True story: For safety, I had put chlorine powder in my spa water and my guests were coming soon. But, when I used my water testing kit, I realized I had over-chlorinated. I had thrown in about n times too much powder, and so I needed to dilute the water by adding fresh water, and fast. Would I better off first draining out some of the over-chlorinated water and then filling the spa back up with fresh water and mixing, or would it be faster emptying and filling at the same time? The garden hose fills the spa at the same rate that the drain faucet at the bottom empties the spa. Assuming that the mixing of fresh water and chlorinated water is instantaneous and that the drain rate does not change with water height, which way is faster and how does the answer depend on n?
In the "simultaneous" fill/drain method, as t--> 0, the dilution rate (dn/dt) must approach -1/T (time to fill), since the mixing at t=0 is negligible. This provides the necessary constraint to solve the differential equation correctly.