Question

A rectangular piece of paper of dimension \[44cm{\text{ }} \times 10cm\] is rolled along the length to form a cylinder. Find the radius of the base of the cylinder so formed. Also find the volume of the cylinder.

Answer 1

Hint: The rectangular piece of paper is rolled along the length to make a cylinder. Hence its breadth will become the height of the cylinder and the length of the rectangular piece will become the circumference of the base of the cylinder. The total curved surface area of the cylinder obtained is equal to the area of the rectangular sheet. So the circumference of the base is equal to the length of the rectangular sheet.

Complete step by step solution:
Given dimension of the rectangular piece of paper is \[44cm{\text{ }} \times 10cm\]
So, length of the rectangular piece of paper \[ = {\text{ }}44cm\]
Breadth of the rectangular piece of paper \[ = {\text{ }}10cm\]
As the paper is folded by its length to make a cylinder so,
Height of the cylinder will be equal to the base of the cylinder, \[h{\text{ }} = {\text{ }}10cm\]
Circumference of the base of the cylinder will be equal to the length of the paper i.e. \[2\pi r{\text{ }} = {\text{ }}44\,cm\]
Where \[r\] is the radius of the base of the cylinder. (Take value of \[\pi \, = \,\dfrac{{22\,}}{7}\] )
On simplifying we get,
 \[2\pi r = 44\]
 \[ \Rightarrow 2 \times \dfrac{{22}}{7} \times r = 44\]
On changing sides for simplification
 \[ \Rightarrow r = \dfrac{{44 \times 7}}{{2 \times 22}}\]
 \[ \Rightarrow r = 7cm\]
Hence the radius of the base of the cylinder formed is equal to \[7cm\]
Now, Volume of the cylinder is given by
Volume, \[V{\text{ }} = {\text{ }}\pi {r^2}h\]
Putting the values we get,
 \[V{\text{ }} = {\text{ }}\dfrac{{22}}{7}{\text{ }} \times 7{\text{ }} \times 7{\text{ }} \times 10\]
On simple multiplication and division
 \[V{\text{ }} = \,\dfrac{{10780}}{7}\, = \,1540\,c{m^3}\]
Hence the volume of the cylinder so formed is equal to \[1540\,c{m^3}\]
So, the correct answer is “ \[1540\,c{m^3}\] ”.

Note: One should observe carefully that the paper is either rolled along length or along the breadth as it gives the values of radius and the height of the cylinder. Questions may vary by changing its unit or its values. Pi should be taken \[\dfrac{{22}}{7}{\text{ }}\] for its precise value until it’s given to take \[3.14\] .