Question

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.

Answer 1

Let $V_1$ and $V_2$ be the volumes of two cylinders. When the sheet is folded along its length, it forms a cylinder of height $h_1=18$cm and perimeter of base equal to $30$cm. Let $r_1$ be the radius of the base. Then,
$2\pi r_1=30\Rightarrow r_1=\frac{15}{\pi }$
$\therefore V_1=\pi r_1^2{h}_1=\pi \times \frac{225}{{\pi }^2}\times 18c{m}^3=\frac{225}{\pi }\times 18c{m}^3$
Similarly, $V_2=\pi r_2^2{h}_2=\pi \times {\left(\frac{9}{\pi }\right)}^2\times 30c{m}^3=\frac{81\times 30}{\pi }c{m}^3$
$\therefore \frac{V_1}{V_2}=\frac{225\times 18}{81\times 30}=\frac{5}{3}$.