A cylindrical tank initially contains 200 liters of brine in which 4 kilograms of salt are dissolved. Starting at time t=0 minutes, brine containing 0.5 kilograms of salt per liter flows into the tank at a constant rate of 3 L/min, and the well-mixed solution drains from the tank at the same rate.
Let S(t) be the amount of salt, in kilograms, in the tank at time t minutes. Assume the volume of liquid in the tank remains constant at 200 liters.
(1) Derive a first-order differential equation that models the rate of change of S(t). Clearly indicate the inflow term and the outflow term in your derivation.
(2) Analyze the differential equation obtained in part (a) and determine whether the variables can be isolated on opposite sides of the equation. Justify your conclusion with explicit algebraic reasoning.
(3) For the modeling approach you selected in part (b), solve for S(t) subject to the initial condition S(0)=4. Present your solution in explicit form if possible, or implicit form if necessary.
(4) The differential equation in part (a) is also linear in S. Discuss the relative advantages and disadvantages of isolating variables versus treating the equation as linear when solving for S(t) in this context.