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 answer is 1 : 1.

Step-by-step Explanation:

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

From the question, we are given two conditions:
1. The sphere's radius is exactly 50% of the cone's radius:

$r_s=0.5\×r_c=\frac{1}{2}{r}_c\⇒r_c=2r_s$

2. Both three-dimensional shapes possess the identical volume ($V_{\text{sphere}}=V_{\text{cone}}$):

$\frac{4}{3}\mathrm{\π}{r}_s^3=\frac{1}{3}\mathrm{\π}{r}_c^{2}h$

We can simplify this volume equation by canceling out common terms $\frac{1}{3}\mathrm{\π}$ on both sides:

$4r_s^3=r_c^{2}h$

Now, substitute $r_c=2r_s$ into the equation:

$4r_s^3={\left(2r_s\right)}^{2}h$

$4r_s^3=4r_s^{2}h$

Divide both sides by $4r_s^2$:

$r_s=h$

Therefore, the ratio of the radius of the sphere to the height of the cone is:

$\frac{r_s}{h}=\frac{1}{1}$

Thus, the ratio is 1 : 1.