A rectangular sheet of paper $30\ cm \times 18\ cm$ 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.

 Given:

A rectangular sheet of paper $30\ cm \times 18\ cm$ 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.

To do:

We have to find the ratio of the volumes of the two cylinders thus formed.

Solution:

Size of the rectangular sheet $= 30\ cm \times 18\ cm$

This implies,

Length of the sheet $= 30\ cm$

Breadth of the sheet $= 18\ cm$

When folded length wise,

Height $= 18\ cm$

Circumference $= 30\ cm$
Therefore,

Radius $=\frac{\text { Circumference }}{2 \pi}$

$=\frac{30}{2 \pi}$

Volume $=\pi r^{2} h$

$=\pi \times \frac{30}{2 \pi} \times \frac{30}{2 \pi} \times 18$

$=\frac{16200}{4 \pi}$

$=\frac{8100}{2 \pi} \mathrm{cm}^{3}$

In the second case,
When folded width wise,

Height $=30 \mathrm{~cm}$

Circumference $=18 \mathrm{~cm}$

Radius $=\frac{C}{2 \pi}$

$=\frac{18}{2 \pi}$

Volume $=\pi(\frac{18}{2 \pi})^{2} \times 30$

$=\pi \times \frac{18}{2 \pi} \times \frac{18}{2 \pi} \times 30$

$=\frac{2430}{\pi} \mathrm{cm}^{3}$

Ratio in the volumes in both cases $=\frac{8100}{2 \pi}: \frac{2430}{2 \pi}$

$=\frac{10}{2}: \frac{3}{1}$

$=5: 3$

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Updated on: 10-Oct-2022

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