Question

Two rectangular sheets of paper each 30 cm × 18 cm are made into two right circular cylinders, one by rolling the paper along its length and the other along the breadth. The ratio of the volumes of the two cylinders, thus formed, is _________.

Answer 1

The length and breadth of each rectangular sheet of paper is 30 cm and 18 cm, respectively.

When the sheet of paper is rolled along its length to form a cylinder, then the circumference of the base of the cylinder so formed is same as the length of the rectangular sheet and the height of the cylinder is same as the breadth of the rectangular sheet.

Let r be the radius and h be the height of the cylinder formed in this case.

∴ h = 18 cm

Circumference of base = 30 cm

⇒ 2$\mathrm{\pi }$r = 30 cm

$\Rightarrow r=\frac{15}{\mathrm{\pi }}$ cm

∴ Volume of the cylinder formed in this case, V₁ = $\mathrm{\pi }{r}^{2}h=\mathrm{\pi }\times {\left(\frac{15}{\mathrm{\pi }}\mathrm{cm}\right)}^2\times 18\mathrm{cm}$ .....(1)

When the sheet of paper is rolled along its breadth to form a cylinder, then the circumference of the base of the cylinder so formed is same as the breadth of the rectangular sheet and the height of the cylinder is same as the length of the rectangular sheet.

Let R be the radius and H be the height of the cylinder formed in this case.

∴ H = 30 cm

Circumference of base = 18 cm

⇒ 2$\mathrm{\pi }$R = 18 cm

$\Rightarrow R=\frac{9}{\mathrm{\pi }}$ cm

∴ Volume of the cylinder formed in this case, V₂ = $\mathrm{\pi }{R}^{2}H=\mathrm{\pi }\times {\left(\frac{9}{\mathrm{\pi }}\mathrm{cm}\right)}^2\times 30\mathrm{cm}$ .....(2)

Now,

$\frac{V_1}{V_2}=\frac{\mathrm{\pi }\times {\left({\displaystyle \frac{15}{\mathrm{\pi }}}\mathrm{cm}\right)}^2\times 18\mathrm{cm}}{\mathrm{\pi }\times {\left({\displaystyle \frac{9}{\mathrm{\pi }}}\mathrm{cm}\right)}^2\times 30}=\frac{5}{3}$

⇒ V₁ : V₂ = 5 : 3

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

Two rectangular sheets of paper each 30 cm × 18 cm are made into two right circular cylinders, one by rolling the paper along its length and the other along the breadth. The ratio of the volumes of the two cylinders, thus formed, is __5 : 3__.