GATE CS 2021 Set 1 — Question 6
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General Aptitude → Quantitative Aptitude → Mensuration & Geometry
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Question
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 __________
Correct answer
(C) (9)/(π)
Solution
Let's analyze the formation of the cylinder and the cube separately.1. Cylinder from Sheet M:
- The dimensions of sheet M are 6 cm x 1 cm.
- It is rolled by bringing the short edges (1 cm) together.
- This means the height of the cylinder will be the length of the short edge, so cm.
- The circumference of the base of the cylinder will be the length of the long edge, so cm.
- The formula for the circumference of a circle is , where is the radius.
- So, cm.
- The volume of a cylinder is given by the formula .
- Substituting the values of and :
- The dimensions of sheet N are 6 cm x 1 cm.
- The total area of sheet N is cm².
- This sheet is used to form the largest possible closed cube.
- A closed cube has 6 identical square faces.
- Let the side length of the cube be 'a'. The area of one face is .
- The total surface area of the cube is .
- The material for this surface area comes from sheet N. So, the total surface area of the cube must be equal to the area of the sheet.
- cm.
- We can cut six 1 cm x 1 cm square patches from a 6 cm x 1 cm rectangular sheet.
- The volume of the cube is given by the formula .
- Substituting the value of :
- The required ratio is the volume of the cylinder to the volume of the cube.
- Ratio = .
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