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in Surface Areas And Volumes by (56.3k points)

A rectangular sheet of paper 30 cm × 18 cm can be transformed into the curved surface of a right circular cylinder in two ways i.e., either by rolling the paper along its length or by rolling it along its breadth. Find the ratio of the volumes of the two cylinders thus formed.

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Best answer

We have,

Dimensions of rectangular sheet = 30 cm × 18 cm

Case (i)

When paper is rolled along its length

2 πr = 30

r = 30 / 2π cm

Height = 18 cm

Volume of cylinder, V1 = πr2h

= π × (30/2π)2 × 18 cm3

Case (ii)

When paper is rolled along its breadth

2πr = 18

r = 18/2π cm

Height = 30 cm

Volume of cylinder, V2 = πr2h

= π × (18/2π)2 × 30 cm3

Hence,

Volume of cylinder,V1 / Volume of cylinder,V2 = {π × (30/2π)2 × 18}/{π × (18/2π)2× 30}

= {π × (30/2π)2 × 18}×1/{π × (2π/18)2× 30}

= 302 × 18 / 182 × 30

= 30/18

= 5/3

∴ The ratio of two volumes is 5:3

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