Chapter 9
89.
Trout are introduced into a stream. Trout is a predator species and therefore has an
influence on the population size of other fish. The first figure shows how the trout and
other fish populations vary over time. The progress of time is shown by the direction
of the arrow. Does the second figure accurately show how the two different
populations vary over time?
Chapter 9
Page 36
90.
Let f be the number of fruit tree blossoms (in ten thousands) and let b be the number of
bees (in hundreds) in an orchard. Suppose f and b satisfy the differential equations
df f fb
dt
= − +
and
db b fb
dt
= − +
,
which correspond to the slope field in the figure. Assume f = 3 and b = 3 when t = 0.
What happens to the number of fruit tree blossoms and bees over time?
A)
The solution approaches the origin.
B)
The solution approaches the line f = b.
C)
f→
, but b does not.
D)
, but f does not.
91.
Let f be the number of fruit tree blossoms (in ten thousands) and let b be the number of
bees (in hundreds) in an orchard. Suppose f and b satisfy the differential equations
df f fb
dt
= − +
and
db b fb
dt
= − +
.
Assume f = 3 and b = 1 when t = 0. Use the differential equations to calculate
/df dt
and
/db dt
when t = 0, and use these to estimate:
A. the number of fruit tree blossoms when t = 1.
B. the number of bees when t = 1.
Part A:
30,000
Part B:
300
of species interactions. difficulty: medium section: 9.6
various types of species interactions. difficulty: medium section: 9.6
Chapter 9
92.
Bees and flowers help each other, and each needs the other in order to survive. Which
(if any) of the following systems of differential equations could model the interaction
between bees and flowers, with either species being x or y? Select all that apply.
A)
0.5 0.04
dx x xy
dt
= − +
,
0.3 0.02
dy y xy
dt
=−
B)
0.14
dx x
dt
=
,
0.6 0.12
dy y xy
dt
= − +
C)
0.4 0.17
dx x xy
dt
= − +
,
0.2 0.11
dy y xy
dt
= − +
D)
None of these
Ans: C Learning Objectives: Interpret or write differential equations that model
various types of species interactions. difficulty: medium section: 9.6
93.
Owls need trees to survive, but trees don’t care one way or the other about owls. Which
(if any) of the following systems of differential equations could model the interaction
between owls and trees, with trees as x and owls as y? Select all that apply.
A)
0.3 0.03
dx x xy
dt
= − +
,
0.6 0.07
dy y xy
dt
=−
B)
0.3
dx x
dt
=
,
0.4 0.17
dy y xy
dt
= − +
C)
0.2 0.33
dx x xy
dt
= − +
,
0.5 0.27
dy y xy
dt
= − +
D)
None of these
Ans: B Learning Objectives: Interpret or write differential equations that model
various types of species interactions. difficulty: hard section: 9.6
94.
Elk and buffalo are in competition with each other. Each would do fine without the
other. Which (if any) of the following systems of differential equations could model
the interaction between elk and buffalo, with either species being x or y? Select all that
apply.
A)
0.4 0.06
dx x xy
dt
= − +
,
0.5 0.03
dy y xy
dt
=−
B)
0.22
dx x
dt
=
,
0.5 0.13
dy y xy
dt
= − +
C)
0.3 0.25
dx x xy
dt
= − +
,
0.4 0.19
dy y xy
dt
= − +
D)
None of these
Ans: D Learning Objectives: Interpret or write differential equations that model
various types of species interactions. difficulty: medium section: 9.6
Chapter 9
95.
The fox eats the hare, so it needs it to survive. The hare would do fine without the fox.
Which (if any) of the following systems of differential equations model the interaction
between the fox and the hare, with the fox as x and the hare as y? Select all that apply.
A)
0.2 0.02
dx x xy
dt
= − +
,
0.3 0.04
dy y xy
dt
=−
B)
0.23
dx x
dt
=
,
0.3 0.14
dy y xy
dt
= − +
C)
0.1 0.26
dx x xy
dt
= − +
,
0.2 0.2
dy y xy
dt
= − +
D)
None of these
various types of species interactions. difficulty: medium section: 9.6
96.
Consider three strains of the flu modeled by the following sets of differential equations.
Which has the infecteds being removed the slowest?
I.
0.01
dS SI
dt
=−
0.01 0.3
dI SI I
dt
=−
II.
0.02
dS SI
dt
=−
0.02 0.5
dI SI I
dt
=−
III.
0.03
dS SI
dt
=−
0.03 0.4
dI SI I
dt
=−
Ans:
I
of species interactions. difficulty: medium section: 9.6
Chapter 9
97.
Two minor league baseball teams in the same city compete with each other for fan
attendance. Both teams would do well in the absence of the other one, but each team
hurts the other team’s attendance at games. Create a system of differential equations to
model this situation.
A)
/
/
dx dt x xy
dy dt y xy
=−
=−
B)
/
/
dx dt x xy
dy dt y xy
= − +
= − +
C)
/
/
dx dt x xy
dy dt y xy
=−
= − +
D)
/
/
dx dt x
dy dt y xy
=
=−
98.
At time t = 0, there are 300 students at a school, 3 of whom have the flu. Given the
differential equation
0.0026 0.5
dI SI I
dt
=−
, will the flu spread?
99.
At time t = 0, there are 400 students at a school, 3 of whom have the flu, and 200 of the
students have been vaccinated against the flu. Given the differential equation
0.0026 0.5
dI SI I
dt
=−
, will the flu spread?
Chapter 9
100.
For a new strain of the flu, the differential equations are:
0.0038
dS SI
dt
=−
and
0.0038 0.9
dI SI I
dt
=−
.
What is
dI
dS
?
A)
236.8 1
S
−
B)
0.0042 1
S
−
C)
236.8 1
S
+
D)
0.0042 1
S
+
101.
For a new strain of the flu, the differential equations are:
0.0038
dS SI
dt
=−
and
0.0038 0.9
dI SI I
dt
=−
.
What is the threshold value for this strain of the flu? Round down to the nearest whole
number.
102.
Consider three strains of the flu modeled by the following sets of differential equations.
Which is the least infectious?
I.
0.01
dS SI
dt
=−
0.01 0.3
dI SI I
dt
=−
II.
0.02
dS SI
dt
=−
0.02 0.5
dI SI I
dt
=−
III.
0.03
dS SI
dt
=−
0.03 0.4
dI SI I
dt
=−
Chapter 9
103.
What is the threshold value for the strain of the flu modeled by the differential equations
0.02
dS SI
dt
=−
0.02 0.5
dI SI I
dt
=−
?
Round to the nearest whole number.
104.
For the strain of the flu modeled by the differential equations
0.02
dS SI
dt
=−
0.02 0.5
dI SI I
dt
=−
,
does the disease spread if initially
028S=
?
105.
In a boarding school of 1000 students, at least _____ students should be vaccinated
against a flu strain satisfying the differential equations
0.0025
dS SI
dt
=−
0.0025 0.35
dI SI I
dt
=−
.
106.
For S and I satisfying the differential equations
dS aSI
dt
=−
dI aSI bI
dt
=−
,
is I increasing or decreasing when
<a
Sb
?
Chapter 9
107.
The following figure gives the slope field for
/dI dS
for an SIR epidemic model.
Estimate the maximum number of people infected at any one time if the number of
susceptibles is initially 400.
A)
About 150
B)
About 120
C)
About 200
D)
About 40
disease. difficulty: easy section: 9.7
108.
A fatal infectious disease is introduced into a growing population. Let S denote the
number of susceptible people at time t and let I denote the number of infected people at
time t. Suppose that, in the absence of the disease, the susceptible population grows at a
rate proportional to itself, with constant of proportionality 0.2. People in the infected
group die at a rate proportional to the infected population with constant of
proportionality 0.05. The rate at which people get infected is proportional to the product
of the number of susceptibles and the number of infecteds, with constant of
proportionality 0.001. Which of the following systems of differential equations are
satisfied by S and I?
A)
0.2 0.05 , 0.2 0.05
dS dI
S SI SI I
dt dt
= − = −
B)
0.001 0.2 , 0.05 0.001
dS dI
S SI SI I
dt dt
= − = −
C)
0.2 0.001 , 0.001 0.05
dS dI
S SI SI I
dt dt
= − = −
D)
0.05 0.001 , 0.001 0.2
dS dI
S SI SI I
dt dt
= − = −
disease. difficulty: easy section: 9.7
Chapter 9
109.
Pollutants are being dumped into a lake at a rate of 9 m3 per day. About 15% of the
lake’s water leaves the lake each day and fresh, unpolluted water flows in to replace it.
The differential equation for the amount of pollutant, Q, in the lake as a function of
time, t, in days is
dQ
dt
=
__________.
Ans:
growth and decay. difficulty: medium section: 9 review
110.
5
yx=
is a solution to the differential equation
5
‘y
yx
=
A)
True
B)
False
difficulty: easy section: 9 review
111.
Find the value of k for which
2
y x kx=+
is a solution to the differential equation
‘ 2 2xy y x−=
.
Ans:
difficulty: easy section: 9 review
112.
Given that
/ 0.4dy dt y=−
and
(0) 175y=
, estimate
(3)y
by first estimating y(1) and
y(2). Assume that the rate of growth given by
/dy dt
is approximately constant over
each unit time interval.
Ans:
37.8
solution. difficulty: easy section: 9 review
Chapter 9
113.
Find the solution to the differential equation
2
dP P
dt
=−
, subject to the initial condition
(0) 8P=
.
A)
1
16
8t
Pe
−
=
B)
1
2
16 t
Pe
−
=
C)
1
2
8t
Pe
−
=
D)
1
8
2t
Pe
−
=
114.
The slope fields for
‘y x y=−
and
‘yy=
are shown in the following figure. Which
slope field goes with the differential equation
‘y
= x
–y
?
Chapter 9
115.
The following slope field has
A)
no equilibrium solution.
B)
a stable equilibrium solution.
C)
an unstable equilibrium solution.
Ans: A Learning Objectives: Verify that a function given by a formula is a
solution of a differential equation. difficulty: easy section: 9 review
116.
The general solution for the differential equation
0.5 2
dy y
dx
=−
is
A)
0.5
2x
y Ce=+
B)
0.5
4x
y Ce=+
C)
2
0.5 x
y Ce=+
D)
4
0.5 x
y Ce=+
Ans: B Learning Objectives: Solve the differential equation dy/dt = ky
difficulty: medium section: 9 review
117.
Money in a bank account earns interest at a continuous rate of 6% per year, and
payments are made continuously out of the account at the rate of $10,000 per year. The
account initially contains $100,000. Write a differential equation for the balance, B, in
the account in t years and use it to find how many years it will take for the account to
run out of money. Round to 1 decimal place.
Ans:
15.3
Chapter 9
118.
The body of a murder victim is found at 9:00 in the morning in a 70 F room. The
temperature of the body when it is found is 87 F, and one hour later it is 80 F. If the
victim had a normal temperature of 98.6 F when he died, how many hours had the
victim been dead when the body was found? Round your answer to one decimal place.
Ans:
1.0
differential equation. difficulty: hard section: 9 review
119.
At time t = 0, there are 700 students in a school, 5 of whom have the flu. No one else
has been exposed yet. Using the SIR model,
0
I
= _____ and
0
S
= _____.
Part A:
5
Part B:
695
difficulty: medium section: 9 review
120.
At time t = 0, there are 500 students in a school, 5 of whom have the flu. No one else
has been exposed yet. Using the SIR model and the differential equation
0.0014 0.6
dI SI I
dt
=−
, will the flu spread?
Ans:
121.
Mark all of the differential equations that are NOT separable.
A)
34
dy ty
dt
=+
B)
3
49
dy xy
dx
=+
C)
9xy
dy e
dx
=
D)
4xy
dy e
dx
+
=
section: focus on theory
Chapter 9
122.
Find the general solution of the separable differential equation:
9
dy yt
dt
=
123.
Solve the initial value problem using separation of variables, and then graph the
solution.
, (0) 8
dy x y
dx y
−
==
124.
Solve the differential equation using separation of variables.
2 5 10
dy xy y x
dx
= + + +
A)
2 2 2
25
10
2 2 2
x y y x
y x C= + + + +
B)
22
5y xy x C= − +
C)
5, 0xy==
D)
2
225
xx
y Ce +
=−