A rectangular sheet of paper `30 cm xx 18 cm` can be transformed into the curved surface of a right circular cylinder in two ways namely, either by rolling the paper along its length or by rolling it along its breadth. Find the ratio of the volume of the two cylinders, thus formed.

Explanation

To solve the problem of finding the ratio of the volumes of two cylinders formed by rolling a rectangular sheet of paper (30 cm x 18 cm) along its length and breadth, we will follow these steps:

### Step 1: Identify the dimensions for the first cylinder (rolling along the length)
When the rectangular sheet is rolled along its length (30 cm):
- Height (h1) of the cylinder = Length of the sheet = 30 cm
- Circumference (C) of the base = Width of the sheet = 18 cm

Using the relationship between circumference and diameter:
\[ C = 2\pi r \]
We can find the radius (r1):
\[ 18 = 2\pi r_1 \]
\[ r_1 = \frac{18}{2\pi} = \frac{9}{\pi} \text{ cm} \]

### Step 2: Calculate the volume of the first cylinder
The volume (V1) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values for the first cylinder:
\[ V_1 = \pi \left(\frac{9}{\pi}\right)^2 \times 30 \]
\[ V_1 = \pi \times \frac{81}{\pi^2} \times 30 \]
\[ V_1 = \frac{2430}{\pi} \text{ cm}^3 \]

### Step 3: Identify the dimensions for the second cylinder (rolling along the breadth)
When the rectangular sheet is rolled along its breadth (18 cm):
- Height (h2) of the cylinder = Width of the sheet = 18 cm
- Circumference (C) of the base = Length of the sheet = 30 cm

Using the relationship between circumference and diameter:
\[ C = 2\pi r \]
We can find the radius (r2):
\[ 30 = 2\pi r_2 \]
\[ r_2 = \frac{30}{2\pi} = \frac{15}{\pi} \text{ cm} \]

### Step 4: Calculate the volume of the second cylinder
Substituting the values for the second cylinder:
\[ V_2 = \pi \left(\frac{15}{\pi}\right)^2 \times 18 \]
\[ V_2 = \pi \times \frac{225}{\pi^2} \times 18 \]
\[ V_2 = \frac{4050}{\pi} \text{ cm}^3 \]

### Step 5: Find the ratio of the volumes
Now we can find the ratio of the volumes \( V_1 \) and \( V_2 \):
\[ \text{Ratio} = \frac{V_1}{V_2} = \frac{\frac{2430}{\pi}}{\frac{4050}{\pi}} \]
The \( \pi \) cancels out:
\[ \text{Ratio} = \frac{2430}{4050} = \frac{243}{405} = \frac{81}{135} = \frac{3}{5} \]

### Final Answer
The ratio of the volumes of the two cylinders is \( 3:5 \).