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We have 2 rectangular sheets of paper. M and N, of dimensions 6 cm x 1 cm each. Sheet M is rolled to from an open cylinder by bringing the short edges of the sheet together. Sheet N is cut into equal patches and assembled to from the largest closed cube. Assuming the ends of the cylinder are closed, the ratio of the volume of the cylinder to that cube is ______
  • a)
    3/π
  • b)
    9/π
  • c)
    π/2
  • d)
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
We have 2 rectangular sheets of paper. M and N, of dimensions 6 cm x 1...
Given:
Two rectangular sheets M and N have the dimension 6 cm × 1 cm each,  in which cylinder will be formed by bringing the short edges of the sheet together. Given that ends of the cylinder are closed.
Concept:
Volume of cylinder = π x r2 x h
The volume of the cube = a3
Calculation:
Let be assume the side of the cube is a,
⇒ The area of the sheet N = 6 x a2
⇒ 6 × 1 = 6 x a2
⇒ a = 1
⇒ Volume of the cube = a3 = 13 = 1
⇒  The circumference  = 6
⇒ 2 × π × r = 6, r = 3/π and height of the cylinder = 1 cm
⇒ Volume of the cylinder = π x (3/π)2 x 1 = 9/π 
⇒ The required ratio = 9/π : 1
∴ The required result will be 9/π .
Free Test
Community Answer
We have 2 rectangular sheets of paper. M and N, of dimensions 6 cm x 1...
Volume of the Cylinder
To find the volume of the cylinder formed by rolling sheet M:
- Dimensions:
- Height (h) = 1 cm (the width of the sheet)
- Circumference = 6 cm (the length of the sheet)
- Radius (r):
- Circumference = 2 * π * r
- r = Circumference / (2 * π) = 6 / (2 * π) = 3 / π cm
- Volume (V_cylinder):
- V_cylinder = π * r² * h
- V_cylinder = π * (3/π)² * 1 = π * (9/π²) = 9/π cm³
Volume of the Cube
Next, calculate the volume of the cube formed by cutting sheet N:
- Dimensions:
- Each patch (cut from sheet N) will be a square with each side of length equal to the side of the cube.
- Total Area of Sheet N:
- Area = length x width = 6 cm x 1 cm = 6 cm²
- Cube Volume:
- Let the side of the cube be 's'.
- Total surface area of a cube = 6 * s²
- Setting the total surface area equal to the area of sheet N: 6 * s² = 6
- Therefore, s² = 1, so s = 1 cm.
- Volume (V_cube):
- V_cube = s³ = 1³ = 1 cm³
Ratio of Volumes
Finally, calculate the ratio of the volume of the cylinder to the volume of the cube:
- Ratio:
- Ratio = V_cylinder / V_cube = (9/π) / 1 = 9/π
Thus, the ratio of the volume of the cylinder to that of the cube is 9/π, confirming that the correct answer is option 'B'.
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We have 2 rectangular sheets of paper. M and N, of dimensions 6 cm x 1 cm each. Sheet M is rolled to from an open cylinder by bringing the short edges of the sheet together. Sheet N is cut into equal patches and assembled to from the largest closed cube. Assuming the ends of the cylinder are closed, the ratio of the volume of the cylinder to that cube is ______a)3/πb)9/πc)π/2d)3πCorrect answer is option 'B'. Can you explain this answer?
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