Final answer:
The rotation of the triangle formed by points A (0,0), B (0,6), and C (3,0) about the y-axis results in a three-dimensional shape known as a cone.
Step-by-step explanation:
When the points A (0,0), B (0,6), and C (3,0) are connected to form a triangle and then rotated about the y-axis, the resulting three-dimensional shape is a cone. This is due to the fact that the rotation around the y-axis of the line segment connecting A and B will form a circular base, while the rotation of point C will trace out the curved surface of a cone. The line segment AC will serve as the slant height, and the line segment AB as the radius of the base of the cone.
To understand this better, consider the coordinate system that has the origin at the center of the cylinder that would be formed if the line AB were to be rotated around the y-axis. The base of this cylinder lies in the xy-plane. However, point C has a non-zero x-coordinate, which means if you rotated A and C around the y-axis, you would create a slope from C down to A, forming the conical shape.