A rectangular piece of paper of dimensions 22 cm by 12 cm is rolled along its length to form a cylinder. The volume (in cu. cm.) of the cylinder so formed is (use `pi = (22)/(7)`)

Options

A. a) 562
B. b) 412
C. c) 462
D. d) 362

Answer

To find the volume of the cylinder formed by rolling a rectangular piece of paper of dimensions 22 cm by 12 cm along its length, we can follow these steps:

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

- Length of the paper (circumference of the cylinder) = 22 cm
- Width of the paper (height of the cylinder) = 12 cm

### Step 2: Calculate the radius of the cylinder
The circumference \( C \) of a cylinder is given by the formula:
\[ C = 2\pi r \]
Where \( r \) is the radius of the base of the cylinder. We can rearrange this formula to find the radius:
\[ r = \frac{C}{2\pi} \]
Substituting the values:
\[ r = \frac{22}{2 \times \frac{22}{7}} = \frac{22 \times 7}{2 \times 22} = \frac{7}{2} \text{ cm} \]

### Step 3: Use the formula for the volume of the 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. Substituting the values we have:
- \( r = \frac{7}{2} \) cm
- \( h = 12 \) cm
\[ V = \frac{22}{7} \left(\frac{7}{2}\right)^2 \times 12 \]

### Step 4: Calculate \( r^2 \)
First, calculate \( r^2 \):
\[ r^2 = \left(\frac{7}{2}\right)^2 = \frac{49}{4} \]

### Step 5: Substitute \( r^2 \) into the volume formula
Now substitute \( r^2 \) into the volume formula:
\[ V = \frac{22}{7} \times \frac{49}{4} \times 12 \]

### Step 6: Simplify the volume expression
Now we can simplify:
\[ V = \frac{22 \times 49 \times 12}{7 \times 4} \]
Calculating \( 7 \times 4 = 28 \):
\[ V = \frac{22 \times 49 \times 12}{28} \]

### Step 7: Further simplify
Now we can simplify \( \frac{22}{28} = \frac{11}{14} \):
\[ V = \frac{11 \times 49 \times 12}{14} \]

### Step 8: Calculate \( 11 \times 49 \)
Calculating \( 11 \times 49 = 539 \):
\[ V = \frac{539 \times 12}{14} \]

### Step 9: Divide \( 539 \) by \( 14 \)
Calculating \( 539 \div 14 = 38.5 \):
\[ V = 38.5 \times 12 = 462 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is \( 462 \) cubic centimeters.

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