(upgrade soon)
Adding rate , it takes T to full the bathtub
Releasing rate, according to the bernouli equation, releasing rate is a function of water height.
Since the bathtub are adding water and releasing water in the same time, the volume change in the bathtub is the adding rate subtract the releasing rate.
Where tau(h) is the time for an empty bathtub to accumulate water to the height of h
In addition, we can use the condition of releasing water from the bathtub contain t.
We can alter the form of eq1 into
Plugin the relation of 2 in to eq1 we can, therefore, derive the relation of
If we change the variable of h into the fulling percentage of the tub, delta=h/H
We have
we derive the relationship between the time tau and the fulling percentage under the condition of full and release the tub at the same time using the variable of t,T
(T is the time to filling the tub without release, t is the time to releasing a full tub without adding)