When the right triangle is rotated about its longer leg (length 8), the shorter leg (length 6) becomes the radius of the circular base, creating a cone with height 8 and radius 6. Volume = $$\frac{1}{3}\pi r^2 h = \frac{1}{3}\pi(6^2)(8) = 96\pi$$. Choice B incorrectly identifies a cylinder. Choice C confuses which leg serves as the axis. Choice D incorrectly suggests a double cone formation.

This question tests understanding of cross-sections and solids of revolution in geometry. The original solid is a right circular cylinder standing upright with circular bases. The slicing plane is vertical, perpendicular to the bases, and passes through the central axis. The plane intersects the cylinder along the height and through the diameter of the bases. This creates a rectangular cross-section with width equal to the diameter and height of the cylinder. A distractor is mistaking it for a circle, which happens with parallel slices. Imagine the slice step by step along the axis to confirm the straight-sided rectangle.

This question tests understanding of cross-sections and solids of revolution in geometry. The original solid is a right circular cone with a base on a horizontal plane. The slicing plane passes through the apex and is perpendicular to the base, containing the axis. The plane cuts along the height and through the base diameter. This produces an isosceles triangular cross-section with the base as the diameter and sides as generators. Confusing it with a circle ignores the axial cut through the apex. Picture the vertical slice step by step from apex to base to see the triangle emerge.

This problem involves finding cross-sections of three-dimensional solids, a key skill in geometry that also relates to solids of revolution. The original solid is a right circular cylinder, which has two parallel circular bases connected by a curved lateral surface. The slicing plane is parallel to the circular bases and intersects the cylinder midway between the top and bottom. When the plane slices parallel to the bases, it intersects the lateral surface in a uniform manner, creating a shape identical to the bases. This results in a circle because the cross-section mirrors the base's shape due to the cylinder's uniform radius. A common misconception is thinking it forms an ellipse, which might occur if the plane were angled, but here it's parallel, so it's a circle. To visualize, imagine slicing a can of soup horizontally step by step; each slice reveals a circular cross-section matching the can's ends.

In a regular tetrahedron, when a plane is parallel to a face at distance $$\frac{1}{4}$$ from that face toward the opposite vertex, it's at distance $$\frac{3}{4}$$ from the vertex. The linear scale factor is $$\frac{3}{4}$$, so the area scales as $$(\frac{3}{4})^2 = \frac{9}{16}$$. Choice A confuses linear and area scaling. Choice C uses the linear scale factor incorrectly. Choice D incorrectly squares $$\frac{1}{4}$$.

This question tests understanding of cross-sections through three-dimensional solids. The original solid is a right circular cone with a circular base and curved surface meeting at a vertex. The slicing plane passes through both the cone's vertex and its central axis, creating a vertical cut through the cone's center. When this plane cuts through the cone, it intersects the circular base along a diameter and the curved surface along two straight lines from the vertex to the base endpoints. The resulting cross-section is an isosceles triangle with the vertex as one corner and the diameter endpoints as the other two corners. Students might incorrectly choose circle or ellipse, not realizing that planes through the vertex create triangular sections. To visualize this, imagine cutting a party hat from top to bottom through the center—you see a triangular shape.

This problem involves finding the cross-section of a 3D solid. The original solid is a right circular cylinder with circular bases. The slicing plane is parallel to the circular bases, making it horizontal. As the plane cuts through the cylinder, it intersects the curved surface evenly at every point. The resulting cross-section is a circle identical in shape to the bases. A common misconception is assuming it forms a rectangle, which occurs with a perpendicular slice instead. To visualize, imagine slicing a tube parallel to its ends step by step to reveal the circular shape.

This question examines cross-sections of 3D objects, specifically cylinders. The solid is a right circular cylinder with circular bases and a curved lateral surface. The cutting plane is perpendicular to the bases and passes through the cylinder's center axis, creating a vertical slice through the middle. This plane intersects the top circle along a diameter, the bottom circle along a diameter, and connects these with straight lines along the cylinder's height. The resulting cross-section is a rectangle with width equal to the cylinder's diameter and height equal to the cylinder's height. Students might incorrectly choose circle, confusing this with horizontal cuts. To visualize, imagine cutting a paper towel tube lengthwise through the center—you see a rectangular shape.

This problem involves finding cross-sections of three-dimensional solids, a key skill in geometry that also relates to solids of revolution. The original solid is a right circular cylinder, featuring circular bases and a rectangular height when unrolled. The slicing plane is perpendicular to the bases and passes through the central axis of the cylinder. As the plane cuts vertically through the center, it intersects the two bases along their diameters and the lateral surface along the height. This produces a rectangle, with the width equal to the diameter of the base and the height matching the cylinder's height. A distractor misconception is assuming a semicircle, perhaps confusing it with a non-central slice, but the central axis ensures a full rectangular shape. To transfer this, imagine unfolding the cylinder and tracing the plane's path step by step to see the rectangular outline emerge.

When a semicircle rotates about its diameter, it sweeps out a complete sphere (not just a hemisphere). After $$180°$$ rotation, the semicircle has moved to the opposite side of the diameter from where it started. Choice A correctly identifies the sphere but incorrectly states the final position. Choice B incorrectly identifies only a hemisphere. Choice D incorrectly suggests no complete sphere is formed.