Suppose a right circular cone and a solid sphere possess equal volumes, and the radius of the sphere is identical to the radius of the cone's circular base. What is the ratio of the cone's vertical height to its base radius?

The correct option is 4:1.

Let us solve the problem step-by-step by using the geometric volume formulas for a cone and a sphere.

Step 1: Define the variables

Let $r$ be the radius of the circular base of the right circular cone.
According to the question, the radius of the solid sphere is identical to the radius of the cone's base, so the radius of the sphere is also $r$.
Let $h$ be the vertical height of the cone.

Step 2: Write down the volume formulas

The volume of a right circular cone ($V_{\text{cone}}$) is given by:

$V_{\text{cone}}=\frac{1}{3}\pi r^{2}h$

The volume of a solid sphere ($V_{\text{sphere}}$) is given by:

$V_{\text{sphere}}=\frac{4}{3}\pi r^3$

Step 3: Equate the two volumes

Since the problem states that the cone and the sphere have equal volumes:

$V_{\text{cone}}=V_{\text{sphere}}$

$\frac{1}{3}\pi r^{2}h=\frac{4}{3}\pi r^3$

Step 4: Simplify to find the ratio of height to base radius

Dividing both sides of the equation by $\frac{1}{3}\pi r^2$:

$h=4r$

Now, divide both sides by $r$ to find the ratio $\frac{h}{r}$:

$\frac{h}{r}=\frac{4}{1}$

Thus, the ratio of the cone's vertical height to its base radius is 4:1.