The tidy toilet bowl has been used with a blue disinfectant in the tank. The sou

Question

The tidy toilet bowl has been used with a blue disinfectant in the tank. The source of the blue antibacterial mixture is removed from the tank, but the five gallons inside the tank still have the blue dye dissolved in the concentration of c0 = 4.18 ppm. It is not known how much water one flush removes from the tank but after each flush the blue water is replaced by clear tap water, i.e. water with no dye in it. Measurement of the concentration of dye in the tank after one flush gave c1=3.344 ppm. Assuming perfect mixing between flushes, what is the concentration after the second flush? c2= What is the concentration of the dye after the tenth flush? c10= What is the volume of water removed from the tank by one flush? What is the concentration of the dye in the bowl after very many flushes?

Explanation / Answer

We can set up a recursive sequence to solve this problem. Each time we flush, we replace 1/4 of the current concentration with a concentration of .7. So if c(n) is the concentration after n flushes, we can write:

c(n) = (3/4)*c(n-1) + (1/4)*(.7) = .75*c(n-1) + .175

where c(0) = 3.8. Now, since we're both multiplying and adding, the sequence is neither arithmetic nor geometric. So we'll write a few iterations to get an idea what's going on:

c(0) = 3.8

c(1) = .75*(3.8) + .175

c(2) = .75*(.75*(3.8) + .175) + .175

= .75

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