Problem 26 · 2016 Math Kangaroo
Stretch
Geometry & Measurement
area-decomposition
In a solid cube P is a point on the inside. We cut the cube into 6 (sloping) pyramids. Each pyramid has one face of the cube as its base and point P as its top. The volumes of five of these pyramids are 2, 5, 10, 11 and 14. What is the volume of the sixth pyramid?
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Answer: C — 6
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Hint 1 of 2
Pair up pyramids on opposite faces of the cube.
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Hint 2 of 2
Two opposite pyramids have heights adding to the cube's edge, so each opposite pair has the same total volume.
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Approach: pair opposite pyramids (equal pair-sums)
- Two pyramids on opposite faces share base area and have heights summing to the cube's edge, so every opposite pair has the same volume sum.
- Pairing the known values: 2+14 = 16 and 5+11 = 16, so the pair containing 10 must also total 16.
- The sixth volume is 16 - 10 = 6 (C).
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