Question

which object shown below could we slice perpendicular to its base/face to create a cross - section whose shape has two edges, one straight and one curved? choose 1 answer: (a)
cone (b)
cube (c)
cylinder (d)
sphere

Explanation:

Step1: Analyze Option A (Cone)

A cone has a circular base. When sliced perpendicular to its base (which is a circle), the cross - section is a triangle? No, wait. Wait, if we slice a cone perpendicular to its base (the base is a circle, and the axis of the cone is perpendicular to the base). Wait, no, let's think again. The base of the cone is a circle, and the lateral surface is curved. If we slice the cone perpendicular to its base (so the slice is along a plane that is perpendicular to the circular base, passing through the apex). Wait, no, the base is a circle, and the plane perpendicular to the base would be a plane that is vertical (if the base is horizontal). The cross - section of a cone when sliced perpendicular to its base (through the apex) is a triangle (all straight edges). But if we slice the cone not through the apex, but perpendicular to the base? Wait, maybe I made a mistake. Let's analyze each option:

Step2: Analyze Option B (Cube)

A cube has all flat faces. Any cross - section of a cube, when sliced perpendicular to its face, will be a polygon with all straight edges (like a square, rectangle, etc.). So it cannot have a curved edge.

Step3: Analyze Option C (Cylinder)

A cylinder has two circular bases and a curved lateral surface. When we slice a cylinder perpendicular to its base (the bases are circles, and the plane of the slice is perpendicular to the bases), the cross - section will be a rectangle (if the cylinder is right - circular) with two straight vertical edges (parallel to the axis of the cylinder) and two curved top and bottom edges? No, wait, no. Wait, the base of the cylinder is a circle. If we slice the cylinder perpendicular to its base (so the slice is along a plane that is perpendicular to the circular bases, parallel to the axis of the cylinder). The cross - section will be a rectangle with two straight vertical sides (parallel to the axis) and two straight horizontal sides (the top and bottom, which are the edges of the circular bases? No, the top and bottom of the cross - section would be straight? Wait, no, the cylinder's lateral surface is curved. Wait, maybe I messed up. Let's re - examine the cone.

Wait, let's start over. The question is about a cross - section with one straight and one curved edge.

- Cone: The base is a circle (curved edge), and the lateral surface is curved. If we slice the cone perpendicular to its base (the base is a circle, and the slice is a plane that is perpendicular to the base, not passing through the apex). Wait, the base is a circle, and the plane perpendicular to the base will intersect the lateral surface of the cone. The cross - section will be a shape with one straight edge (the edge along the plane's intersection with the base, which is a straight line segment, since the base is a circle and the plane is perpendicular to the base, so the intersection with the base is a straight line) and one curved edge (the intersection with the lateral surface of the cone). Wait, maybe I was wrong earlier. Let's think of the cone: the base is a circle. A plane perpendicular to the base will intersect the base in a straight line (a diameter, if the plane passes through the center of the base) and intersect the lateral surface in a curved line. So the cross - section will have one straight edge (from the base) and one curved edge (from the lateral surface).

Wait, no, let's take a cone with a circular base (radius r) and height h. The base is in the xy - plane, centered at the origin, and the apex is at (0,0,h). A plane perpendicular to the base (say the xz - plane, x = 0). The intersection of the plane x = 0 with the base (xy - plane) is the line segment from (0, - r, 0) to (0, r, 0) (straight edge). The intersection of the plane x = 0 with the lateral surface of the cone (the equation of the cone is \(x^{2}+y^{2}=(\frac{r}{h}(h - z))^{2}\), for \(z\in[0,h]\)). When x = 0, we have \(y^{2}=(\frac{r}{h}(h - z))^{2}\), so \(y=\pm\frac{r}{h}(h - z)\), which is a straight line. Wait, that's a triangle. So maybe my initial thought was wrong.

Wait, let's analyze the sphere (Option D):

Step4: Analyze Option D (Sphere)

Any cross - section of a sphere is a circle, which has only a curved edge (no straight edges).

Wait, maybe I made a mistake in the cone analysis. Let's re - evaluate:

The key is to find a 3D object where a perpendicular slice to the base/face gives a cross - section with one straight and one curved edge.

- Cube: all straight edges in cross - section. Eliminate.
- Sphere: all curved edge in cross - section. Eliminate.
- Cylinder: If we slice the cylinder perpendicular to its base (the base is a circle), the cross - section is a rectangle (two straight vertical edges, two straight horizontal edges? No, the horizontal edges are the intersections with the circular bases, which are straight? Wait, no, the bases are circles. The intersection of a plane perpendicular to the base with the circular base is a straight line (a diameter). The intersection with the other base is also a straight line (a diameter), and the intersection with the lateral surface is two straight lines (parallel to the axis of the cylinder). So the cross - section of a cylinder sliced perpendicular to its base is a rectangle (all straight edges). Eliminate.

- Cone: Wait, maybe the base of the cone is a circle, and if we slice the cone perpendicular to its base, but not through the apex. Wait, the base is a circle, and the plane perpendicular to the base will intersect the base in a straight line (a chord? No, if the plane is perpendicular to the base, and the base is a circle, the intersection of the plane and the base is a straight line (a diameter, if the plane passes through the center of the base). The intersection of the plane and the lateral surface of the cone: the lateral surface of the cone is a curved surface. The cross - section will have one straight edge (from the base) and one curved edge (from the lateral surface). Wait, maybe I was wrong earlier about the cone. Let's think of a cone as a circular base and a curved surface going up to the apex. If we slice the cone with a plane that is perpendicular to the base (so vertical, if the base is horizontal) and does not pass through the apex. The intersection with the base is a straight line segment (part of the base's diameter), and the intersection with the lateral surface is a curved line. So the cross - section has one straight and one curved edge.

Wait, but the options are A (Cone), B (Cube), C (Cylinder), D (Sphere). Let's check again:

- Cube: all straight edges in cross - section. So B is out.
- Sphere: all curved edges. D is out.
- Cylinder: when sliced perpendicular to base, cross - section is rectangle (all straight edges). C is out.
- Cone: when sliced perpendicular to base (not through apex), cross - section has one straight (from base) and one curved (from lateral surface) edge. So A is the answer.

Answer:

A. Cone