Problem 9 · 2014 Math Kangaroo
Medium
Geometry & Measurement
areaarea-decomposition
The curved surfaces of two identical cylinders are cut open along the vertical dotted line, as shown, and then stuck together to create the curved surface of one big cylinder. What can be said about the volume of the resulting cylinder compared to the volume of one of the small cylinders?
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Answer: D — It is 4 times as big.
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Hint 1 of 2
The two lateral surfaces are joined side by side, so what doubles — the height or the circumference?
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Hint 2 of 2
Same height, double circumference means double the radius. How does radius affect volume?
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Approach: track how radius scales the volume
- Joining two equal lateral surfaces edge to edge keeps the height the same but doubles the circumference, so the new radius is twice the old: R = 2r.
- Volume ∝ radius², so the big cylinder's volume is (2)² = 4 times one small cylinder's.
- It is 4 times as big.
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