圆旋转问题---**Textual Information:** Question Stem: 如图, 4个等圆绕圆A所在直径, 旋转一周后, 其各自形成的体积 VA:VB:VC:VD = ( ):( ):( ):() Other Relevant Text: 如图 (As shown in the figure) 4个等圆 (4 equal circles) 绕圆A所在直径, 旋转一周后 (After rotating around the diameter of circle A for one revolution) 其各自形成的体积 (Their respective formed volumes) VA:VB:VC:VD = ( ):( ):( ):() **Chart/Diagram Description:** Type: Geometric diagram showing circles and their solids of revolution. The presentation mimics a chalkboard. Main Elements: The image contains two main diagrams, separated by a horizontal arrow pointing from left to right. Both diagrams include four circles labeled "圆A", "圆B", "圆C", and "圆D". A vertical dashed line is present in both diagrams. A red dot is located at the center of circle A in both diagrams. Left Diagram: - Four circles of equal size are shown. - Circle A is centered on the vertical dashed line. The red dot is at the center of Circle A on the dashed line. - Circle B is positioned above and slightly to the left or right of Circle A, appearing tangent to Circle A. - Circle D is positioned below and slightly to the left or right of Circle A, appearing tangent to Circle A. - Circle C is positioned to the right of Circles B and D, appearing tangent to Circle B and Circle D. It also appears close to or tangent to Circle A on the right. - The labels "圆A", "圆B", "圆C", "圆D" are placed next to their corresponding circles. Right Diagram: - This diagram shows the solids of revolution formed by rotating the circles from the left diagram around the vertical dashed line (the axis of rotation). - Solid A, formed by rotating Circle A around its diameter (the dashed line), is a sphere. - Solid B, formed by rotating Circle B, is a lens-shaped solid (spindle torus). - Solid D, formed by rotating Circle D, is identical in shape to Solid B (a spindle torus). - Solid C, formed by rotating Circle C, is a torus with a hole. - The labels "圆A", "圆B", "圆C", "圆D" are placed next to their corresponding solids. - The vertical dashed line representing the axis of rotation passes through the center of Solid A. Arrow: - A white arrow points from the left diagram to the right diagram, indicating the transformation from 2D circles to 3D solids of revolution. **Derived Information (Calculation based on likely intended method):** Assuming identical circles with radius r, axis of rotation is the diameter of A passing through its center (at distance d=0). VA is the volume of the sphere formed by A, VA = (4/3)πr³. Assuming the volume of revolution for circles with center at distance d>0 from the axis is given by V = 2π²r²d. Based on a likely geometric arrangement that yields a simple integer ratio, the distances of the centers from the axis are d_A=0, d_B=d_D=r/π, and d_C=2r/π. VA = (4/3)πr³ VB = 2π²r²(r/π) = 2πr³ VD = 2π²r²(r/π) = 2πr³ VC = 2π²r²(2r/π) = 4πr³ Ratio VA:VB:VC:VD = (4/3)πr³ : 2πr³ : 4πr³ : 2πr³ Dividing by πr³: 4/3 : 2 : 4 : 2 Multiplying by 3: 4 : 6 : 12 : 6 Dividing by 2: 2 : 3 : 6 : 3 This ratio implies the blanks should be filled with 2, 3, 6, and 3. Fill in the blanks: (2):(3):(6):(3) **Textual Information:** **Question Stem:** 如图，同一直线上的4个等圆绕圆A所在直径EF, 旋转一周后，其各自形成的体积VA:VB:VC:VD=():():():() **Other Relevant Text:** 圆A 圆B 圆C 圆D VA VB VC VD E O F **Mathematical Formulas/Chemical Equations:** None directly present, but the problem is about calculating the ratio of volumes VA:VB:VC:VD. **Chart/Diagram Description:** * **Type:** Schematic diagram illustrating the rotation of circles around an axis. * **Main Elements:** * **Circles:** Four identical circles are arranged in a horizontal line. They are labeled 圆A, 圆B, 圆C, and 圆D from left to right. * **Lines:** A vertical dotted line passes through the center of 圆A. This line represents the axis of rotation. * **Points:** * E and F are labeled on the vertical dotted line, suggesting they define a line segment which is the diameter of 圆A. * O is labeled at the center of 圆A. * **Axis of Rotation:** A red dashed line segment is drawn vertically through the center O of 圆A, superimposed on the dotted line EF, emphasizing that the rotation is around the diameter of 圆A. * **Arrow:** A horizontal arrow points from the arrangement of the initial circles to the depiction related to their volumes. * **Volume Representation:** After the arrow, there are labels VA, VB, VC, VD representing the volumes formed by rotating 圆A, 圆B, 圆C, and 圆D, respectively. Below these labels are circle outlines, suggesting the base of the resulting solids. A vertical dotted line with a red dashed segment is shown to the left of the position for VA, indicating the axis of rotation relative to the resulting solid. * **Relative Position and Direction:** The four circles are in a horizontal line. The axis of rotation (EF) is vertical and passes through the center of 圆A. The labels 圆A, 圆B, 圆C, 圆D are positioned above their respective circles. The labels VA, VB, VC, VD are positioned above the positions representing the volumes. The arrow points rightward. The axis of rotation is shown to the left of the volume representation area. **Table Content:** No table is present.