AP Calculus AB · Integration and accumulation of change
A well-stirred tank initially contains 80 liters of pure water
Problem
A well-stirred tank initially contains \(80\) liters of pure water. Brine containing \(2\) grams of salt per liter enters at \(4\) liters per minute, and the mixture leaves at the same rate. Let \(A(t)\) be the number of grams of salt in the tank after \(t\) minutes. (a) Derive the differential equation and initial condition for \(A\). Explain why the volume of liquid in the tank remains \(80\) liters. (b) Solve for \(A(t)\) and identify \(\lim_{t\to\infty}A(t)\). (c) Find the first time at which the tank contains \(80\) grams of salt, and explain why that time is unique. Attach units to the answer.
Hint
Rate in minus rate out; equal flow rates keep the volume at \(80\) L.
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