A 500–liter tank initially contains 20 kg of dissolved salt thoroughly mixed in the solution. Beginning at time t=0 minutes, a brine whose salt concentration (in kg L−1) varies with time according to cin(t)=0.10+0.02sin(0.1t) flows into the tank at a rate of 4 L min−1. The well-stirred mixture leaves the tank at the same rate, so the total volume in the tank remains constant.
Assume perfect mixing at every instant and ignore any change in temperature or density.
(1) Part 1: (a) Let Q(t) denote the mass of salt (in kg) in the tank at time t. Write a differential equation satisfied by Q(t). Clearly state the expressions you use for the inflow and outflow rates of salt.
(2) Part 2: (b) Solve the differential equation from part (a) subject to the initial condition Q(0)=20. Express your answer in exact form and include appropriate units.
(3) Part 3: (c) Calculate the concentration of salt in the tank, in kg L−1, after 30 minutes. Round your answer to three decimal places.
(4) Part 4: (d) Determine t→∞limQ(t) and interpret its meaning in the context of the mixing process.