To determine which solid is bounded by the set of all points at a given distance from a given point, let's explore the characteristics of each option provided:
A. Sphere:
- A sphere is defined as the set of all points in three-dimensional space that are at a fixed distance (called the radius) from a given point (called the center). This definition directly matches the description given in the question.
B. Cone:
- A cone consists of a circular base and a vertex. The points forming the surface of a cone are not at a fixed distance from a single point, as they vary in distance depending on their position along the height of the cone.
C. Cube:
- A cube is a solid with six equal square faces. The points on a cube's surface can be at various distances from any given point within or outside the cube. Therefore, a cube does not satisfy the condition of having all points at a fixed distance from a given point.
D. Cylinder:
- A cylinder has two parallel circular bases connected by a curved surface. The points on the surface of a cylinder are not all at a fixed distance from any given point within the cylinder or outside it.
Considering these definitions, we can see that the solid bounded by the set of all points at a given distance from a single, given point is indeed a sphere.
Thus, the correct answer is A. Sphere.
