Question

$500$ persons are taking a dip into a cuboidal pond which is $80m$ long and $50m$ broad. Find the rise of water level in the pond in millimetres, if the average displacement of water by a person is $0.04{m^3}$.

Answer 1

Hint: Using the average displacement and number of persons we can find the total displacement of water. Then this can be equated with the rise in volume of water in the pond. Using the equation of volume of a cuboid we can calculate the rise in water level.

Formula used: Volume of a cuboid, $V = lbh$, where $l,b,h$ are the length, breadth and height respectively.
Average of some value is calculated by dividing the total by its number.

Complete step-by-step solution:
Given that,
Length of the pond, $l = 80m$
Breadth of a pond, $b = 50m$
Number of persons taken dip, $n = 500$
Average displacement of water by a person is $0.04{m^3}$.
Average of some value is calculated by dividing the total by its number.
Therefore, total displacement of water by $500$ persons is equal to $500 \times 0.04 = 20{m^3}$.
We can see that the displacement of water is equal to the rise in volume of water in the pond.
And this volume will be equal to $80 \times 50 \times h$ where $h$ is the change in height occurred by rise in water level.
This is because the water level increased is equally distributed to a cuboidal area and volume of a cuboid, $V = lbh$
So we have,
$\Rightarrow$$80 \times 50 \times h = 20$
This gives
$\Rightarrow$$h = \dfrac{{20}}{{80 \times 50}}$
$\Rightarrow$$h = \dfrac{1}{{200}}$
Thus the rise in water level is $\dfrac{1}{{200}}m$.
We know, $1m = 1000mm$
This gives the rise in water level equal to $\dfrac{1}{{200}} \times 1000 = 5mm$.

$\therefore $ The answer is $5mm$.

Note: When an object falls into a water body, the level of the water rises. Also this water is equally distributed over the surface whatever the shape of the body. So we can equate it with rise in volume. This is the important point used here.