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GATE CSE 2021 Set 1 Q6: Ratio of volumes: rolled cylinder vs cube from the same sheet

Question

We have 2 rectangular sheets of paper, M and N, of dimensions 6 cm × 1 cm each. Sheet M is rolled to form an open cylinder by bringing the short edges of the sheet together. Sheet N is cut into equal square patches and assembled to form the largest possible closed cube. Assuming the ends of the cylinder are closed, the ratio of the volume of the cylinder to that of the cube is __________

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Answer: C (official GATE 2021 final answer key)

Concept used: Rolling along the long side makes the 6 cm edge the circumference; six 1×11 \times 1 squares make a unit cube.

First step: Circumference 2πr=62\pi r = 6 gives r=3/πr = 3/\pi with height 1; the cube has edge 1 cm.

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Source: official GATE 2021 Set 1 CS question paper and final answer key, organised by IIT Bombay. Question paper · Answer key

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