Salt water containing 0.4 kg of salt per liter flows into a well-mixed tank at a constant rate of 5 liters per minute. The tank initially contains 200 liters of pure water. Brine is removed from the tank at the same rate that it enters so that the volume of liquid in the tank remains constant.
Let y(t) denote the mass of salt (in kilograms) in the tank t minutes after the flow begins.
(a) Derive a differential equation for y(t) that features a single parameter k multiplying a function of y. Express any auxiliary quantities (for example, incoming concentration) numerically, but do not evaluate k.
(b) Identify k explicitly in terms of the given flow rates and volume. State its sign and interpret the physical meaning of that sign in this context.
(c) Without solving the differential equation, determine whether y(t) is increasing, decreasing, or remaining constant at the instant just after the process begins. Justify your answer using the differential equation from part (a).