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

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the relationship between the hollow cylinder and the rectangular sheet.
The lateral surface area of the hollow cylinder is equal to the area of the rectangular sheet formed when the cylinder is cut along its height.

### Step 2: Write down the formula for the lateral surface area of a cylinder.
The lateral surface area (LSA) of a hollow cylinder is given by the formula:
\[ \text{LSA} = 2\pi rh \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 3: Use the given lateral surface area to find the dimensions of the rectangular sheet.
We know the lateral surface area of the hollow cylinder is 4224 cm². When the cylinder is cut, it forms a rectangular sheet where:
- Width (breadth) = 33 cm
- Length (l) = ?

The area of the rectangular sheet is given by:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Thus, we can set up the equation:
\[ 4224 = l \times 33 \]

### Step 4: Solve for the length (l).
To find the length, we rearrange the equation:
\[ l = \frac{4224}{33} \]

Calculating this gives:
\[ l = 128 \text{ cm} \]

### Step 5: Calculate the perimeter of the rectangular sheet.
The perimeter (P) of a rectangle is given by the formula:
\[ P = 2 \times (\text{Length} + \text{Width}) \]
Substituting the values we found:
\[ P = 2 \times (128 + 33) \]

Calculating this gives:
\[ P = 2 \times 161 = 322 \text{ cm} \]

### Final Answer:
The perimeter of the rectangular sheet is **322 cm**.

---