that

$\sqrt{3} s \le 2 r$

Using the value we found above for s,

$\sqrt{3} (\frac 2 3 \pi)^{1/3} r \le 2 r$

$\sqrt{3} (\frac 2 3 \pi)^{1/3} \le 2$

$2.22 \le 2$

which is false. We can therefore conclude that the original cube of wax would not fit inside the spherical mold.