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Water level in containers - Differential equations. See attached file for full problem description.

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SOLUTION

Problem 13 10'
10'

10'
dh
h
(h-dh)

Circular hole (radius 2")

As the water flows out of the hole, the level of water progressively falls in the tank. Let the height of the water column at any instant t be h and let after a time interval dt the height be (h - dh) [that is the water level has fallen by dh in time dt]. Then,

Rate at which the water is flowing out of the tank = Total volume of water flown out/Total time taken = Awdh/dt where, Aw = Area of water surface ........(1)
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Rate at which water is flowing out of the hole = cAhâˆš2gh where, Ah = Area of c/s of the hole, c is an empirical constant (0 < c < 1) .......(2)
___
Equating (1) and (2) we get : - Awdh/dt = cAhâˆš2gh (negative sign on left hand side signifies reduction in the volume)
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dh/dt + c(Ah/Aw)âˆš2gh = 0 .......(3)

(3) is the required ...

Solution Summary

Detailed step-by-step solutions provided for the given differential equations.

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