Your swimming pool containing 60,000 gal of water has been contaminated by 7 kg
ID: 1879061 • Letter: Y
Question
Your swimming pool containing 60,000 gal of water has been contaminated by 7 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 () be the amount of dye in the pool at any time t. What is the initial amount of the dye in the pool? Enter the exact, fully simplified answer. Use q asa(t) d q Qg/min (0)-Number grams (b) Solve for q (t) from part (a) Do not use thousands separator in the answer field. a(t) a e gramsExplanation / Answer
(a) The rate at which q(t) is changing is given as
dq/dt = q(t) / 60000 * 250
dq/dt = -1/240 Kg/min
dq/dt = -4.166 g/min ( negative sign indicates dye being removed)
q(0) = 7 kg
q(0) = 7000 g
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(b) q(t) = 7000e-t/240
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(c) No, it won't be able to filter as
t = 5*60 = 300
q(t) = 7000e-300/240
Answer is No
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(d) concentration is given as 0.02 g/gal
mass = 0.02*60000 = 1200 g
Therefore
1200 = 7000e-t/240
1200/7000 = e-t/240
12/70 = e-t/240
Taking ln on both sides
ln(12/70) = -t/240
t = 423.26 minutes
t = 7.05433333 hours
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