Question

Your swimming pool containing 75,000 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 250 gal/min. Assume that the dye is uniformly distributed throughout the pool.

(a) What is the initial value problem for the filtering process; let q(t) be the amount of dye in the pool at any time t. What is the initial amount of the dye in the pool?

(b) Solve for q(t) from part (a).

(c) You have invited several dozen friends to a pool party that is scheduled to begin in 5 hours. You have determined that the effect of the dye is imperceptible if its concentration is less than 0.017 g/gal. Is your filtering system capable of reducing the dye concentration to this level within 5 h?

(d) Find the time T at which the concentration of the dye first reaches the value of 0.017 g/gal.

(e) Find the flow rate r that is sufficient to achieve the concentration 0.017 g/gal within 4 h.

Answer 1