Two very large tanks A and B are each partially filled with 100 gallons of brine. Initially, 85 pounds of salt are dissolved in the solution in tank A and 65 pounds of salt are dissolved in the solution in tank B. The system is closed in that the well-stirred liquid is pumped only between the tanks, as shown in the figure below.
Two cylindrical tanks are shown with two pipes connecting them in a system. The first tank is labeled A 100 gal. The second tank is labeled B 100 gal. The first pipe connects tank A and B on the bottom of the tanks and has an arrow pointing from tank A to tank B labeled mixture 2 gal/min. The second pipe connects tank A and B on the top of the tanks and has an arrow pointing from tank B to tank A labeled mixture 3 gal/min.
(a) Use the information given in the figure to construct a mathematical model for the number of pounds of salt x1(t) and x2(t) at time t in tanks A and B, respectively.
dx1/dt = dx2/dt = x1(0) = lbs x2(0) = lbs
(b) Find a relationship between the variables x1(t) and x2(t) that holds at time t. (Write an equation using x1 for x1(t) and x2 for x2(t).) Explain why this relationship makes intuitive sense.
Since the system is closed, salt enters and leaves the system at the same rate. Since the system is closed, no salt enters or leaves the system. Since the system is closed, salt is not exchanged between the two tanks. Since the system is closed, salt enters the system at a constant rate. Since the system is closed, salt leaves the system at a constant rate.
Use this relationship to help find the amount of salt in tank B at t = 25 min. (Round your answer to one decimal place.)
x2(25) = lbs