The rain water from a 22 m × 20 m roof drains into a cylindrical vessel of diameter 2 m and height 3.5 m.

We have,

the length of the roof, $l=22 \mathrm{~m}$,

the width of the roof, $b=20 \mathrm{~m}$,

the base radius of the cylindrical vessel, $R=\frac{2}{2}=1 \mathrm{~m}$ and

the height of the cylindrical vessel, $H=3.5 \mathrm{~m}$

Let the height of the rainfall be $h$.

Now,

Volume of rainfall = Volume of rain water collected in the cylindrical vessel

$\Rightarrow l b h=\frac{4}{5} \times$ Volume of cylindrical vessel

$\Rightarrow 22 \times 20 \times h=\frac{4}{5} \times \pi R^{2} H$

$\Rightarrow 440 h=\frac{4}{5} \times \frac{22}{7} \times 1 \times 1 \times 3.5$

$\Rightarrow h=\frac{4}{5} \times \frac{22}{7} \times \frac{3.5}{440}$

$\Rightarrow h=0.02 \mathrm{~m}$

$\therefore h=2 \mathrm{~cm}$

So, the height of the rainfall is 2 cm.