Cross-Section Analysis of Solid Objects

The question asks us to identify a solid object based on its cross-sections along two mutually perpendicular axes. We are told that one cross-section is a circle and the other is a square.

Let's understand what a cross-section is. A cross-section is the shape obtained when a solid is sliced by a plane. The shape of the cross-section depends on the orientation of the cutting plane relative to the solid and its axes.

Solid Object Properties from Cross-Sections

We are given two specific cross-sections along mutually perpendicular axes. Let's consider typical axes for common shapes. For shapes with rotational symmetry, like a cylinder or cone, standard axes might be the axis of rotation and an axis perpendicular to it.

• A cross-section that is a circle suggests some form of rotational symmetry around an axis perpendicular to the plane of the circle.
• A cross-section that is a square along a perpendicular axis gives more information about the shape's dimensions in that direction.

Analyzing the Options

Let's examine each option to see which one fits the description:

1. A truncated cone: A truncated cone is a cone with the top sliced off by a plane parallel to the base. Cross-sections parallel to the base are circles. Cross-sections perpendicular to the base can be parabolas, hyperbolas, or ellipses, depending on the angle. A cross-section through the axis is an isosceles trapezoid. It is not possible to get a square cross-section along a standard perpendicular axis.
2. A cylinder: A cylinder is a solid with two parallel circular bases and a curved surface connecting them.
   • A cross-section parallel to the base is a circle.
   • A cross-section perpendicular to the base and passing through the central axis is a rectangle. If the height of the cylinder is equal to the diameter of its base, then this rectangular cross-section will be a square.
   This object fits the description of having a circular cross-section along one axis (parallel to the base) and potentially a square cross-section along a perpendicular axis (through the center, if height equals diameter).
3. A rhomboid: A rhomboid is a 3D shape bounded by six parallelogram faces. Cross-sections would typically be polygons like parallelograms, rectangles, or triangles. It cannot produce a circular cross-section.
4. A cube: A cube is a solid with six square faces. Cross-sections of a cube are polygons, such as squares, rectangles, triangles, or hexagons. A cube cannot produce a circular cross-section.

Conclusion

Based on the analysis, only a cylinder can have a circular cross-section (parallel to the base) and a square cross-section (perpendicular to the base through the axis, if height equals diameter) along mutually perpendicular axes. The question states *a* square cross-section along a perpendicular axis, which is possible for a cylinder of specific dimensions.

Therefore, the object described is a cylinder.