sol3 - MAP 2302 Fall-2011 SOLUTIONS: Quiz 3 1. A swimming...

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MAP 2302 Fall-2011 SOLUTIONS: Quiz 3 1. A swimming pool whose volume is 10,000 gal contains water that is 0.01% chlorine. Starting at t = 0 city water containing 0.001% chlorine is pumped into the pool at a rate of 5 gal/min. The pool water flows out at the same rate. What is the percentage of chlorine in the pool after 1 h? When will the pool water be 0.002% chlorine? SOLUTION. Let x ( t ) be the amount of chlorine in the pool at time t . Then the concentration of chlorine is c ( t ) = x ( t )10 - 4 . The input rate is 5 × 0 . 001 100 = 5 × 10 - 5 . The output rate is 5 × c ( t ) = 5 × 10 - 4 x ( t ). The initial condition is x (0) = c (0)10 4 = 0 . 01 100 10 4 = 1. The problem is to find c (60) in percents and to find t such that x ( t ) 10 4 = 0 . 002 / 100. Remember, 1 h = 60 minutes. The initial value problem is dx dt = 5 × 10 - 5 - 5 × 10 - 4 x x (0) = 1 . The equation is separable. It can be rewritten as dx x - 10 - 1 = - 5 × 10 - 4 dt. The integration of both sides gives us ln
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sol3 - MAP 2302 Fall-2011 SOLUTIONS: Quiz 3 1. A swimming...

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