1992 AMC 12 Problem 19
Attempt Problem 19 of the 1992 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1992 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
19.
For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a cuboctahedron. The ratio of the volume of the cuboctahedron to the volume of the original cube is closest to which of these?
Answer: D
Small Hint:
Choose a convenient side length for the cube and compute one removed corner tetrahedron
Big Hint:
Each removed tetrahedron has three mutually perpendicular edges equal to half the cube’s side
Solution:
Take the cube’s side length to be Its volume is Each corner tetrahedron has three perpendicular edges of length so its volume is The eight removed tetrahedra have total volume leaving The ratio is which is closest to
Thus the correct answer is D.
Problem 19 in Other Years
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