Problem 27 · 2020 Math Kangaroo
Stretch
Geometry & Measurement
areaarea-fraction
The submerged part of an iceberg shaped like a cube makes up 96.4% of the iceberg's volume, and the part above the water has the same three edges meeting at a corner. What percentage of the total surface area of the iceberg is in contact with the air?
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Answer: A — 9%
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Hint 1 of 2
Only 3.6% of the cube pokes above water, as a small corner with three equal edges.
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Hint 2 of 2
A corner tetrahedron's volume is (edge)³/6 — find that edge.
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Approach: corner tetrahedron above water
- The above-water part is a corner cut off by the waterline, a tetrahedron with three equal edges x along the cube's edges; its volume x³/6 = 0.036a³.
- So x³ = 0.216a³, giving x = 0.6a.
- Exposed area = three right triangles 3·(x²/2) = 0.54a²; over the cube's 6a² that is 9%.
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