Question

Water is being pumped into a tank at a rate P(t), where P is measured in gallons per minute. There is a hole near the bottom of the tank causing the water to leak out at a rate of L(t), where L is also measured in gallons per minute. If the tank initially contained 20 gallons of water, which expression would find the volume of water after 10 minutes?

Answer 1

Let V(t) be the volume of water at time t, with V(0) = 20.

For t >= 0, and assuming that the tank does not empty within the
interval t in [0, 10], then we have:

V(t) = 20 + int_{x=0}^{x=t}[P(x) - L(x)]dx

In general, if it is possible for the tank to empty (which depends on the
size of the P(x) and L(x) functions), we would have:

V(t) = 20 + int_{x=0}^{x=t}[(P(x) - L(x))*F(x) + max[P(x) - L(x),0]E(x)] dx

where:

F(x) = 1 if V(x)>0
F(x) = 0 if V(x) =0.

E(x) = 1-F(x).

Thus, F(x)=1 if and only if the tank is non-empty at time x (and 0 else),
and E(x)=1 if and only if the tank is empty at time x (and 0 else).
Intuitively, the integral equation above says that V(x) increases
with rate P(x) - L(x) at a time x such that V(x)>0, and increases
at a rate max[P(x)-L(x),0] if the tank is empty at time x.