Determine the ratio of the radius of the sphere to the height of the cone, given that both three-dimensional shapes possess the identical volume, and the sphere's radius is exactly 50% of the cone's radius.

The correct option is 1 : 1.

To find the ratio of the radius of the sphere to the height of the cone, let us define the variables for each shape:

Let $r_s$ be the radius of the sphere.
Let $r_c$ be the radius of the base of the cone.
Let $h$ be the height of the cone.

From the problem statement, we are given two conditions:

1. The sphere's radius is exactly 50% of the cone's radius:
$r_s=0.5r_c=\frac{1}{2}{r}_c$
This implies that:
$r_c=2r_s$

2. Both shapes have identical volumes ($V_s=V_c$).

The formula for the volume of a sphere is:
$V_s=\frac{4}{3}\pi r_s^3$

The formula for the volume of a cone is:
$V_c=\frac{1}{3}\pi r_c^{2}h$

Equating the two volume expressions:
$\frac{4}{3}\pi r_s^3=\frac{1}{3}\pi r_c^{2}h$

Simplifying the equation by canceling $\frac{1}{3}\pi$ from both sides:
$4r_s^3=r_c^{2}h$

Now, substitute $r_c=2r_s$ into the equation:
$4r_s^3={\left(2r_s\right)}^{2}h$
$4r_s^3=4r_s^{2}h$

Dividing both sides by $4r_s^2$ (since radius is non-zero):
$r_s=h$

Therefore, the ratio of the radius of the sphere to the height of the cone is:
$\frac{r_s}{h}=\frac{1}{1}$ or 1 : 1.