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,

Dimension of rectangular sheet of paper = 30cm × 18cm

Case (i)

When paper is rolled along its length,

Then, 2nr = 30 = r = \(\cfrac{30}{2\pi}\) = \(\cfrac{15}{\pi}\) cm

= h = 18 cm

So, volume of cylinder V₁ = πr²h = π \(\times\big(\cfrac{15}{\pi}\big)^2\times\) 18

= \(\cfrac{225}\pi\times\) 18 cm³

Case (ii)

When paper is rolled along its width,

Then 2πr = 18 = r = \(\cfrac{18}{2\pi}=\cfrac{9}{\pi}\) cm

= h = 30 cm

So, volume of cylinder thus form 

= \(\cfrac{22}7\times\big(\cfrac{9}{\pi}\big)^2\times\) 30

= \(\cfrac{81}{\pi}\times\) 30 cm³

Ratio of volumes V₁ : V₂ = \(\cfrac{225\times18}{81\times30}\) = \(\cfrac{5}3\)