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Student 1.D.
Math 2250-1
Exam #1
September 27, 2012
Please show all work for full credit. This exam is closed book and
closed note. You may use a
scientific calculator, but not a “graphing calculator” i.e. not one which is
capable of integration, taking
derivatives, or matrix algebra. In order to receive full or partial
credit on any problem, you must
show all of your work and justify your conclusions. There are 100
points possible. The point values
for each problem are indicated in the right-hand margin. Good
Luck!
___
____
_____
Score
POSSIBLE
25
2
25
3
15
4
15
20
5
TOTAL
1). Consider a boat which starts t rest at time I = 0
,
100
is accelerated in a straight path by an engine that
provides a
consta€of Force, and w ich is also subject to drag 1brc0Nforeah
velocity. The boat has mass 400 kg.
--
of
j.g). Use your modeling ability to show that the boat velocity v(l) (in
meters per second) satisfies the initial
value problem
v’(t)2.5—.Iv
vyV
I
4ov v’
--
t1-(A
= (co C
(5 points)
—
1-0 V
j Solve the initial value problem in JiL
/
Vt.
e .ltiv / ÷.iv)
zc
t
—
-
(15 points)
-
(€v)c
vJ
2.ce4+
2&
id
v
i-C’
25
how lar does the boat travel in the first 20 seconds’?
2L
(5 points)
.i-L
2-o
—it
-2ce
Jo
v5t+-2’Oe
(
4- iSO
-
501 -2S0
(‘, 2’4J
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2.0
Consider the following input-output model: A brine tank initially contains 1000 gallons of very salt)
water - the initial concentration is I lb of salt for each gallon of water. At time 1 = 0 less salty water begins
to flow into the tank, at a rate of 10 gallons per minute. This water contains 0.2 pounds of salt per gallon.
Also at time 1= 0 water begins to flow out of the tank, at 20 gallons per minute. (There is space below for
you to sketch the configuration, as it may help you with the rest of the problem.)
(0
.2-
).
20
c.
jE)
When does the tank become empty?
(5 points)
r-y::: IQ—o---tL’
Cfr’LL
V- VUD
-
OVD
t Lct (OP
Find a formula lbr the volume V(1) of water in the tank at time 1 minutes, until the time of emptying.
(5 points)
v
(r )
‘:
Use your modeling ability tojusti\ why the initial value problem lbr the pounds x(f) of salt in the
tank at time 1 minutes turns out to be
x(1)2
x’(i) +
x(0)
=
1000
(until the tank empties).
(5 points)
rjL —Vo
-2-O 4(1
Vii)
,LL
ik’CL
-l
+i—’s
‘“T’Q
(\1.1UI
i
I
‘I
V I
a
1
Solve the IVP in 2c(until the tank empties). For your convenience, the IVP is repeated here:
x’(t) +
x(0)
x(/)2
1000
(10 points)
2
-
c- (,‘#
tp—t
Z(Lap-t’
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(tov-t’
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C. Wv-i)
-
1p2.oa
4. C.10,001?
c:
+
•oc
j Consider the differential equation for x(t):
x’(l)
Find the equilibrium solutions.
4 —4x
x
2
(ê)
,
( _j)()3 points)
Construct the phase diagram for this differential equation, and use
it to determine the stabiliiy of the
equilibrium solutions.
F
2
(x-Lxi -‘
(7 points)
t(ty_s
Let x(1) be the solution to the initial value problem for this difiCrential
equation. with x(O)
Without solving for x(1) deduce the value of limx(i). Lxplai
n.
(5 points)
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