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 namely, 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

The dimensions of the rectangular sheet of paper are 30 cm × 18 cm.

Let V₁ and V₂ be the volumes of the cylinders formed by rolling the rectangular sheet of paper along its length (i.e. 30 cm) and breadth (i.e. 18 cm), respectively.

Suppose r₁ and h₁ be the radius and height of the cylinder formed by rolling the rectangular sheet of paper along its length, respectively.

$\therefore 2\mathrm{\pi }{r}_1=30\Rightarrow r_1=\frac{30}{2\mathrm{\pi }}\mathrm{cm}$

h₁ = 18 cm

$\therefore V_1=\mathrm{\pi }{r}_1^2{h}_1=\mathrm{\pi }{\left(\frac{30}{2\mathrm{\pi }}\right)}^2\times 18{\mathrm{cm}}^3$

Also, suppose r₂ and h₂ be the radius and height of the cylinder formed by rolling the rectangular sheet of paper along its breadth, respectively.

$\therefore 2\mathrm{\pi }{r}_2=18\Rightarrow r_2=\frac{18}{2\mathrm{\pi }}\mathrm{cm}$

h₂ = 30 cm

$\therefore V_2=\mathrm{\pi }{r}_2^2{h}_2=\mathrm{\pi }{\left(\frac{18}{2\mathrm{\pi }}\right)}^2\times 30{\mathrm{cm}}^3$

Now,

$\frac{V_1}{V_2}=\frac{\mathrm{\pi }{\left({\displaystyle \frac{30}{2\mathrm{\pi }}}\right)}^2\times 18}{\mathrm{\pi }{\left({\displaystyle \frac{18}{2\mathrm{\pi }}}\right)}^2\times 30}=\frac{5}{3}$

⇒ V₁ : V₂ = 5 : 3

Thus, the ratio of the volumes of the two cylinders thus formed is 5 : 3.