Question

The rain water from a roof of $22{\text{m}}$×$20{\text{m}}$ drains into a cylindrical vessel having a diameter of base $2{\text{m}}$ and height$3.5{\text{m}}$. If the vessel is just full, find the rainfall in cm.

Answer 1

Hint: Here, the quantity of rainfall will be the height of the water body filled in the roof. The roof is like a rectangle with length and breadth of $22{\text{m}}$×$20{\text{m}}$.The volume of water collected on the roof will be equal to the volume of a cylindrical vessel of the base $2{\text{m}}$ and height $3.5{\text{m}}$.

Complete step-by-step answer:
Given, length and breadth of roof= $22{\text{m}}$×$20{\text{m}}$
The base of cylindrical vessel=$2{\text{m}}$ =diameter
Height of cylindrical vessel (H) =$3.5{\text{m}}$. We have to find the quantity of rainfall that drains from the roof to the cylindrical vessel, given the vessel is just full. Here, we need to think of the roof as a rectangle filled with rainwater up to h height (suppose). This height will be the quantity of rainfall that we have to find.

The diameter of cylindrical vessel=$2{\text{m}}$. So the radius(r) =$\dfrac{{{\text{diameter}}}}{2} = \dfrac{2}{2} = 1{\text{m}}$
We know that the volume of a cylindrical vessel=$\pi {{\text{r}}^2}{\text{H}}$. On putting the given values we get,
Volume of cylindrical vessel=$\pi \times {1^2} \times 3.5 = \dfrac{{22}}{7} \times \dfrac{{35}}{{10}} = 11$ m³
Now according to the question, the volume of water collected over the roof=Volume of rectangle=${\text{l}} \times {\text{b}} \times {\text{h}}$
On putting given values we get,
Volume of water collected over roof=$22 \times 20 \times {\text{h = 440h}}$ m³
Since the cylindrical vessel is just full of the water drained from the roof, so the volume of water collected over the roof=volume of the cylindrical vessel. Now on putting values, we get,
$
   \Rightarrow 440{\text{h = 11}} \\
   \Rightarrow {\text{h = }}\dfrac{{11}}{{440}} = \dfrac{1}{{40}}{\text{m}} \\
 $
Since $1{\text{m = 100cm}}$
$ \Rightarrow \dfrac{1}{{40}} \times 100 = 2.5{\text{cm}}$
Hence the rainfall in cm is$2.5{\text{cm}}$.
Note: You can also solve this question by first finding the area of roof$\left( {22{\text{m}} \times {\text{20m}}} \right)$ and then finding the base area of vessel$\left( {\pi {{\text{r}}^2}} \right)$ , where r=$1{\text{m}}$. Then according to the question put,
$\dfrac{{{\text{area of roof}}}}{{{\text{base area of vessel}}}} = \dfrac{{{\text{height of water in vessel}}}}{{{\text{height of water in roof}}}}$
Put the given values and you’ll get the answer.