Question

A rectangular sheet of paper 30 cm × 18 cm 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 × 0.18 cm

Case (i)

When sheet is rolled along its length:

= 2πr = 30

= r = \(\frac{30}{2π}\)

height of cylinder = 18 cm

Hence, Volume of cylinder thus formed = V₁ = \(\frac{22}{7}\) x \((\frac{30}{2π})^2\) x 18 cm³

Case (ii)

When sheet is rolled along its breadth:

 = 2πr = 18

= r = \(\frac{18}{2π}\)

Height of cylinder = 30 cm

Volume of cylinder = V₂ = \(\frac{22}{7}\) x \((\frac{18}{2π})^2\) x 30 cm³

Hence,

\(\frac{V_1}{V_2}\) = \(\frac{[\frac{22}{7}\times 900\times 4π^2\times 18]}{[\frac{22}{7}\times 324\times 4π^2\times 30]}\) = \(\frac{30}{18}\) = \(\frac{5}{3}\)