Water leaks from a uniform cylindrical tank 50 m high at the rate of 10*h m^3/s. How much water leaks out while water is emptied using a tap in 50 s and the height of the water level falls from 50 to 0 m.
The rate at which the water leaks from the tank is related to its height by 10*h m^3/s. The initial height of the water level in the tank is 50 m. It is emptied through a tap in 50 s. The water that leaks out in this duration has to be determined. It is assumed that the water is emptied at a uniform rate and the leakage is included in it. As the tank is emptied in 50 s, the drop in the water level is 1 m/s
In an infinitesimal duration of time dt the water that leaks out is 10*h*dt m^3.
The tank is emptied in 50 s. This gives the volume of water that leaks out as the integral `int_(0)^50 10*(50 - t) dt`
`int_(0)^50 10*(50 - t) dt`
=> `500t - 5t^2` from t = 0 to t = 50
=> 500*50 - 5*50^2 - 500*0 + 5*0^2
=> 25000 - 12500
=> 12500 m^3
The water that leaks out while the tank is being emptied is 12500 m^3