Page 110 - Matrix Analysis & Applied Linear Algebra
P. 110
104 Chapter 3 Matrix Algebra
3.5.11. Consider three tanks each containing V gallons of brine. The tanks are
connected as shown in Figure 3.5.2, and all spigots are opened at once.
As fresh water at the rate of r gal/sec is pumped into the top of the
first tank, r gal/sec leaves from the bottom and flows into the next
tank, and so on down the line—there are r gal/sec entering at the top
and leaving through the bottom of each tank.
r gal / sec
r gal / sec
r gal / sec
r gal / sec
Figure 3.5.2
Let x i (t) denote the number of pounds of salt in tank i at time t, and
let
x 1 (t) dx 1 /dt
dx
and = dx 2 /dt .
dt
x = x 2 (t)
x 3 (t) dx 3 /dt
Assuming that complete mixing occurs in each tank on a continuous
basis, show that
−1 0 0
dx r
= Ax, where A = 1 −1 0 .
dt V
0 1 −1
Hint: Use the fact that
lbs lbs
dx i
= rate of change = coming in − going out.
dt sec sec
