Problem 6 · 2021 Math Kangaroo
Medium
Geometry & Measurement
arearatio
A rectangular sheet of paper has length x and width y where \(x > y\). The rectangle may be folded to form the curved surface of a circular cylinder in two different ways. What is the ratio of the volume of the longer cylinder to the volume of the shorter cylinder?
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Answer: B — \(y : x\)
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Hint 1 of 2
Rolling the sheet either way fixes which side becomes the circular edge and which becomes the height.
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Hint 2 of 2
Volume of a cylinder is (circumference)² / (4π) times height; compare the two volumes.
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Approach: compute both volumes and take the ratio
- Roll so circumference = x, height = y: volume = x²y/(4π). Roll so circumference = y, height = x: volume = y²x/(4π).
- Since x > y, the taller (longer) cylinder is the second one, height x, volume y²x/(4π).
- Ratio longer : shorter = (y²x) : (x²y) = y : x.
Mark:
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