401
Chapter 6: Differential Equations
____
11.
Match the logistic equation and initial condition with the graph of the solution.
a. d.
b. e.
c.
6.4 The Logistic Equation
402
____ 12. A conservation organization releases 40 coyotes into a preserve. After 4 years, there
are 70 coyotes in the preserve. The preserve has a carrying capacity of 175. Write a logistic
function that models the population of coyotes in the preserve.
a.
b.
c.
d.
e.
____ 13. A conservation organization releases 50 foxes into a preserve. After 5 years, there are 85
foxes in the preserve. The preserve has a carrying capacity of 225. Determine the population after 10
years. Discard any fractional part of your answer.
126
118
139
131
205
____ 14. A conservation organization releases 30 panthers into a preserve. After 3 years, there are 50
panthers in the preserve. The preserve has a carrying capacity of 150. Determine the time it takes for
the population to reach 110.
13.139 years
8.994 years
10.378 years
7.811 years
12.003 years
403 Chapter 6: Differential Equations
6.4 The Logistic Equation
Answer Section
6.5 First-Order Linear Differential Equations
404
6.5 First-Order Linear Differential Equations
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Solve the first order linear differential equation.
a.
b.
c.
d.
e.
____ 2. Solve the first-order linear differential equation .
a.
b.
c.
d.
e.
____ 3. Solve the first order linear differential equation.
a.
b.
c.
d.
e.
405 Chapter 6: Differential Equations
____
4.
Find the particular solution of the differential equation
that
satisfies the initial condition
.
a.
b.
c.
d.
e.
____
5.
Find the particular solution of the differential equation
that
satisfies the boundary condition
.
a.
b.
c.
d.
e.
____
6.
Find the particular solution of the differential equation
that
satisfies the boundary condition
.
a.
b.
c.
d.
e.
6.5 First-Order Linear Differential Equations
406
____
7.
Find the particular solution of the differential equation
that
satisfies the boundary condition
.
a.
b.
c.
d.
e.
____
8.
Solve the Bernoulli differential equation
.
a.
b.
c.
d.
e.
____
9.
Find the particular solution of the differential equation
passing
through the point
.
a.
b.
c.
d.
e.
407
Chapter 6: Differential Equations
____
10.
A 300-gallon tank is full of a solution containing 35 pounds of concentrate. Starting
at time
distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred
solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a
function of t.
a.
b.
c.
d.
e.
____ 11. A 200-gallon tank is full of a solution containing 25 pounds of concentrate. Starting
at time , distilled water is added to the tank at a rate of 10 gallons per minute, and the
well-stirred solution is withdrawn at the same rate. Find the time at which the amount of
concentrate in the tank reaches 15 pounds. Round your answer to one decimal place.
18.3 min
3.6 min
20.4 min
10.2 min
5.1 min
____ 12. A 100-gallon tank is full of a solution containing 25 pounds of concentrate. Starting
at time , distilled water is added to the tank at a rate of 10 gallons per minute, and the
well-stirred solution is withdrawn at the same rate. Find the quantity of the concentrate in the
solution as .
a.
10
b.
26
c.
25
d.
0
e.
1
____ 13.
A 300-gallon tank is half full of distilled water. At time
, a solution containing
0.5 pound of concentrate per gallon enters the tank at the rate of 8 gallons per minute, and the
well-stirred mixture is withdrawn at the rate of 6 gallons per minute. At what time will the tank
be full?
150 minutes
76 minutes
75 minutes
301 minutes
600 minutes
6.5 First-Order Linear Differential Equations
408
____ 14.
A 200-gallon tank is half full of distilled water. At time
, a solution containing
0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minute, and the well-
stirred mixture is withdrawn at the rate of 3 gallons per minute. At the time the tank is full, how
many pounds of concentrate will it contain? Round your answer to two decimal places.
11.61 lbs
64.64 lbs
71.34 lbs
49.39 lbs
82.32 lbs
____ 15. Suppose an eight-pound object is dropped from a height of 5000 feet, where the air
resistance is proportional to the velocity. Write the velocity as a function of time if its velocity after
7 seconds is approximately –75 feet per second. Use a graphing utility or a computer algebra system.
Round numerical answers in your answer to four places.
a.
b.
c. 0.8018
d.
e.
____ 16. Assume an object weighing 7 pounds is dropped from a height of 9,000 feet, where the air
resistance is proportional to the velocity. Round numerical answers in your answer to two places.
Write the velocity as a function of time if the object’s velocity after 6 seconds is 3.50 feet
per second.
What is the limiting value of the velocity function?
a.
(i)
; (ii) 0
b.
(i)
; (ii) 0
c.
(i)
; (ii) 3.5000
d.
(i)
; (ii) 3.5000
e.
(i)
; (ii) limit does not exist
409 Chapter 6: Differential Equations
____ 17.
Use
as a integrating factor to find the general solution of the
differential equation
.
a.
b.
c.
d.
e.
6.5 First-Order Linear Differential Equations
410
6.5 First-Order Linear Differential Equations
Answer Section
411 Chapter 6: Differential Equations
6.6 Predator-Prey Differential Equations
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. We modeled populations of aphids and ladybugs with a Lotka-Volterra
system. Suppose we modify those equations as follows:
,
a.
b.
c.
d.
e.
____ 2. A phase trajectory is shown for populations of rabbits and foxes .
Describe how each population changes as time goes by.
Select the correct statement.
At the population of foxes reaches a minimum of about 30.
At the number of rabbits rebounds to 500.
At the number of foxes reaches a maximum of about 2400.
6.6 Predator-Prey Differential Equations
412
6.6 Predator-Prey Differential Equations
Answer Section