Question

A rectangular sheet of foil is 88cm long and 20 cm wide. A cylinder is made out of it, by rolling the foil along the width. Find the volume of the cylinder.

Answer 1

Hint: The length and width of the rectangular foil are 88 cm and 20 cm. The cylinder is rolled along the width so, the circumference of the circular base is equal to the length of the foil and the height of the cylinder is equal to the width of the rectangular foil. Assume that the radius of the circular base is \[r\] cm. Use the formula, Circumference = \[2\pi \left( radius \right)\] and calculate the circumference. Now, make it equal to the length of the foil and get the value of \[r\] . Finally, use the formula, Volume = \[\pi {{\left( radius \right)}^{2}}\left( height \right)\] and calculate the volume of the cylinder.

Complete step-by-step solution:
According to the question, we are given,
The length of the rectangular foil = 88 cm …………………………………………….(1)
The width of the rectangular foil = 20 cm ………………………………………….(2)
Now, it is given that the rectangular foil is rolled along the width to obtain a cylinder.
Let us assume that the radius of the circular base is \[r\] cm …………………………………………..(3)
We know the formula for the circumference of the circle, Circumference = \[2\pi \left( radius \right)\] ……………………………….(4)
From equation (3) and equation (4), we get
The circumference of the circular base = \[2\pi r\] cm …………………………………….(5)

Since the cylinder is rolled along the width so, we can say that the circumference of the circular base is equal to the height of the foil and the width of the cylinder is equal to the length of the rectangular foil ………………………………………….(6)

Now, from equation (1), equation (2), equation (5), and equation (6), we get
The circumference of the circular base = \[2\pi r\] cm = 88 cm ………………………………(7)
The height of the cylinder = 20 cm …………………………………………..(8)
On solving equation (7), we get
\[\begin{align}
  & \Rightarrow 2\pi r=88 \\
 & \Rightarrow 2\times \dfrac{22}{7}\times r=88 \\
 & \Rightarrow r=\dfrac{88\times 7}{2\times 22} \\
 & \Rightarrow r=2\times 7 \\
\end{align}\]
\[\Rightarrow r=14\] ……………………………………..(9)
Now, we have
The radius of the circular base of the cylinder = 14 cm (from equation (9))
The height of the cylinder = 20 cm
We know the formula for the volume of the cylinder, Volume = \[\pi {{\left( radius \right)}^{2}}\left( height \right)\]
On calculating, we get
The volume of the cylinder = \[\dfrac{22}{7}\times {{\left( 14 \right)}^{2}}\times 20\,c{{m}^{3}}=12320\,c{{m}^{3}}\]
Therefore, the volume of the cylinder is \[12320\,c{{m}^{3}}\].

Note: In this type of question the process to obtain the radius and height of the cylinder is important. For instance, one might make a silly mistake and take the width of the foil equal to the circumference of the circular base and the height of the cylinder equal to the length of the foil. This is wrong. Since the cylinder is rolled along the width so, we can say that the circumference of the circular base is equal to the height of the foil and the height of the cylinder is equal to the width of the rectangular foil.