Question 21 How many ways are there of slicing a unit cube by a plane into two parts of...
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How many ways are there of slicing a unit cube by a plane into two parts of equal volume given that the plane has to pass through at least one vertex?
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Any plane passing through the center of symmetry of a cube bisects its volume into two equal parts. For any chosen vertex V, any plane containing the line connecting V to the center of the cube splits the volume equally, and there are infinitely many such planes.
Step-by-step Derivation:
Step 1: A cube has point reflection symmetry about its center C = (1/2, 1/2, 1/2).
Step 2: Any plane that passes through center C divides the cube into two congruent halves of equal volume.
Step 3: Given that the plane must pass through a vertex V, any plane containing the line passing through V and C bisects the volume.
Step 4: Infinitely many planes contain this line (rotating around the axis VC), so the number of ways is infinite.