We can approximate the continuous-time tank model of theprevious problem by a discrete model as follows.
Assume that we only observe the tank contents each minute (timeis now discrete). During each minute, 20 liters (or 10% of eachtank’s contents) are transferred to the other tank.
Let x1(t) and x2(t) be the amounts of salt in each tank at timet. We then have:
x1(t + 1) = 9 /10 x1(t) + 1 /10 x2(t)
x2(t + 1) = 1 /10 x1(t) + 9 /10 x2(t)
Formulate the problem in the form x(t + 1) = Ax(t) where A is a2 Ă— 2 matrix, then solve for the amount of salt in each tank as afunction of time using the eigenvalues and eigenvectors of A.
Sketch the graphs of the amount of salt in each tank asfunctions of time.
How does your solution compare to the continuous time model?
