A rectangular sheet of paper 30 cm × 18 cm can be transformed into the curved surfaced 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.

#### Solution

Given data is as follows:

Dimensions of the rectangular sheet of paper = `30 "cm" xx 18 "cm"`

We have to find the ratio of the volumes of the cylinders formed by rolling the sheet along its length and along its breadth.

Let V_{1} be the volume of the cylinder which is formed by rolling the sheet along its length.

When the sheet is rolled along its length, the length of the sheet forms the perimeter of the base of the cylinder. Therefore, we have,

`2pir_1 = 30`

`r_1 = 15/pi`

The width of the sheet will be equal to the height of the cylinder. Therefore,

h_{1 }=18 cm

Therefore,

`V_1 = pir_1^2 h_1`

`= pi xx 15/pi xx 15/pi xx 18`

`V_1 = 225/pi xx 10 cm^3`

Let V_{2} be the volume of the cylinder formed by rolling the sheet along its width.

When the sheet is rolled along its width, the width of the sheet forms the perimeter of the base of the cylinder. Therefore, we have,

`2pir_2`=18

`r_2 = 9/pi`

The length of the sheet will be equal to the height of the cylinder. Therefore,

h_{2}= 30 cm

Now,

V_{2}= `pir_2^2h_2`

= `pi xx 9/pi xx 9/pi xx 30`

`V_2 = (81xx30)/pi`

Now that we have the volumes of the two cylinders, we have,

`V_1/V_2 = (225xx18)/(18 xx 30)`

`V_1/V_2 = 5/3`

Therefore, the ratio of the volumes of the two cylinders is 5:3