Question

The rain water from a 22 m $\times$ 20 m roof drains into a cylindrical vessel of diameter 2 m and height 3.5 m. If the rain water collected from the roof fills $\frac{4}{5}$th of the cylindrical vessel, then find the rainfall in centimetre. [CBSE 2015]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{the}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{roof},l=22\mathrm{m}, \\ \mathrm{the}\mathrm{width}\mathrm{of}\mathrm{the}\mathrm{roof},b=20\mathrm{m}, \\ \mathrm{the}\mathrm{base}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{vessel},R=\frac{2}{2}=1\mathrm{m}\mathrm{and} \\ \mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{vessel},H=3.5\mathrm{m} \\ \mathrm{Let}\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{rainfall}\mathrm{be}h. \\ \mathrm{Now}, \\ \mathrm{Volume}\mathrm{of}\mathrm{rainfall}=\mathrm{Volume}\mathrm{of}\mathrm{rain}\mathrm{water}\mathrm{collected}\mathrm{in}\mathrm{the}\mathrm{cylindrical}\mathrm{vessel} \\ \Rightarrow lbh=\frac{4}{5}\times \mathrm{Volume}\mathrm{of}\mathrm{cylindrical}\mathrm{vessel} \\ \Rightarrow 22\times 20\times h=\frac{4}{5}\times \mathrm{\pi }{R}^{2}H \\ \Rightarrow 440h=\frac{4}{5}\times \frac{22}{7}\times 1\times 1\times 3.5 \\ \Rightarrow h=\frac{4}{5}\times \frac{22}{7}\times \frac{3.5}{440} \\ \Rightarrow h=0.02\mathrm{m} \\ \therefore h=2\mathrm{cm} \end{gathered}$$

So, the height of the rainfall is 2 cm.