Question

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.

Answer 1

We have,

Dimensions of rectangular sheet = 30 cm × 18 cm

Case (i)

When paper is rolled along its length.

2πr = 30

r = `30/(2pi)` cm

Height = 18 cm

Volume of cylinder, V₁ = πr²h

= `pi xx (30/(2pi))^2 xx 18` cm³

Case (ii)

When paper is rolled along its breath.

2πr = 18

r = `18/(2pi)` cm

Height = 30 cm

Volume of cylinder, V₂ = πr²h

= `pi xx (18/(2pi))^2 xx 30` cm³

Hence,

`("Volume of cylinder", "V"_1)/("Volume of cylinder", "V"_2) = {{pi xx (30/(2pi))^2 xx 18}}/{{pi xx (18/(2pi))^2 xx 30}}`

= `{pi xx (30/(2pi))^2 xx 18} xx 1/{{pi xx ((2pi)/18)^2 xx 30}`

= `(30^2 xx 18)/(18^2 xx 30)`

= `30/18`

= `5/3`

∴ The ratio of two volumes is 5 : 3.