Let's solve each part of the problem step-by-step.

Part (i): Finding the rainfall in cm

First, we need to calculate the volume of the cylindrical vessel when it is filled with water. The volume $V$ of a cylinder is given by:

$V = \pi r^2 h$

where $r$ is the radius and $h$ is the height. The diameter of the vessel is 2 m, so the radius $r$ is:

$r = \frac{\text{diameter}}{2} = \frac{2 \, \text{m}}{2} = 1 \, \text{m}$

The height $h$ of the vessel is 3.5 m. Therefore, the volume $V$ of the cylindrical vessel is:

$V = \pi (1 \, \text{m})^2 (3.5 \, \text{m})$ $V = \pi \times 1 \times 3.5$ $V = 3.5\pi \, \text{m}^3$ $V \approx 3.5 \times 3.1416$ $V \approx 10.9956 \, \text{m}^3$

Next, we need to calculate the volume of rainwater collected from the roof. The area of the roof is:

$\text{Area} = 22 \, \text{m} \times 20 \, \text{m}$ $\text{Area} = 440 \, \text{m}^2$

Let $h$ be the height of the rainfall in meters. The volume of the rainwater collected is then:

$\text{Volume of rainwater} = \text{Area of roof} \times \text{Height of rainfall}$ $10.9956 \, \text{m}^3 = 440 \, \text{m}^2 \times h$

Solving for $h$:

$h = \frac{10.9956 \, \text{m}^3}{440 \, \text{m}^2}$ $h \approx 0.025 \, \text{m}$

Converting this to cm (since 1 m = 100 cm):

$h \approx 0.025 \, \text{m} \times 100 \, \text{cm/m}$ $h \approx 2.5 \, \text{cm}$

So, the rainfall is approximately $2.5 \, \text{cm}$.

Part (ii): Finding the height of water collected in the cylindrical vessel if the rainfall is 1.5 cm

First, convert the rainfall from cm to meters:

$\text{Rainfall} = 1.5 \, \text{cm} = 1.5 \, \text{cm} \times \frac{1 \, \text{m}}{100 \, \text{cm}} = 0.015 \, \text{m}$

Next, calculate the volume of rainwater collected from the roof with this rainfall:

$\text{Volume of rainwater} = \text{Area of roof} \times \text{Height of rainfall}$ $\text{Volume of rainwater} = 440 \, \text{m}^2 \times 0.015 \, \text{m}$ $\text{Volume of rainwater} = 6.6 \, \text{m}^3$

Now, we need to determine the height of water collected in the cylindrical vessel. Using the volume of a cylinder formula:

$V = \pi r^2 h$

We know the volume $V$ is 6.6 m³, the radius $r$ is 1 m. We need to find the height $h$:

$6.6 = \pi (1)^2 h$ $6.6 = \pi h$ $h = \frac{6.6}{\pi}$ $h \approx \frac{6.6}{3.1416}$ $h \approx 2.1 \, \text{m}$

So, the height of water collected in the cylindrical vessel is approximately $2.1 \, \text{m}$.

Would you like any further details or have any questions about the solution?

Here are five questions you can ask next:

1. How do you convert between different units of volume and height?
2. Can you explain the concept of volume conservation in more detail?
3. How do you solve similar problems with different roof and vessel dimensions?
4. What are some real-world applications of these calculations in water management?
5. How can you calculate the volume if the shape of the vessel changes?

Tip: Always double-check unit conversions and the dimensions used in calculations to ensure accuracy.