A rectangle of length 44 cm and breadth 12 cm is rolled along its length to form a cylinder. Find the volume of the cylinder.

Options

A. `504 cm^3`
B. `1848 cm^3`
C. `1924 cm^3`
D. `1644 cm^3`

Answer

To find the volume of the cylinder formed by rolling a rectangle of length 44 cm and breadth 12 cm along its length, we can follow these steps:

### Step 1: Identify the dimensions of the cylinder
When the rectangle is rolled along its length, the length of the rectangle becomes the circumference of the base of the cylinder, and the breadth becomes the height of the cylinder.

- Length of rectangle (circumference of cylinder) = 44 cm
- Breadth of rectangle (height of cylinder) = 12 cm

### Step 2: Use the formula for circumference to find the radius
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
Where \( r \) is the radius of the base of the cylinder. We can set this equal to the length of the rectangle:
\[ 2\pi r = 44 \]

### Step 3: Solve for the radius \( r \)
To find \( r \), we rearrange the equation:
\[ r = \frac{44}{2\pi} \]
\[ r = \frac{22}{\pi} \]

### Step 4: Use the volume formula for a cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Where \( h \) is the height of the cylinder. We already know \( h = 12 \) cm.

### Step 5: Substitute \( r \) and \( h \) into the volume formula
Now we substitute \( r \) and \( h \) into the volume formula:
\[ V = \pi \left(\frac{22}{\pi}\right)^2 \times 12 \]

### Step 6: Simplify the expression
Calculating \( r^2 \):
\[ r^2 = \left(\frac{22}{\pi}\right)^2 = \frac{484}{\pi^2} \]
Now substitute this back into the volume formula:
\[ V = \pi \times \frac{484}{\pi^2} \times 12 \]
\[ V = \frac{484 \times 12}{\pi} \]

### Step 7: Calculate the volume
Now we can calculate the volume:
\[ V = \frac{5808}{\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ V \approx \frac{5808 \times 7}{22} \]
\[ V \approx \frac{40656}{22} \approx 1848 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is approximately \( 1848 \text{ cm}^3 \).

---