2. (12 marks) At time t = 0, tanks X and Y are each lled with 10 L of brine at a salt concentration of 5 g/L. [Thus at t = 0, each tank has 50 g ofsalt.] After every minute (at t = 1,2, 3...)4 L of the well- 0'6 0_4 mixed mixture are taken out of each tank and replaced as follows. I: In tank X we put 3 L of pure water and 1 L of the tank Y mixture, T and in tank Y we put all 4 L ofthe tank X mixture. The remaining 3 L from tank Y is thrown into the sink. Let x,, and y... be the number of grams of salt in the two tanks at any time it (immediately after the nth ushingthus you could take x0 =yn = 50]. A node diagram for the system is drawn at the right We might also draw a 3-node diagram with a node S for the sink, but we don't really need that for our calculations. With the Z-node diagram, the sum of the arrows leaving '1' is less than one. (a) Let bi = average number of moves until a salt molecule starting in tankX hits the sink ty = average number of moves until a salt molecule starting in tank Y hits the sink. Use a recursive approach to calculate these. Note: the nal journey to the sink counts as a move. [b] Let a = the average length of time a molecule starting in tank X spends in tank X E = the average length of time a molecule starting in tank X spends in tank Y I should be a bit more precise. Here's an example. A molecule that starts in X at time 0, goes to Y at t=1, stays in Y at i=2, returns to X at t=3, goes to Y at t=4, hits the sink at t=5, spends 2 minutes in X and 3 minutes in Y. Find a recursive system for aand l9 and solve it to nd their values. (c) There is a very close relationship between the answers to [a] and (b). Explain and use this to get a check on one of the answers to [a]. Math 111 assignment 7 winter 2022 4 (d) Find a matrix such that X, and y, change according to the linear recursive system 0.6 0.4 0.6 X Y (e) Solve the system of (d) and use this solution to obtain a second calculation of a, the average length of time a molecule starting in tank X spends in tank X. Math 111 assignment 7 winter 2022 5