Maximum cube shadow
You hold up a cube with a side length of 2 so that it casts a shadow on a flat surface.
What is the maximum shadow area that the cube can cast on that surface? Round your answer to the nearest tenth.
You should assume the incoming light rays are parallel to each other and perpendicular to the surface, and you should neglect the effects of diffraction (i.e. light rays only travel straight paths).
The answer is 6.9.
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1 solution
Light against one cube's face (with the source of light shining exactly perpendicular to the center of one of the six faces)
==> Shadow would take the exact form and size of that square face
==> area = 2² = 4.
Light against two cube's faces (with the source of light shining exactly perpendicular to the middle of one of the twelve edges and 'aiming' to the cube's center)
==> Shadow would take the form of a rectangle with the shorter sides equal the cube's edges and it's longer sides equal the cube's face diagonals
==> area = 2 × 2√2 = 4√2 = 5.66
Light against three cube's faces (with the source of light shining exactly towards one of the eight vertices and 'aiming' to the cube's center)
==> Shadow would take the form of a regular hexagon with three looping face diagonals (as opposed to a radial one) on the three shone faces making up the largest equilateral triangle in it (the hexagonal shadow)
==> area
= 2 × area of equilateral triangle with sides of 2√2
= 2 × 2√3
= 4√3
= 6.93