solutions2bd - tank and 3 gallons leave per minute the...

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Math 33b, Quiz 2bd, January 24, 2008 Name: UCLA ID: 1. A tank initially containing 100 gallons of pure water is being fed by a pipe which pours in a solution at the rate of 2 gallons per minute. The solution contains 0.5 pounds of salt per gallon. At the same time, the tank is being drained at the rate of 3 gallons per minute. Assume that the salt in the tank is being continuously mixed. Let x ( t ) be the amount of salt in the tank, where x is measured in pounds and t is measured in minutes, and t = 0 corresponds to the initial state of the tank. Find x ( t ). Solution. We can write dx dt = (rate of salt being poured in) - (rate of salt being drained out) The rate at which salt being poured in is (0.5 pound salt/1 gallon water)(2 gallons water/minute) = (1 pound salt/minute). The rate at which salt is leaving the tank is (concentration of salt in tank)(rate of water leaving tank). The concentration of salt in the tank is x 100 - t , since x is the amount of salt in the tank and (100 - t ) is the amount of water in the tank. (Since 2 gallons of water enter the
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Unformatted text preview: tank and 3 gallons leave per minute, the number of gallons of water in the tank is (100-t ).) The rate at which water is leaving the tank is 3 gallons per minute, so the rate of salt being drained out is simply 3 x 100-t . We now have dx dt = 1-3 x 100-t dx dt + 3 x 100-t = 1 This is a linear differential equation, which can be solved by using the integrating factor e R 3 100-t dt = e-3log | 100-t | = (100-t )-3 : (100-t )-3 dx dt + 3 x (100-t )-4 = (100-t )-3 d dt ( x (100-t )-3 ) = (100-t )-3 x (100-t )-3 = 1 2 (100-t )-2 + C x = 1 2 (100-t ) + C (100-t ) 3 Since the tank contains no salt at the beginning, we have the initial condition x (0) = 0. Therefore we have 0 = 50 + C (100) 3 , so C =-50 100 3 =-1 20000 and 1 x ( t ) = 1 2 (100-t )-1 20000 (100-t ) 3 . (Note: at t = 100 minutes, the tank is empty, so our differential equation no longer applies. It is not necessary to note this for the solution though.) 2...
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