The problem asks to find the height of a right cone that can be inscribed inside a sphere with a radius of 9 cm, such that the volume of the cone is maximized.

Key Information:

• The radius of the sphere is 9 cm.
• The cone is right, meaning it has a circular base and a perpendicular height.
• We need to determine the height of this cone that will give the maximum possible volume.

Steps to solve:

1. Relation between the sphere and cone dimensions:

   • If the cone is inscribed in the sphere, the diameter of the sphere is the slant height of the cone. So, the slant height of the cone is $2 \times 9 = 18 \, \text{cm}$.

2. Maximizing the volume of the cone: The volume $V$ of a cone is given by: $V = \frac{1}{3} \pi r^2 h$ where:

   • $r$ is the radius of the base of the cone.
   • $h$ is the height of the cone.

3. Use of Pythagoras Theorem: In a right cone inscribed in a sphere, the slant height $l$ of the cone, the height $h$, and the radius $r$ of the base are related by: $l^2 = r^2 + \left(\frac{h}{2}\right)^2$ Since $l = 18 \, \text{cm}$, we get: $18^2 = r^2 + \left(\frac{h}{2}\right)^2$

4. Optimization: To maximize the volume, we use calculus or geometrical reasoning to find the height that gives the maximum volume. It is known that the height that maximizes the volume of the cone inscribed in a sphere is $\frac{4}{3} \times \text{radius of the sphere}$.

5. Final answer: The radius of the sphere is 9 cm, so the height of the cone is: $h = \frac{4}{3} \times 9 = 12 \, \text{cm}$

Therefore, the height of the cone that maximizes its volume is 12 cm (Option B).

Let me know if you need further details or clarifications!

Follow-up questions:

1. How does the slant height of a cone relate to its radius and height?
2. Why does maximizing the height of a cone not necessarily maximize its volume?
3. Can a cone have the same height and radius when inscribed in a sphere?
4. How does the formula for the volume of a cone differ from that of a cylinder?
5. How would the result change if the radius of the sphere were different?

Tip:

When working with 3D objects like cones and spheres, always pay attention to the relationships between their dimensions (e.g., slant height, radius, and height). These relationships are often key to solving optimization problems.