Problem:
The largest possible right circular cone is cut out of a cube of edge r cm. What is the volume of the cone?

Solution:
To find the volume of the largest possible right circular cone, we need to understand the relationship between a cone and a cube. Let's break down the solution into the following steps:

Step 1: Understanding the problem
We have a cube with an edge length of r cm. The task is to find the largest possible right circular cone that can be cut out from this cube.

Step 2: Visualizing the situation
Imagine a cube with an edge length of r cm. Now, visualize a right circular cone inscribed inside the cube. The base of the cone is in contact with one face of the cube, and the apex of the cone lies at the center of the opposite face of the cube.

Step 3: Determining the dimensions of the cone
In this situation, the radius of the base of the cone is equal to half the edge length of the cube, which is r/2 cm. The height of the cone is equal to the diagonal of the cube, which can be found using the Pythagorean theorem.

Step 4: Calculating the volume of the cone
The volume of a cone can be calculated using the formula V = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height of the cone.

In this case, the radius (r) of the base is r/2 cm, and the height (h) is the diagonal of the cube. The diagonal of a cube can be calculated using the formula d = √(3) * r, where d is the diagonal and r is the edge length of the cube.

So, the height (h) of the cone is √(3) * r cm.

Now, substituting the values of r and h into the formula, we have:

V = (1/3) * π * (r/2)^2 * (√(3) * r)

Simplifying the equation further:

V = (1/3) * π * r^3 * (√3 / 4)

Step 5: Simplifying the result
The volume of the largest possible right circular cone is (1/3) * π * r^3 * (√3 / 4) cubic units.

Conclusion:
The volume of the largest possible right circular cone cut out of a cube with an edge length of r cm is (1/3) * π * r^3 * (√3 / 4) cubic units.