Find the volume of the largest cylinder formed when a rectangular piece of paper 22 cm by 15 cm is rolled along its longer side.

Answer

To find the volume of the largest cylinder formed when a rectangular piece of paper measuring 22 cm by 15 cm is rolled along its longer side, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the dimensions of the rectangle:**
- Length = 22 cm
- Breadth = 15 cm

2. **Determine the circumference of the cylinder:**
- When the paper is rolled along its longer side (22 cm), this length becomes the circumference of the cylinder.
- Circumference (C) = 22 cm.

3. **Use the circumference to find the radius:**
- The formula for the circumference of a cylinder is given by:
\[ C = 2\pi r \]
- Rearranging the formula to find the radius (r):
\[ r = \frac{C}{2\pi} = \frac{22}{2\pi} \]

4. **Substitute the value of π:**
- Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{22}{2 \times \frac{22}{7}} = \frac{22 \times 7}{2 \times 22} = \frac{7}{2} \text{ cm} \]

5. **Identify the height of the cylinder:**
- The height (h) of the cylinder is equal to the breadth of the paper, which is 15 cm.

6. **Calculate the volume of the cylinder:**
- The formula for the volume (V) of a cylinder is:
\[ V = \pi r^2 h \]
- Substituting the values of \( r \) and \( h \):
\[ V = \pi \left(\frac{7}{2}\right)^2 \times 15 \]
- Calculating \( r^2 \):
\[ r^2 = \left(\frac{7}{2}\right)^2 = \frac{49}{4} \]
- Now substituting \( r^2 \) into the volume formula:
\[ V = \pi \times \frac{49}{4} \times 15 \]
- Simplifying:
\[ V = \frac{49 \times 15 \times \pi}{4} = \frac{735\pi}{4} \]
- Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{735 \times \frac{22}{7}}{4} = \frac{735 \times 22}{28} \]
- Simplifying further:
\[ V = \frac{16170}{28} = 577.5 \text{ cm}^3 \]

### Final Answer:
The volume of the largest cylinder formed is **577.5 cm³**.