Part C: Triple tanks You decide to have a more elaborate setup and connect three small fish tanks in a loop. You run pipes from tank C to tank D, from tank D to tank E5 and nally from tank E back to tank C. Pumps are installed so that water is constantly cycling through all three pipes at the rate of 1 litre per minute. Suppose initally tank 6' is lled with 40 litres of salt water with a concentration 80 grams per litre. tank D with 20 litres of fresh water. and tank E 40 litres of fresh water. 1. Sketch a diagram of the tank setup with arrows for ows entering and leaving each tank. [1 mark] 2. Let X (t), Y(t), Z (t) denote the amount of salt in tanks C, D and E respectively. Write down a system of ODE's for X, Y and Z. [2 marks] 3. Rewrite your system of ODE's in the form of X'(t) X (t) Y'(t) = A Y(t) Z ' (t) Z (t) for some matrix A. [1 mark] 2 1 i 1 + 1' 4. Given that 1 , 1 + 12 . and 1 i are eigenvectors of the matrix A. determine the corresponding 2 2 2 eigenvalues. [2 marks] X (t) 5. Hence write down the general solution of Y9) . From the general solution, deduce whether the Z 0) salt concentrations in the tanks settle to a nal steady state as time approaches innity, or oscillate endlessly without settling. Justify your answer. [3 marks]