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

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

$\therefore$ Length of sheet = 30 cm

and breadth = 18 cm

By folding length wise,

Height = 18 cm and circumference = 30 cm

$\therefore Radius=\frac{Circumference}{2\pi }=\frac{30}{2\pi }$

Volume = $\pi r^{2}h=\pi \times \frac{30}{2\pi }\times \frac{30}{2\pi }\times 18$

=$\frac{16200}{4\pi }=\frac{8100}{2\pi }c{m}^3$

In second case,

By folding width wise,

Height = 30 cm and circumference = 18 cm

$\therefore Radius=\frac{C}{2\pi }=\frac{18}{2\pi }$

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

=$\pi \times \frac{18}{2\pi }\times \frac{18}{2\pi }\times 30$ = $\frac{2430}{\pi }c{m}^3$

$\therefore$ Ratio in their volumes =$\frac{8100}{2\pi }:\frac{2430}{\pi }$ = 5:3