Problem 2: (50 points total) Consider the three-tank system shown in the picture below. You may assume that the volume of material V_i = 1, 2, 3 in each tank is constant, that the flow rate F is constant, and that the concentrations C_i = 1, 2, 3 and the inlet concentration C are zero initially.
1) (20 points) Write a mathematical model describing the evolution of concentrations in the tanks.
2) (20 points total) Find an expression for C(t) in the case where the inlet concentration increases to C at t = 0. Consider two specific cases (10 points each):
a) V_1, V_2, and V_3 are the same.
b) Note that V_1, V_2, and V_3 are not the same.
Use Laplace transforms to solve this problem.