3. [6marks] At time / = 0, tanks X and Y are each filled with 10 L of brine at a salt concentration of 5 g/L. [Thus at / = 0, each tank has 50 g of salt.] After every minute (att = 1, 2, 3...) 4 L of the well- mixed mixture are taken out of each tank and replaced as follows. In tank X we put 3 L of pure water and 1 L of the tank Y mixture, and in tank Y we put all 4 L of the tank X mixture. The remaining 3 L from tank Y is thrown down the drain. Let X, and ye be the number of grams of salt in the two tanks at any time a (immediately after the ath flushing-thus you could take xo = ) = 50). A node diagram for the system is drawn at the right. We might also draw a 3-node diagram with a node D for the drain, but we 3 Lim 4Lm. don't really need that for our calculations. With the 2-node 10L I L/m TOL diagram, the sum of the arrows leaving Y is less than one. 3Lim X (a) Let a = the average length of time a molecule starting in tank X spends in tank X # = the average length of time a molecule starting in tank X spends in tank Y We should be a bit more precise. Here's an example. A molecule that starts in X at time 0, goes to Y at # 1, stays in Y at /=2, returns to X at =3, goes to Y at f=4, hits the sink at =5, spends 2 minutes in X and 3 minutes in Y. Use the backward recursion method of Example 512 to find a recursive system for a and # and solve it to find their values. (b) Argue that In and In change according to a linear recursive system and find the matrix. (c) Solve the system of (b) (i.e. find expressions for Xe and y's) (d) Use this solution of (c) to obtain a second calculation of a, the average length of time a molecule starting in tank X spends in tank X. 5