The rain water from a roof `22 m xx 20 m` drains into a cylindrical vessel having a diameter of 2m and height 3.5 m. If the vessel is just full, then the rainfull (in cm) is :

Options

A. (a) 2
B. (b) `2.5`
C. (c) 3
D. (d) `4.5`

Answer

To solve the problem step by step, we will calculate the volume of rainwater collected in the cylindrical vessel and then relate it to the area of the roof to find the rainfall.

### Step 1: Calculate the volume of the cylindrical vessel
The formula for the volume \( V \) of a cylinder is given by:

\[ V = \pi r^2 h \]

Where:
- \( r \) is the radius of the cylinder
- \( h \) is the height of the cylinder

Given:
- Diameter of the cylinder = 2 m, hence radius \( r = \frac{2}{2} = 1 \) m
- Height \( h = 3.5 \) m

Substituting the values into the formula:

\[ V = \pi (1)^2 (3.5) = \pi \times 1 \times 3.5 = 3.5\pi \text{ m}^3 \]

Using \( \pi \approx \frac{22}{7} \):

\[ V = 3.5 \times \frac{22}{7} = 3.5 \times 3.14 = 11 \text{ m}^3 \]

### Step 2: Calculate the area of the roof
The area \( A \) of the roof can be calculated using the formula:

\[ A = \text{length} \times \text{width} \]

Given:
- Length = 22 m
- Width = 20 m

Substituting the values:

\[ A = 22 \times 20 = 440 \text{ m}^2 \]

### Step 3: Relate the volume of water to the rainfall
Let \( x \) be the height of the rainfall in meters. The volume of rainwater collected from the roof can also be expressed as:

\[ \text{Volume} = \text{Area} \times \text{Height of rainfall} = A \times x \]

Substituting the area:

\[ \text{Volume} = 440 \times x \text{ m}^3 \]

### Step 4: Set the volumes equal
Since the volume of rainwater collected is equal to the volume of the cylindrical vessel:

\[ 440x = 11 \]

### Step 5: Solve for \( x \)
To find \( x \):

\[ x = \frac{11}{440} = \frac{1}{40} \text{ m} \]

### Step 6: Convert \( x \) to centimeters
To convert meters to centimeters, we multiply by 100:

\[ x = \frac{1}{40} \times 100 = 2.5 \text{ cm} \]

### Final Answer
The rainfall is **2.5 cm**.

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