We have 2 rectangular sheets of paper, M and N, of dimensions 6 cm x 1 cm…
2021
We have 2 rectangular sheets of paper, M and N, of dimensions 6 cm x 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 __________
- A.
\(\frac{\pi}{2} \) - B.
\(\frac{3}{\pi} \) - C.
\(\frac{9}{\pi} \) - D.
\(3\pi\)
Attempted by 32 students.
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Correct answer: C
Key idea: determine the cylinder dimensions from how the sheet is rolled and use the sheet area to find the cube side.
Cylinder:
Interpreting the intended construction as joining the longer edges (length 6 cm) to form the circular direction, the circumference of the cylinder is 6 cm. Thus r = 6/(2π) = 3/π cm, and the height is the remaining dimension 1 cm.
Cylinder volume = π·r²·h = π·(3/π)²·1 = 9/π cm³.
Cube:
The sheet area is 6 cm². If a cube has side s, its total surface area is 6s². Using all the sheet area to make the cube faces gives 6s² = 6, so s = 1 cm.
Cube volume = s³ = 1 cm³.
Ratio of volumes (cylinder : cube) = (9/π) : 1 = 9/π.
Note on wording: if the sheet were instead rolled by joining the shorter edges (circumference 1 cm, height 6 cm), the cylinder volume would be 3/(2π) cm³, which does not match the given choices. Therefore the intended interpretation is joining the longer edges, yielding the ratio 9/π.