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Unformatted text preview: Exam 1  February 13, 2004  Solution 1 1. Suppose that a fish tank containing 100 liters of water is originally free of f ormalin. Beginning at time t = 0 contaminated water containing 10% of formalin is introduced to the tank at a rate of 10 liters/sec, and the wellmixed water leaves the tank at the same rate. (a) [18 pts] Find an expression for the concentration x ( t ) of formalin in the tank at time t ≥ 0. [Hint: Begin with a differential equation model for Q ( t ), the volume of formalin in the tank at time t , rather than x ( t ).] Let Q ( t ) be the volume, measure in liters, of formalin in the tank at time t . Then Q =rate in rate out = 10 liter sec × 1 10 10 liter sec × Q ( t ) liter 100 liter , since the total volume is always 100 liters = 1 1 10 Q ¶ liter sec , and Q (0) = 0 liter, since the tank is originally free of formalin . Hence we need to solve Q + 1 10 Q = 1 , Q (0) = 0 . The integrating factor is e t/ 10 . So we have [ e t/ 10 Q ] = e t/ 10 Q + 1 10 e t/ 10 Q = e t/ 10 . By integrating, we get e t/ 10 Q = 10 e t/ 10 + C ⇒ Q = 10 + Ce t/ 10 . Using the initial condition Q (0) = 0, we have C = 10, i.e., Q ( t ) = 10(1 e t/ 10 ) liter . From the definition of the concentration, we finally get x ( t ) = Q ( t ) liter 100 liter = 1 10 (1 e t/ 10 ) . MATH2400  Introduction to Differential Equations  Spring 2004  JeongRock Yoon Exam 1  February 13, 2004  Solution 2 (b) [4 pts] What is the limit value of the concentration x ( t ) as t → ∞ . lim t →∞ x ( t ) = lim t →∞ 1 10 (1 e t/ 10 ) = 1 10 = 10% , since lim t →∞ e t/ 10 = 0 . (c) [8 pts] All the fish will be killed if they are exposed to a formalin concentration higher than 5%. Find the time T at which this concentration is reached. [Hint: ln2 ≈ . 69. Do not forget the unit in your final answer.] We need to find T that satisfies . 05 = 5% = x ( T ) = 1 10 (1 e T/ 10 ) ....
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This homework help was uploaded on 02/24/2008 for the course MATH 2400 taught by Professor Yoon during the Spring '04 term at Rensselaer Polytechnic Institute.
 Spring '04
 Yoon
 Differential Equations, Equations

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