Differential Equations and Matrix Algebra II (MA 222), Winter Quarter, 2000—2001
Exam 2 (100 points)
Name
Box
This is the last question on exam 2. Hand in this page (along with the Maple code used
in part (c)) at the start of class on Friday. You may use the text, Maple, notes, maple
worksheets, and any other written material you have. But you may not discuss any part
of this question with any other person (classmates or others). This means that you cannot
compare answers, or give any hint or direction about how you solved the problem. If there
is any hint of collaboration, I will automatically give 0 points to all involved.
10 pts 8. Consider the following tank conÞguration where x(t), y(t), and z(t) are the amounts of salt
in tanks 1, 2, and 3 respectively. Furthermore assume that the tanks are full at time t = 0
with the totals in tank 1 being 100, tank 2 being 200, and tank 3 being 300. Also at time 0,
x(0) = 40, y(0) = 80, and z(0) = 30. Notice that 1/2 lb of salt per gallon is entering tank 1.
The ßow rates between the tanks are given in the picture.
2 pts a) Please give the system of differential equations which describe x(t), y(t), and z(t) (that is,
x (t) = . . . , y (t) = . . . , and z (t) = . . .). Put your answer in both equation form and matrix
form. Do your scratch work elsewhere and put only your Þnal answer here (you need not
include the units).
2 pts b) Find lim x(t) =
t→∞
lim y(t) =
t→∞
and lim z(t) =
t→∞
Although Maple is not needed in Þnding these limits, you may use your work in part c if you
wish. In either case, explain how you found your answers.
6 pts c) Find x(30) =
, y(30) =
and z(30) =
Gve your answers to 1 decimal place. Staple to this page a copy of the Maple code that you
used to get your answers.