Question

An opaque pyramid (shown below), with a square base and isosceles faces, is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The pyramid can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible? 

 

• A. P
• B. Q
• C. R
• D. S

Hint:

When analyzing shadows of 3D objects, remember a crucial rule: the shadow of a single convex object (like a sphere, cube, pyramid, or cone) must always be a convex 2D shape. If you see a shadow with an inward-pointing angle, it cannot be from a single convex object.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question tests spatial reasoning and the understanding of how 3D objects cast 2D shadows. A key property of a shadow cast by a single convex object (like a pyramid) is that the shadow itself must be a convex shape. A convex shape has no "dents" or inward-pointing corners. Any line segment connecting two points within the shape must lie entirely within the shape.
Step 2: Detailed Explanation:
Let's analyze the possibility of each shadow shape:
- Shadow S (Square): This shadow is formed when the parallel light beam is aimed directly at the base of the pyramid, perpendicular to the base. The square base will cast a square shadow. This is possible.
- Shadow R (Pentagon): This shadow can be formed by tilting the pyramid. For example, if the light beam is directed towards one of the triangular faces but at an angle, the outline of the shadow will be formed by the edges of that face plus parts of the adjacent faces and the base. A pentagonal shadow is possible when the pyramid is tilted such that the shadow is cast by two of the base edges and three of the slanted edges. This is possible.
- Shadow P (Irregular Hexagon): This shadow can also be formed by tilting the pyramid. If the pyramid is oriented such that the light shines towards one of the slanted edges, the shadow's outline can be formed by the two faces meeting at that edge, along with parts of the other two faces. This can result in a hexagonal (arrow-like) shape. This is possible.
- Shadow Q (Non-convex shape): This shape is non-convex. It has an interior angle greater than 180 degrees (an inward-pointing corner). A pyramid is a convex polyhedron. The shadow cast by a single convex object from a parallel light source must always be a convex polygon. Since shape Q is non-convex, it is impossible to create this shadow from a single pyramid.
Step 3: Final Answer:
The shadow that is NOT possible is Q.
Step 4: Why This is Correct:
The fundamental principle is that the projection (shadow) of a convex set is always convex. The pyramid is a convex object. Shapes P, R, and S are all convex polygons. Shape Q is a non-convex (or concave) polygon. Therefore, it cannot be the shadow of the pyramid.