A large mixing tank can hold up to 200 liters of liquid
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A large mixing tank can hold up to 200 liters of liquid. Initially, the tank contains 40 liters of pure water (no salt). Starting at t=0 minutes, brine containing 0.50 kg of salt per liter flows into the tank at a rate of 3 L min−1. Simultaneously, the well-stirred mixture is pumped out at a rate of 2 L min−1. The density of the solution is assumed to remain constant, and thorough mixing occurs instantly.
(a) Let Q(t) be the mass of salt (in kg) in the tank at time t minutes. Write expressions for both the volume V(t) of liquid in the tank and the rate of change dtdQ. Do not solve the equation in this part.
(b) Solve the differential equation obtained in part (a) for Q(t), using the initial condition Q(0)=0. Your final expression should be valid for times before the tank reaches capacity.
(c) Determine the concentration of salt, in kg L−1, when the volume in the tank first reaches 150 liters.
(d) At what time (to the nearest tenth of a minute) will the concentration of salt in the tank reach 0.25 kg L−1? Make sure your answer falls within the interval during which the model is valid.