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

Given,

Dimensions of rectangular sheet = 30 cm× 18 cm

Case (i)

when paper is rolled along its length:

= 2πr = 30

= r = \(\frac{30}{2π}\) cm

= height = 18 cm

Volume of cylinder thus formed = πr²h = π x (\(\frac{30}{2π}\))² x 18 cm³

Case (ii)

When paper is rolled along its breadth:

= 2πr = 18

=  r = \(\frac{18}{2π}\) cm

= Height = 30 cm

Volume of cylinder thus formed = πr²h = π(\(\frac{18}{2π}\))² x 30 

Hence,

= \(\frac{volume\,of\,cylinder\,1}{volume\,of\,cylinder\,2}\) = \(\cfrac{[π(\frac{30}{2π})^2\times 18]}{[π(\frac{18}{2π})^2\times 30]}\) = \(\frac{30}{18}\) = \(\frac{5}{3}\)

Answer 2

As we roll the sheet, one of the dimensions is the height of the cylinder, the other is circumference of the base circle.

Therefore:

C₁ = 2 π r₁ = 18cm ⇒ r₁ = 9cm/π

h₁ = 30cm

V₁ = π (r₁)² h₁ = 81cm*30cm/π 

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C₂ = 2 π r₂ = 30cm ⇒ r = 15cm/π

h₂ = 18cm

V₂ = π (r₂)² h₂ = 225cm*18cm/π 

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V₁/V₂ = (81cm*30cm)/(225cm*18cm) = 0.6