Question

select the correct answer from each drop-down menu. a circle is rotated around a vertical axis. the circle does not intersect the axis. complete the following statement. the three - dimensional shape resulting from the rotation is a ▼ with a ▼ center.

Explanation:

Step1: Recall Rotation of Shapes

When a circle is rotated around a vertical axis that it does not intersect, we use the concept of solid of revolution. A circle rotated around an axis it doesn't intersect forms a torus? Wait, no, wait. Wait, if a circle is rotated around a vertical axis (a line) and the circle does not intersect the axis, the resulting 3D shape is a torus? Wait, no, actually, if the circle is in a plane, and we rotate it around an axis in the same plane (vertical axis) that is parallel? Wait, no, the axis is vertical, and the circle is in a plane. Wait, the standard case: if a circle is rotated around an axis that is at a distance \(d\) from the center of the circle (and \(d > r\), where \(r\) is the radius of the circle), then the resulting solid is a torus. But wait, the problem says "the circle does not intersect the axis". So the distance between the center of the circle and the axis is greater than the radius of the circle. Then, when we rotate the circle around the vertical axis, the 3D shape is a torus, which is a doughnut - shaped object. A torus has a center (the center of the doughnut - the axis of rotation is the central axis of the torus) and a radius (the radius of the circular cross - section, and the distance from the center of the torus to the center of the cross - section circle).

Wait, maybe I made a mistake. Wait, another approach: when a circle is rotated around an axis that is parallel to its plane? No, the axis is vertical, and the circle is in a plane. Let's think of the circle in the \(xy\) - plane, and the vertical axis is the \(z\) - axis. If the center of the circle is at \((h,0,0)\) where \(h>r\) ( \(r\) is the radius of the circle), then rotating the circle around the \(z\) - axis (vertical axis) will create a torus. The torus has a major radius \(R = h\) (distance from the \(z\) - axis to the center of the circular cross - section) and a minor radius \(r\) (radius of the circular cross - section).

So the three - dimensional shape resulting from the rotation is a torus, which can be described as a "torus" with a "center" (the center of the torus, along the axis of rotation) and a "radius" (the radius of the circular cross - section, and the distance from the center of the torus to the center of the cross - section circle). Wait, but maybe the problem is simpler. Wait, if the circle is rotated around an axis that it does not intersect, the resulting solid is a torus, which is a type of surface of revolution. The torus has a center (the axis of rotation is the central axis) and a radius (the radius of the circular generating curve and the distance from the center of the torus to the center of the generating circle).

But maybe the intended answer is: the three - dimensional shape resulting from the rotation is a "torus" with a "center" (the axis of rotation is the center line of the torus) and a "radius" (the radius of the circular cross - section and the distance from the center of the torus to the center of the cross - section circle). Wait, perhaps the problem is using simpler terms. Let's re - examine the problem.

The problem says "the three - dimensional shape resulting from the rotation is a [blank] with a [blank] center". Wait, maybe I misread. Wait, the first blank: when a circle is rotated around an axis it doesn't intersect, the 3D shape is a torus. A torus has a center (the axis of rotation is the central axis of the torus) and a radius (the radius of the circular cross - section, and the distance from the center of the torus to the center of the cross - section circle).

So, step 1: Identify the solid of revolution. A circle rotated around an axis it does not intersect forms a torus. Step 2: The torus has a center (the axis of rotation is the central axis) and a radius (the radius of the circular cross - section and the distance from the center of the torus to the center of the cross - section circle).

Answer:

The three - dimensional shape resulting from the rotation is a \(\boldsymbol{\text{torus}}\) with a \(\boldsymbol{\text{central}}\) (or appropriate term related to the axis - based center) center. (Note: If the drop - down options include "torus" for the first blank and "central" or "axis - aligned" for the second, those would be the correct choices. Since the exact options are not visible, but based on the concept, the shape is a torus, and it has a center related to the axis of rotation.)