As in problems 1 and 2, you are the manager of a swimming pool which contains 10
ID: 2857457 • Letter: A
Question
As in problems 1 and 2, you are the manager of a swimming pool which contains 10^6 liters water in which are mixed 10^6 mg chlorine (Cl). You drain the pool at a rate of 2000 liters/min, and simultaneously pump in "fresh" water which contains only O.1 mg Cl/liter. If water were to be pumped in at the same rate of 2000 liters/minute, this would be the same mixing problem solved earlier. However, the fresh water is pumped in at a rate of only 1900 liters/min. so the water level is slowly going down. a. Find a formula for the volume of water in the pool after t minutes b. Set up a differential equation which models the amount of chlorine in the pool after t minutes. c. Solve this differential equation. d. After how mam minutes should you shut the drain valve (so that you can finish refilling the tank with the hose water and have the right amount of chlorine in the tank when it fills)?Explanation / Answer
a) Volume of water in pool at time t = V(t) = V0 + (1900 - 2000)t = 106 - 100t
b) c'(t) = rate of change of chlorine
= rate of chlorine added - rate of chlorine pumped out
= Concentration/lt * Pumping rate - (c(t)/V(t)) * 2000
= .1*1900 - c(t)/500 = 190 - 20c(t)/(104- t)
The DE is c'(t) = 190 - 20c(t)/(10000- t) = 190 + 20c(t)/ (t - 10000)
c) The DE is of the from c'(t) + p(t)c(t) = q(t) which is first order linear eqn.
Solution is obtained by integrating factor I = ep(t)dt
Solving the DE yields, c(t) = k1(t -10,000)20 - 10t + 100,000. The amount of Chlorine at t= 0 is 106 or c(0) = 10^6
=> 10^6 = k1 * 10^80 + 10^5 or k1 = 900000/10^80 = 9*10-75
c(t) = 9*10-75 (t - 10,000)20 - 10t + 100,000
d) Criteria is c(t)/v(t) = 1/2. Solve for t.
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