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? \begin{center} \includegraphics[width=0.5\textwidth]{03.jpeg} \end{center}

• A. P
• B. Q
• C. R
• D. S

Hint:

For shadow problems, always think about orthographic projections. A pyramid can project into triangles, trapeziums, pentagons, or a square, but not into an incomplete square with one side missing.

Answer 1

The Correct Option is A

Step 1: Pyramid Geometry.
\nThe pyramid features a square base and four identical isosceles triangular faces that converge at the apex. \n\n \n

Step 2: Projection Variations.
\n- Axial illumination (top view) results in a square shadow (consistent with option S).
\n- Illumination angled to reveal two adjacent triangular faces and the apex produces a pentagonal shadow (consistent with option R).
\n- Further tilting of the pyramid can yield an irregular quadrilateral shadow (consistent with option Q). \n\n \n

Step 3: Evaluation of Option P.
\nShadow P, a square with a single straight cut, is not geometrically feasible because: \n- The pyramid's apex is pointed. Any angled shadow must taper, not exhibit a flat, truncated edge as seen in P. \n- A projected square base (S) is a complete square; it cannot manifest as the partial, irregular shape depicted in P. \n\n \n

Step 4: Determination.
\nConsequently, shadow P is impossible. \n\n \n\[\n\boxed{\text{P is not possible.}}\n\]