Question

The lateral surface area of a hollow cylinder is 5632 $c{{m}^{2}}$. It is cut along its height and a rectangular sheet of width 44 cm is formed. Find the perimeter of the rectangular sheet?
A. 344 cm
B. 388 cm
C. 320 cm
D. 300 cm

Answer 1

Hint: Here, we can find the radius of the cylinder using the width of the rectangular sheet because the perimeter of the base of the cylinder will be equal to the width of the sheet. The length of the sheet will be equal to the height of the cylinder and it is calculated by using the lateral surface area of the hollow cylinder given.

Complete step-by-step answer:
We know that the formula for lateral surface area of a cylinder is given as:
\[LSA=2\pi \times r\times h...........\left( 1 \right)\]
Now, we will try to find the radius of the cylinder.
When the hollow cylinder is cut along its height the width of the rectangular sheet obtained is equal to the perimeter of the base of the cylinder or we can say circumference of the base of the cylinder.
Let us consider this radius to be = r
We know that the formula for circumference is = $2\pi \times r$ . So, we have:
$\begin{align}
  & 2\pi \times r=44cm \\
 & r=\dfrac{44}{2\pi }cm \\
 & r=\dfrac{44}{2\times \dfrac{22}{7}}cm \\
 & r=7cm \\
\end{align}$
So, the radius of the cylinder is = 7cm.
Now, let the height be = h
On substituting the values of radius and lateral surface area of the cylinder in equation (1), we get:
$\begin{align}
  & 2\pi \times 7\times h=5632 \\
 & 2\times \dfrac{22}{7}\times 7\times h=5632 \\
 & h=\dfrac{5632\times 7}{2\times 22\times 7} \\
 & h=128cm \\
\end{align}$
So, the height of the cylinder is 128 cm.
Since, the height of the cylinder is equal to the length of the sheet. So, the length of the rectangular is also = 128 cm.
Since, the formula for the perimeter of the sheet is = $2\times \left( length+breadth \right)$
So, $\begin{align}
  & P=2\times \left( 128+44 \right) \\
 & \,\,\,\,\,=2\times 172 \\
 & \,\,\,\,\,=344cm \\
\end{align}$
Therefore, the perimeter of the rectangular sheet is 344 cm.
Hence, option A is the correct answer.

Note: It is important to note here that height of the cylinder is equal to the length of the rectangular sheet. Also the width of the sheet is equal to the perimeter of the base of the cylinder.Students should remember the formula of lateral surface area of hollow cylinder.