A rectangular piece of paper is 24 cm long and 22 cm wide. A cylinder is formed by rolling the paper along its length. The volume of the cylinder is

Options

A. `924" cm"^(3)`
B. `462" cm"^(3)`
C. `264" cm"^(3)`
D. `528" cm"^(3)`

Answer

To find the volume of the cylinder formed by rolling a rectangular piece of paper, we can follow these steps:

### Step 1: Identify the dimensions of the rectangle
The dimensions of the rectangular piece of paper are:
- Length (L) = 24 cm
- Width (W) = 22 cm

### Step 2: Understand the formation of the cylinder
When the paper is rolled along its length, the height (h) of the cylinder will be equal to the width of the paper, and the circumference (C) of the base of the cylinder will be equal to the length of the paper.

- Height of the cylinder (h) = Width of the paper = 22 cm
- Circumference of the base of the cylinder (C) = Length of the paper = 24 cm

### Step 3: Relate circumference to the radius
The circumference of a circle is given by the formula:
\[ C = 2\pi r \]
Where \( r \) is the radius of the base of the cylinder.

From the circumference, we can find the radius:
\[ 24 = 2\pi r \]
\[ r = \frac{24}{2\pi} = \frac{12}{\pi} \]

### Step 4: Calculate the volume of the cylinder
The volume (V) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]

Substituting the values of \( r \) and \( h \):
\[ V = \pi \left(\frac{12}{\pi}\right)^2 \times 22 \]

### Step 5: Simplify the expression
Calculating \( r^2 \):
\[ r^2 = \left(\frac{12}{\pi}\right)^2 = \frac{144}{\pi^2} \]

Now substituting back into the volume formula:
\[ V = \pi \times \frac{144}{\pi^2} \times 22 \]
\[ V = \frac{144 \times 22}{\pi} \]

### Step 6: Calculate the numerical value
Now, using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{144 \times 22 \times 7}{22} \]
\[ V = 144 \times 7 \]
\[ V = 1008 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is **1008 cm³**.

---