A large mixing vat initially contains 200 L of water in which 10 kg of salt are dissolved. Beginning at time t=0 min, a brine solution containing 0.50 kg of salt per liter flows into the vat at a constant rate of 3 L/min. The well-stirred mixture leaves the vat at the same rate so that the volume in the vat remains constant.
Let y(t) be the amount of salt, in kilograms, present in the vat t minutes after the inflow begins.
(a) Write a differential equation that models y(t) and specify the initial condition.
(b) Solve the differential equation obtained in part (a) to obtain an explicit expression for y(t).
(c) Determine the time, to the nearest tenth of a minute, at which the concentration of salt in the vat first reaches 0.30 kg/L.
(d) Evaluate t→∞lim200y(t) and interpret the result in the context of the problem, including appropriate units.