Consider a closed two-tank system in which connecting pipes allow for circulating liquid flow between the tanks at a con

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Consider A Closed Two Tank System In Which Connecting Pipes Allow For Circulating Liquid Flow Between The Tanks At A Con 1 (324.59 KiB) Viewed 24 times

Consider a closed two-tank system in which connecting pipes allow for circulating liquid flow between the tanks at a constant rate of 3 gal/min. Tank A initially contains 200 gal of water in which 41 lbs of salt are dissolved. Tank B initially contains 200 gal of water in which 19 lbs of salt are dissolved. The mixture is kept uniform by stirring. The ultimate goal will be to find the salt amounts x(t) and y(t) in tanks A and B, respectively, as functions of time t, in min. This is similar to the two-tank mixing problem in the video: "Two Tanks / Mixing Problem / Linear System of DES # 1 " - Zero Eigenvalue (10:17) by math et al (description in ASULearn on April 1). (a) Set up a DE system for the salt amounts in the two tanks. (b) Convert the system to matrix-vector form. (c) Find the eigenvalues of the associated 2 by 2 matrix A. (d) Find eigenvectors for each eigenvalue. (e) Build the general solution of the DE system in vector form. (f) Use the initial data to find the constants C1 and. C2. Find formulas for x(t) and y(t). What are their long-term trends? (h) Graph x(t) and y(t), and label any horizontal asymptotes. (i) Add the DEs, followed by integration with respect to time. Then graph the corresponding solution path in the xy-phase plane, starting from the point (41, 19) and going forward in time to infinity. Restrict to Quadrant I. (j) The path corresponds to conservation of (rather than energy). (k) Find all equilibrium points of the matrix-vector system and graph all EQ in Quadrant I of the phase plane. (1) For the given initial salt amounts, which EQ point is the attractor? (m) Graph the phase portrait for this system - including the path for the given initial data point as well as paths for two other starting points. Stay in Quadrant I.