The lateral surface area of a hollow cylinder is 4224 c{m^2} . It is cut along its height and formed a rectangular sheet of width 33 cm. Find the perimeter of the rectangular sheet.

We are given that the lateral surface area of a hollow cylinder is 4224 cm², and when the cylinder is cut along its height, it forms a rectangular sheet of width 33 cm. We are asked to find the perimeter of the rectangular sheet.

Step 1: Formula for Lateral Surface Area of a Hollow Cylinder
The lateral surface area (L.A.) of a hollow cylinder is given by the formula:

L.A. = 2π * (R + r) * h

where:

R = outer radius of the cylinder
r = inner radius of the cylinder
h = height of the cylinder
We are also given that the lateral surface area is 4224 cm², so we can write the equation as:

2π * (R + r) * h = 4224

Step 2: Relation between the Width of the Rectangular Sheet and the Circumference
When the cylinder is cut along its height, the circumference of the outer surface (2πR) becomes the width of the rectangular sheet. We are given that the width is 33 cm, so:

2π * R = 33

Solving for R:

R = 33 / (2π)

Step 3: Find the Height of the Cylinder
Substitute the value of R into the lateral surface area equation:

2π * (R + r) * h = 4224

Substitute R = 33 / (2π) into the equation:

2π * ((33 / (2π)) + r) * h = 4224

Simplify the equation:

(33 + 2π * r) * h = 4224

We do not have enough information to find the exact value of r or h individually, but we can solve for the perimeter of the rectangular sheet without needing to find r and h explicitly.

Step 4: Perimeter of the Rectangular Sheet
The rectangular sheet formed by cutting the cylinder has:

Width = 33 cm (circumference of the outer surface of the cylinder)
Length = height of the cylinder, which we will call h.
The perimeter (P) of the rectangular sheet is given by the formula:

P = 2 * (Length + Width)

Substitute the values:

P = 2 * (h + 33)

This is the general formula for the perimeter. To get the exact perimeter, we would need additional information such as the height or the inner radius of the cylinder. However, based on the information provided, this is the best representation of the perimeter.