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Question: The pressure in the water pipe at the ground level of a building is 120000Pa, whereas, the pressure ...

The pressure in the water pipe at the ground level of a building is 120000Pa, whereas, the pressure on a third floor is 30000Pa30000Pa. What is the height of the third floor?
(Takeg=10ms2g = 10\,m{s^{ - 2}}, density of water=1000kgm3 = 1000\,kg{m^{ - 3}}).

Explanation

Solution

Firstly we will calculate the height of the ground floor and third floor from the tank using the suitable formula and then, we will calculate the actual height of the building. Just remember, the level of floors from the ground floor, assuming the level of the ground floor to be 1.

Formula used:
We will use the formula of variation of pressure with depth, which is given by
P=hρgP = h\rho g
Where PP is the pressure, hh is the height, ρ\rho is the density, and gg is the gravity.

Complete step by step solution:
Let us first write the given terms in the question
P1=120000Pa{P_1} = 120000\,Pa(Pressure in water pipe on the ground floor)
P2=30000Pa{P_2} = 30000\,Pa(Pressure in water pipe on the third floor)
ρ=1000kgm3\rho = 1000\,kg{m^{ - 3}}(Density of water)
g=10ms2g = 10\,m{s^{ - 2}}(Gravity)
Now, for calculating the height of the tank, we will use the formula which is given by
P1=hρg{P_1} = h\rho g
120000Pa=h×1000×10120000\,Pa = h \times 1000 \times 10
h=1200001000×10\therefore \,\,h = \dfrac{{120000}}{{1000 \times 10}}
h=12m\Rightarrow \,h = 12m
Therefore, the height of the ground floor from the tank is 12m12\,m.
Therefore, the building is 12m12\,m tall.
Now, we will calculate the height of the third floor from the tank as
P2=hρg{P_2} = h\rho g
30000=h×1000×1030000 = h \times 1000 \times 10
h=300001000×10\therefore \,\,h = \dfrac{{30000}}{{1000 \times 10}}
h=3m\Rightarrow \,h = 3m
Therefore, the height of the tank from the third floor is 3m3m.
Now, assume that the ground level is 11, so, the level of the third floor will be 44.
Now, the height of the third floor from the ground floor can be calculated as
heightofthirdfloor=heightofthebuildinglevelofgroundfloor×levelofthirdfloorheight\,of\,third\,floor = \dfrac{{height\,of\,the\,building}}{{level\,of\,ground\,floor}} \times \,level\,of\,third\,floor
h=124×3h = \dfrac{{12}}{4} \times 3
h=3×3\Rightarrow \,h = 3 \times 3
h=9m\therefore \,h = 9m

Therefore, the height of the third floor is9m9m.

Additional Information:
You might have observed that while on a plane flight your ears may pop or ears may ache during a deep dive in a swimming pool, this is because of the effect on depth of pressure of a fluid. Now, on the Earth’s surface, the pressure exerted on you will be the result of the pressure of the air above you. This pressure of air will be reduced as you climb up in an altitude because the weight of air in the high altitude reduces.

Also, in the case of water, when we will go underwater, the pressure of water increases with an increase in depth. Also, the pressure on you will be the result of both the pressure of water and the pressure of the atmosphere above you.

Note: As we all know that liquids are incompressible, therefore, the equation used above i.e. P=hρgP = h\rho g can be used to great depth.
But, in the case of gases, which are compressible, we can apply this equation only when the density change is small over the depth considered.