Melting a sphere of wax

Let $s$ be the length of a side of the cubical mold, and $r$ be the radius of the sphere

The statement that the sphere of wax fills half the cubical mold tells us the relationship between the volumes:

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

Therefore, using the formulas for those volumes,

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

Solving for s,

$s = (\frac 8 3 \pi)^{1/3} r$

The question of whether or not the sphere would fit inside the cube can be answered by checking if the sphere's diameter $2r$ is smaller than a side of the cube $s$ . Therefore we have to check if it's true that

$2 r \le s$

Using the value we found above for s,

$2 r \le (\frac 8 3 \pi)^{1/3} r$

$2 \le (\frac 8 3 \pi)^{1/3}$

$2 \le 2.03$

which is true. We can therefore conclude that the original sphere of wax would fit inside the cubical mold.