Question

A rectangular sheet of paper $36\times 22$, is rolled along its length to form a cylinder. Then find the volume of the cylinder.
A. 2268
B. 226.8
C. 226
D. 23.4

Answer 1

Hint: We first try to understand how a figure of a rectangle creates a figure of a cylinder. We find the relation between the dimensions of those two figures. From the perimeter of the base of the cylinder, we find its radius. Then using the formula of volume of a cylinder $v=\pi {{r}^{2}}h$, we find the solution to the problem.

Complete step-by-step solution
The rectangular sheet of paper $36\times 22$ has been rolled along its length to form a cylinder. So, the cylinder’s height is the breadth of the rectangle which is 22 cm. The length creates the outer line of the base of the cylinder. The perimeter of the base of the cylinder is 36 cm.

We need to find the volume of the cylinder. We know the formula to find the volume of the cylinder is $v=\pi {{r}^{2}}h$, where h is the height of the cylinder and r is the radius of the base.
For our given problem $h=22$. We don’t have the radius but we got the perimeter of the base which is in the shape of a circle of value 36.
The perimeter of the base is of formula $2\pi r$ cm. So, $2\pi r=36$.
From the equation, we find the value of r as
$\begin{align}
  & 2\pi r=36 \\
 & \Rightarrow r=\dfrac{36}{2\pi } \\
\end{align}$
So, we have both the values of r and h. We place the values to get the volume.
The volume of the cylinder is $v=\pi {{r}^{2}}h$ where $h=22,r=\dfrac{36}{2\pi }$.
$\begin{align}
  & v=\pi {{r}^{2}}h=\pi \times {{\left( \dfrac{36}{2\pi } \right)}^{2}}\times 22 \\
 & \Rightarrow v=\dfrac{36\times 36\times 7\times 22}{4\times 22}=2268 \\
\end{align}$.
The volume of the cylinder is 2268 $c{{m}^{3}}$. The correct option is A.

Note: There are two dimensions of the rectangle. We need to understand which one changes into the height of the cylinder. The best way to remember is that the part which is rolled changes into the perimeter of the cylinder. We can’t confuse between those two as that will give different results.