Problem 22 · 2000 AMC 8
Stretch
Geometry & Measurement
surface-area
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Answer: C — About 17%.
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Hint 1 of 2
Don't recompute the whole new solid — just ask what surface *changed*. Gluing covers a 1×1 patch of the big top, but the small cube's own top reappears directly above it. So the upward-facing area is exactly the same as before.
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Hint 2 of 2
Net new surface = the parts you can newly see minus the parts now hidden. Here the tops trade evenly, leaving only the small cube's 4 side walls as genuinely new area.
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Approach: count only the faces that actually change
- Original surface area: 6 · 2² = 24.
- Gluing on the unit cube hides a 1×1 patch of the big top, but the unit cube's top (also 1×1) now sits right above it — so total upward-facing area is unchanged. The glued-down bottom face contributes nothing. The only new surface is the 4 side walls of the small cube: +4.
- Increase = 4 / 24 = ⅙ ≈ 16.7%, closest to 17%.
- The principle: for 'how much did surface area change,' never re-add everything — track only what got covered vs. newly exposed. A bump glued flat on top trades its footprint for its own top (a wash) and adds only its sides.
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