Extracting an initial value: filtering water in a pool

Oneiromancy

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Your swimming pool, containing 60,000 gal of water, has been contaminated by 5 kg of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of 200 gal/min.

Write down the initial value problem for the filtering process; let q(t) be the amount of dye (in grams) in the pool at any time t (in minutes).

The answer the book gave me was q' = -(1/300)t. That doesn't seem right because it doesn't say anything about the dye. Where does the 5000 g of dye come into play?
 
Re: Extracting an initial value problem.

the 5 kg is an initial condition ... the rate of change of dye in the pool is a function of the variable concentration of dye in the pool at any time t,

\(\displaystyle \frac{dq}{dt} = -\frac{q \, kg}{60000 \, gal} \cdot 200 \frac{gal}{min} = -\frac{q}{300} \, \frac{kg}{min}\)

\(\displaystyle \frac{dq}{q} = -\frac{dt}{300}\)

\(\displaystyle \ln{q} = -\frac{t}{300} + C\)

\(\displaystyle q = Ae^{-\frac{t}{300}}\)

at t = 0 , q = 5 kg

\(\displaystyle q = 5e^{-\frac{t}{300}}\)
 
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