Question 1181746
Here's how to analyze this problem and determine if there are any solutions:

**1. Visualize the Setup:** Imagine the cone with the four spheres inside. The first sphere sits at the bottom, and the other three rest on top of it, forming a small pyramid.  Crucially, all spheres are tangent to each other and the cone.

**2. Key Geometric Relationships:**

*   **Tangency:**  The centers of tangent spheres are separated by a distance equal to twice their radius (2R).
*   **Cone and Sphere:** The centers of the three upper spheres form an equilateral triangle.  The distance from the center of any of these spheres to the apex of the cone is related to the cone's dimensions and the sphere's radius.

**3. Analyze the Vertical Distances:**

*   The center of the bottom sphere is at a height R from the cone's apex.
*   The centers of the upper spheres are at a height 3R from the cone's apex (R from the apex to the center of the bottom sphere, plus 2R separating the centers of the bottom and top spheres).

**4. The Crucial Condition:**

The problem states that the upper spheres are tangent to the *top* of the vessel. This is where the issue arises. For the top spheres to be tangent to the top of the vessel, the top of the vessel must be at the same height as the top of the spheres. The top of the spheres is located at a height 4R from the apex of the cone (3R from the apex to the center of the top spheres, plus R from the center to the top).

**5. The Contradiction:**

For the three top spheres to also be tangent to the *sides* of the cone, the angle of the cone must be such that the spheres can fit snugly. The height of the cone required for three spheres of radius R to be tangent to each other and the sides of the cone is *less than* 4R. The condition that the spheres must be tangent to the *top* of the vessel forces the height to be 4R. These two requirements are incompatible.

**6. Conclusion:**

There are *no* solutions for the radius and altitude of the cone that satisfy all the given conditions. The requirement that the top spheres be tangent to both the sides and the top of the vessel creates a geometric contradiction.