Problem 18 · 2022 Math Kangaroo
Hard
Geometry & Measurement
area
A cuboid with surface area X is cut up along six planes parallel to the sides (see diagram). What is the total surface area of all 27 thus created solids?
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Answer: C — \(3X\)
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Hint 1 of 2
Every cut parallel to a face creates two new faces equal to that cross-section.
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Hint 2 of 2
Two cuts in each of the three directions; total new area relates simply to X.
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Approach: count area added by the cuts
- Let the three face-pair areas be S1, S2, S3, so X = 2(S1+S2+S3).
- Two cuts perpendicular to each direction add 4S1 + 4S2 + 4S3 = 2X of new surface.
- Total = X + 2X = 3X.
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