Question

4. A rectangular strip of aluminum, 30 cm long and 10 cm wide, is rolled into the form of a cylinder.

Which would have the greater volume, a cylinder 10 cm high with circumference of 30 cîn or a cylinder
30 cm high with circumference of 10 cm?

Answer 1

A cylinder 10 cm high with a circumference of 30 cm would have a greater volume

Explanation:

Volume of a Cylinder

Given a cylinder of radius r and height h, the volume is calculated as:

`V=\pi r^2h`

It's given a rectangular strip of aluminum 30 cm long and 10 cm wide. If we roll it to form a cylinder we can do it in two ways.

Rolling it across its length will make the cylinder have a height of h=10 cm and the circumference of the base equal to 30 cm.

The circumference of a circle of radius r is:

`C=2\pi r`

Thus the radius is:

`\displaystyle r=(C)/(2\pi)`

`\displaystyle r=(30)/(2\pi)=(15)/(\pi)`

Thus the volume of the cylinder is:

`V=\pi ( (15)/(\pi))^2*10`

`V=\pi ( (225)/(\pi^2))*10`

`V= (2250)/(\pi)`

If we rolled the strip across its width, then the height will be h=30 cm and the circumference C=10 cm, thus the radius is:

`\displaystyle r=(10)/(2\pi)=(5)/(\pi)`

And the volume is:

`V'=\pi ( (5)/(\pi))^2*30`

`V'=\pi ( (25)/(\pi^2))*30`

`V'= (750)/(\pi)`

Dividing V/V' = 3. The first volume it 3 times as much as the second volume.

A cylinder 10 cm high with a circumference of 30 cm would have a greater volume