Solution manual for college algebra 12th edition by gustafson

Let B œthe liters of skimmed milk added.!Þ!$& Liters of butterfat Liters of Liters of butterfat at start butterfat added at end 2.25 liters of skimmed milk should be added.. Let B œthe l

Exercises 0.1 (page 13)

4.

3.

2.

25. Every natural number is a whole number, 26. Every rational number is a real number,

51. oñqñqqñqñp"" "$ "( "* 52. oñqñqñqñqñqñqñp# " ! " # $ %

$( )

oq ïï qp

Ð Ñ Ñ Ð

74 and , ,

, ,

oq ïï qp

Ò Ñ Ñ Ò

, ,

oq ïï qp

Ò Ó

Ò Ó

, ,

oq ïï qp

Ð Ó

97. distanceœ )  $ œ & œ &k k k k 98. distanceœ "#  & œ "( œ "(k
k k k

99. distanceœ $  ) œ & œ &k
k k k 100 distanceœ #!  ' œ #' œ #'k k k k

natural numbers

Since theSince population must be positive and

never has a fractional part, the set of

103 Since temperatures are usually reported

without fractional parts and may be eitherpositive or negative (or zero), the set of

integers

104.

should be used

Since the financial condition of a business

is usually described in terms of dollars andcents (fractional parts of a dollar), the set of

109. The statement is always true 110 The statement is always true

111. The statement is not always true 112 The statement could be

interpreted to mean thatFor example, let and

-113. False There are 5 integers: , , , , 114. False is not a rational number because

115. False ∞ is not a number at all 116 True.

117. True (You cannot find an element in the 118 True (You cannot find an element in the

1st set that is1st set that is not in the 2nd set.) not in the 2nd set.)

119. False There are eight subsets 120 True.

+++ œ œ + œ + œ+ < œ œ < œ <

cd

m/min m/mi

œ "Þ*)' ‚ "!%

mm mm

mi

8 œ $ œ * #(& #!! $ #  %

,,,

, mi

8 œ % œ * #(& #!! $ #  %

8135equatorial radius

129. B B œ B 130. B œ B

B

$ 7$



BB

136. B &&œ "
&&† B&&œ B&&

is not a number between 1 and 10

"Î$ "Î$

"Î$ $ "Î$

21.
'%"Î#Ênot a real number 22.
"#&"Î$œ
& $ "Î$‘ œ &

23.
"'+#"Î#œ
%+ # "Î#‘ œ % +k k 24.
#&+%"Î#œ’
&+# #“"Î#œ & + œ &+k k# #

55
B C"! &$Î&œ B$!Î& "&Î&C œ B C' $

56
'%+ ,' "#&Î'œ '%&Î' $!Î' '!Î'+ , œ '%ˆ "Î'‰&+ ,& "!œ # + ,& & "!œ $#+ ,& "!

"#& œ & œ &$ $

69. È$ "#& œÉ$
& $œ & 70. È& #%$ œÉ&
$ &œ $

,

œœ

# BB

C

) CC

ÈÈ

102 ( ( '- ( '- (

'-

$'-$'- œÈ †È'- œÈ#"'- œ È

†œ

"! œ "! † & œ "! & œ# & $ œ $ † C œ$ C

C

ÈÈÈ

ÈÈ

115 Ê Ê Ê ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ

ÈÈ

È

ÈÈ

È

ÈÈ

#

#

#B

BB

$#

#

BB

BB

BB

))

CC

CC

% $# &!! % # $# # &!! #

œ #C #C #C) '% " !!!

129. To rationalize a denominator means to write 130.

an equivalent fraction with a denominatorequal to a rational number

131 Œ BC œ BC œBC †BB CC œ CB œ

C œŒBC  œÊCB

B

"Î8 7Î8

7Î8 7Î8

7Î8

"Î8 7

7

8

132 Consider the case when is even, is odd and is negative Then

Thus, must be a real number fo

9. coefficients, variables 10. $ÈB  # 11. yes, trinomial, 2nd degree

12. yes, binomial, 3rd degree 13. no 14. no

15. yes, binomial, 3rd degree 16. yes, monomial, 5th degree 17. yes, monomial, 0th degree

18. no 19. yes, monomial, no degree 20. yes, none, 3rd degree

21
B  $B  &B  )B œ B  $B  &B  )B œ B  &B  $B  )B œ 'B  $B  )B$ #
$  $ # $ $ $ # $ #

ŒŒ

72 $+ #+  $+ œ $+ #+  $+ $+ œ '+  *+ œ '+  *+

œ ' *+

84 " " &  # &  # &  #

&  # œ &  #† &  # œ œ œ œ &  #

+  ,+  +,  +,  ,

105. &B C  "&B C &B C "&B C 106. *7 8  '7 8 *7 8 '7 8

117. B  #  118. B  # 

B  $ B  !B  #B  $B  !B  *

B  # B  !B  #B  $B  !B  *

 $BB

 #BB

123 Areaœlength width† œ B  & B  #

 ft#œ B  #B  &B  "!
#  ft#œ B  $B  "!
#  ft#

heightheightheightheight

129 Answers may vary.

130 Multiply the numerator and denominator by the conjugate of the numerator ˆÈB  #‰

131 Check the formula with + œ " and , œ # 132 Check the formula with + œ $ and , œ %

133. False Some polynomials are trinomials 134. True

135 True
"#B  &C œ "#B  &C "#B  &C œ "%%B  '!BC  '!BC  #&C



71. 'B  #*BC  $&C œ $B  (C #B  &C# #

 72. "!B  "(BC  'C œ &B  'C #B  C# #



73 "#:  &):;  (!; œ # ':  #*:;  $&;# #
# #œ # ':  $&; :  ;



74 $B  'BC  *C œ $ B  #BC  $C# #
# #œ $ B  $C B  C



75 '7  %(78  $&8 œ  '7  %(78  $&8# #
# #œ  '7  &8 7  (8



76 "%<  ""<=  "&= œ  "%<  ""<=  "&=# #
# #œ  (<  &= #<  $=



77 'B  #$B  $&B œ B 'B  #$B  $& œ B 'B  ( B  &$ #
# 



115-118 Answers may vary.

119. $B  # œ #ˆ$B#‰œ #ˆ$B  "‰ 120. &B  $ œ &ˆ&B$‰œ & B ˆ $‰

‰ˆ

Œ

‰ˆ

135. True 136. False is the sum of

two squares and is thus prime

25. %  B #  B #  B B  # 26. #&  B &  B &  B B  &

B  "!B  #& B  & B  & B  &

B  "')

41 $B  &B  # 'B  "$B  & $B  &B  # #B  &B

B  $

#"B  &!B  "' † ƒ œ † †

œ B  $ B  " †$B  )† (

B  $(B  # $B  )

51. # " # " " 52. $ # $ # &

B  '

B  ''  B

+  "'+  (+  "# œ 

œ "  +

+  % +  %+  %

œ " +  %  +

+  % +  %+  % +  %

+  % +  % +  % +  %

œ #+  %+

58 , % , % , , % ,  #

, ,  # ,  #, ,  # ,  #

, ,  #,  # ,  #

,  #,

, ,  # ,  #, ,  # ,  #

œ ,  %,  ), ,  # ,  #

œ B  C ƒC  B œB  C † +, œ "

C  B+,

+,+,

B B

+ $B  %+,

$+ B  %+ ,

+B  ,+B  ,

'

B ' B

' '

B B

90 #B  % & #B  % "!B  #! # &B  "! &B  "!

#  œ & #  œ "!  %B œ # &  #B œ &  #B

# B

# B

B" B#

B B"

111. False The denominator can never equal 0 112. True

113. False for all values of 114 False for all values of

C

B CC

B

"

B

B œ '

120 The domain is the set of all real numbers.

1. natural: $ ' ), , 2. whole: ! $ ' ), , ,

3. integers: ' $ ! $ ' ), , , , , 4. rational: ' $ !, , , , , ,"# $ ' )

5. irrational: ,1 È& 6. real: ' $ !, , , , , , "# $ 1 È& ' ), ,

9. even integers: ' ! ' ), , , 10. odd integers: $ $,

11. Associative Property of Addition 12. Commutative Property of Addition

13. Associative Property of Multiplication 14. Distributive Property

15. Commutative Property of Multiplication 16. Commutative Property of Addition

17 Double Negative Rule

34. 8 8 œ 8# % #%œ 8' 35.
:$ #œ :$†#œ :'

 œœ

#)

)

& &

œ # #

83. " " % % % 84. # # & # & # &

# œ # † % œ ) œ # È#& œ È#&†È& œ È"#& œ &

89. 3rd degree, binomial 90. 2nd degree, trinomial

91. 2nd degree, monomial 92. 4th degree, trinomial

‰ˆ

$B C 'B C

"

"

B C

B C

1. odd integers: ( " $, , 2. prime numbers: $

3. Commutative Property of Addition 4. Distributive Property

B C

conditional equation

B

#B

: œ (

# (B  & œ #

#

Œ

#(B  "! œ #B  "&

''

47 48.

77

'  B  " œ $

B  ( œ $

$B

(8  $œ

"

$'

œ #A,

#

#+

11

11

111

1

11

.55

5.5

œ B

00

=3

=3

00

=3

=3

00

=3

=33

+

0 :; † " œ 0 :; ""

;:0

:; œ 0 ;  0 ::; œ 0 ;  ::; 0 ;  :

;  :

;  : œ:;

;  : œ 0

BC

+,

B+

+, B C+

‹Š

Š ,‹œ +, † ",B œ +,  +C,B œ + ,  C,B + ,  C

,  C,  C œ,B,  C œ +

11

4. break point 5. rate, time !Þ!& $! œ "Þ&


7 Let the score on the final exam Sincethe final is weighted as two test grades, itcounts as two test grades

#B  $*' œ &'!

#B œ "'%

B œ )#

B œ

His score on the final exam needs to be 82

Let the score on the final

9 Let the score on the first exam.

Then the score on the midterm,and the score on the final

His score on the first exam was 84

Let the score on the first exam Then herscores on

13. Let the number of locks replaced 14. Let th

7 locks can be changed for $236

Then the height

18 Total Fence Length Square Fence Length

The total fencing required is

B œ "#

he walkway has a width of 12 yards

21. Total Area Triangular Area 22. Sum of

The dimensions are 8 feet by 20 feet

New

Area Area

$7250 was invested at 4% and $8750 was invested at 6%.

26 Let the amount invested at 7% Then the amount invested at 6%.Interest at 7% Interest at 6% Total interest

$10,000 was invested at 7% and $12,000 was invested at 6%

27 Let the amount invested at 7%

Interest at 7% Interest at 9% Total interest

She needs to invest $45,714.29 at 7% to reach her goal

28 Let the amount invested at 8% Then , the amount invested at 9 %.Interest at 9 % Interest at 8%

29 Let the amount invested at each rate.

Interest at 6% Interest at 7% Interest at 8% Total interest

$9,700 was invested at each rate, for a total investment of $ ,100

30 Let the amount invested at each rate

Interest at 5% Interest at 7% Interest at 9% Interest at 11% Total interest

$4500 was invested at each rate, for a total investment of $18,000

31 Let B œthe number of full-price tickets sold Then the number of student tickets sold

There were 327 student tickets sold

32 Let B œthe cost of a student ticket

$$'

Cost of # of Cost of # of full-price full-price student student

the original price

Discount

The original price was $79.95

New Original

price price

35. Let the wholesale cost 36. Let

price cost

37. Let # of plates for equal costs 38.

The break point is 200 plates

B œœ'!!  $B œ )!!  #B

B œ #!!

Cost of 2nd Cost of 1st

machine machine

Let # of fasteners for equal costs

machine machine

int is about 85,714 fasteners

39. Let # of computers to break even 40

Income Expenses

21 computers need to be sold to break even

B œœ

"#(&B œ )*#&  )&!B

Man in Roofer in Total in

1 day 1 day 1 day

They can roof the house in 2 days

Let hours for both working together

6 11

Total in Crew 2

43 Let hours for both working together.

Woman Man in Total in

in 1 hour 1 hour 1 hour

ey can mow the lawn in 1 hours

Let days for both hoses to fill the pool

The pool can be filled in 1 days.15

45 Let B œhours for pool to fill with drain open

Pipe in Drain in Total in

1 hour 1 hour 1 hour

The pool can be filled in 21 hours

Let hours for sister to stuff 1000 shrimp

"

*

46.

Siste Sam in

Oz of alc Oz of alc Oz of alc.

at start added at end

10 oz of water should be added

Let the ml of water removed

#Þ%  !Þ'B œ $

!Þ'B œ !Þ'

B œ " Ê

Liters of a.f at end

1 liter should be replaced with pure antifreeze

50 Let B œthe liters of skimmed milk added.

!Þ!$&

Liters of butterfat Liters of Liters of butterfat

at start butterfat added at end

2.25 liters of skimmed milk should be added

51 Let B œthe liters of pure alcohol added

15of a liter of pure alcohol should be added.

52 Let B œthe cubic centimeters of water added

œ

 Cubic centimeters of Cubic ce

Cubic centimeters of

chemical added chemical at start

ntimeters of chemical at end

%!!  ! œ !Þ#& "!!!  B

%!! œ #&!  !Þ#&B

"&! œ !Þ#&B'!! œ B Ê

600 cubic centimeters of water should be added

53 Let B œthe gallons of pure chlorine added

About 4.5 gallons of pure chlorine should be added

54 Let B œthe percentage of substitute fuel used Then the percentage of gasoline used

† Cost per

Percentage of gasoline used

Cost per gallon Cost per gallon

Percentage of gallon

of mixture

of fuel fuel used

55 Let B œthe liters of water evaporated.

B œ % Ê 4 liters of water should be evaporated

56 Let B œthe ml of water boiled away

œ

ml of salt ml of salt ml of salt

at start removed at end

!Þ!' 53 ml of water should be boiled away.

57 Let B œthe pounds of extra-lean hamburger used

50 pounds of the extra-lean hamburger should be used

58 Let B œthe gallons of cream used Then the gallons of milk used

2 gallons of cream should be used

59 Let B œthe gallons of 5% solution used

60 Let B œthe ounces of % cream used Then " the ounces of 5% cream used.

0.75 ounces of the 1% cream should be used with 0.25 ounces of the 5% cream

61 Since the mixture is to be 25% barley, there will be pounds of barley used.Thus, the other 1800 pounds will b

!Þ#& #%!! œ '!!


e either oats or soybean meal

Let B œthe number of pounds of oats used Then the number of pounds of meal used

Pounds of protein Pounds of protein Pounds of protein Total poun ds

of protein from soybean meal

The farmer should use 600 pounds of barley, 1,637 pounds of oats and 163 pounds of soybean meal

62 Since the mixture is to be 20% barley, there will be pounds of barley used.Thus, the other 0 pounds will b

!Þ#! #%!! œ %)!

"*#



e either oats or soybean meal

Let B œthe number of pounds of oats used Then the number of pounds of meal used

Pounds of protein Pounds of protein Pounds of protein Total poun ds

of protein from soybean meal

The farmer should use 480 pounds of barley, 1,757 pounds of oats and 163 pounds of soybean meal

63 Let his first rate Then his return rate

Distance to city Return distance

64 Let the time Allison and Austin travel.

for 5 hours, so Allison traveled 300 miles

65 Let the time the cars travel

car travels car travels

ill be 310 miles apart after 2.5 hours

66 Let > œthe hours the robbers travel Then "! " œthe hours the police travel

The police will catch up &) hours after the robbery

67 Let the time the runners run

them (in miles) 1st runs 2nd runs

seconds

ur minutes minuteThey will meet after 50 seconds

69 Let the speed of the boat in still water.

Then the speed of the boat is downstream and upstream

Time upstream Time

<  #

#% œ #<

"# œ < Ê f the boat is 12 mph

70 Let the speed of the wind

Then the speed of the plane is downwind and upwind

Time upwind Time downwind

A œ

$%!  Aœ

71 72. Since the diameter of the semicircle

The hole is about 11.2 millimeters deep

Z œ < 2

%Þ& ("#Þ&" œ

("#Þ&"

%Þ& œ

""Þ# ¸

111

Total Area of

Area of

area semicircle

2 ¸ "#

1

The height of the window is about 12 feet

73-75 Answers may vary.

11. È È È È'% È# 12. È È È È$' $ œ '3È$

œ #)3 $'

19. Equate real parts: 20. Equate imaginary parts:

Equate imaginary parts:

Equate real parts:

49. 3 % 50. "! "! 3 "!3 "!3 "!

((3

(3 3(3

#

#

# )

' # '

œœ

B œ 3 #& & #& &

B œ &3 & &

ÈÈ

ÈÈ

ÊÊ

ÊÊ

C  #

*)) C  # œ *)

(C  #  %) œ !(C  #(C  # œ

or

œ )

))

62.

or

+  "'+  )# œ !

+  )+  ) œ

ÈÈ

È

ÈÈ

È È

ÈÈ

ÈÈ

È È

ÈÈ

ÈÈ

ÈÈ

ÈÈ

ÈÈ

ÈÈ

ÈÈ

È

84 &B B  "

&

œœ

85 B  #B  # œ ! Ê + œ " , œ # - œ #

œœ

œœ

# %

#+

œ % „ )3 œ " „ 3

#)

ÈÈ

 É



È

#

#

,  C œ +

„ , B œ +,  C

œ +

„,B ,  C,  C

ËÈ

œ + C, œ + C, œ „ + C, œ „+C

‰ˆ

ÊÈ

 É




È

two different nonreal complex numbers

*B  %#B  %* œ !+ œ * , œ %# - œ %

Set the discriminant equal to 0:

True for all values