Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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966 APPENDIX 3 Programs
Goto 6
Lbl 3
J 1 J
If J N 1
Then Goto 7
If End
Goto 6
HP-39 Rectangular/Polar Conversion Program
Description: Enter the rectangular coordinates of a point in the plane; the pro-
gram displays the polar coordinates of the point.
Input X; “RECTANGULAR TO POLAR”; “X ”; “ENTER X”; 0:
Input Y: “RECTANGULAR TO POLAR”; “Y ”; “ENTER Y”; 0:
If X 0
Then (ATAN(Y/X)) C
Else (ATAN(X, Y) p) C:
End:
MSGBOX “R ”
X
2
Y
2
“u ” C:
HP-39 Polar/Rectangular Conversion Program
Description: Enter the polar coordinates of a point in the plane; the program
displays the rectangular coordinates of the point.
Input R; “POLAR TO RECTANGULAR”; “R ”; “ENTER R”; 0:
Input u; “POLAR TO RECTANGULAR”; “u ”; “ENTER u”; 0: [CHARS
menu]
MSGBOX “X ” R(cos u) “Y ” R(sin u):
Chapter 1
Section 1.1, page 13
1.
3. Positive 5. Positive 7. Negative
9.
2
5
0
2
4
3
0
,
1
3
8
7
9
, 27,
4
6
5
9
8
1
7
, 6.735, 47
11. 4 8 13. p 100 15. z 4
17. d 7 19. z 17 21. 5 3
23. 25. 3 27. 5
29. 0 31. b c a
33. a lies to the right of b. 35. a b
37.
39.
41.
43.
[5, 10] 45. (3, 14) 47. [9, )
49. 6.506 10
9
51. 5.91
10
12
53. 2
10
9
m
55. 150,000,000,000 m
57. .00000000000000000016726 kg
59. (a) 8.365 10
12
; 2.984 10
8
(b) $28,032.84
61. 4 63. 6/5 65. 2
67. u
2
69. 11 71. 0
73. 169 75. p 2
77. p
79. 81. 83.
85. 7 87. 14.5 89. p 3
91. 3
2
93. 2.21
95. x 143 21; x 163 26
97. 15 99. 30 101. t
2
103. b 3 105. (c d) d c
107. 0
109. (c d)
2
(c d)
2
c
2
2cd d
2
111. x 5 4 113. x 4 17
4
p
805
161
1−4 −20
132−3 −2 −10
2468−20
10−7 −5 −1 0 2.25
1
2
−4.75
8
3
115. c b
117. The distance from x to 3 is less than 2 units.
119. The distance from x to 7 is at most 3 units.
121. (a) iii (b) i (c) ii (d) v (e) iv
123. x 1 or 1 125. x 1 or 3
127. x p 4 or p 4 129. 7 x 7
131. 3 x 7 133. x 5 or x 1
135. Since a 0, b 0, and c 0, the sum a b c
is positive only when one or more of a, b, c is positive.
But a is positive only when a 0; similarly for b, c.
Special Topics 1.1.A, page 17
1. .7777 3. .8181
5. 3.142857142857 7.
9.
75
9
8
9
,
0
6
0
79
25
3
2
3
,
0
8
0
93
11.
3
5
7
13.
51
9
7
9
,
0
8
0
96
14
2
,
7
3
5
86
15. No
17. Yes 19. No 21. Yes
23. If d .74999
, then 1000d 100d
(749.999
) (74.999
) 675. Hence 900d 675
so that d
6
9
7
0
5
0
3
4
. Also .75000
.75
1
7
0
5
0
3
4
.
25. .0588235294117647058823
27.
2
1
9
.03448275862068965517241379310344
29.
2
4
8
7
3
37
99
Answers
6.02127659574468085106382978723404255319148936170212
31. All of these numbers are approximations of 1/17, but not
equal to 1/17. If you subtract 1/17 from any one of them,
the answer is not 0.
Section 1.2, page 26
1. x 8 3. x
6
5
5. y 32
7. y
x
3
5
9. b
2A
h
hc
(h 0)
11. h
p
4
d
V
2
(d 0) 13. x 3 or 5
967
15. x 2 or 7 17. y
1
2
or 3
19. t 2or
4
1
21. u 2 or
3
2
23. x 1 13 25. x
1
2
5
or
1
2
5
27. 2 29. 2 31. 1
33. x 2 3
35. x 3 2
37. No real solutions 39. x
2
1
2
41. x
1
2
2
43. x 3 or 6
45. x
1
2
2
47. x
2
3
or 5
49. No real solutions 51. No real solutions
53. x 1.824 or 0.470 55. x 13.79
57. y 1 or 6
59. x 7
61. y 2 or
1
2
63. x
5
5
65. No real solution 67. 1
69. x
2
5
71. x
5
8
57
73. x
3
4
or 2 75. 2011
77. 2006 79. 1500
81. About 6.32 sec
83. (a) About 4.38 sec (b) 50 sec
85. (a) 670.5
f
l
t
b
2
(b) 14,400 ft
87. 1960
89. (a) 2003 (b) 2007 (c) mid-2010
91. (a) About 32 and 62 (b) About 17 and 77
93. (a) 73.53% (b) 84.75% (c) 91.74%
95. k 10 or 10 97. k 16
99. Yes 101. No 103. k 4
105. (a) x
5
2
17
(b) Answers vary
(c) x .3068 or .6959 or 1.1183 or 1.3959
Special Topics 1.2.A, page 33
1. x 3 or 6 3. x
3
2
5. x 2
7. x
3
2
9. x 5, 3, 1, 1
11. x
5
2
33
, 1, 4
968 ANSWERS
13. Upper control limit: 0.0497;
Lower control limit: 0.0097
Special Topics 1.2.B, page 37
1. A varies directly as the square of r; the constant of variation
is p.
3. A varies jointly as l and w; the constant of variation is 1.
5. V varies jointly as the square of r and h; the constant of
variation is
p
3
.
7. a
b
k
9. z kxyw 11. d k h
13. v ku; k 4 15. v k/u; k 16
17. t krs; k 4 19. w kxy
2
; k 2
21. T kpv
3
/u
2
; k 16 23. r 4
25. b
9
4
27. u 50
29. r 3 31. $1521.00 33. 12 lb
35. 80 kg/cm
2
37. 400 ft 39. 30 in.
41. 3750 kg
43. (a) 2400 lb (b) No more than 4 ft
Section 1.3, page 48
1. A(3, 3); B(1.5, 3); C(2.5, 0); D(1.5, 3); E(0, 2);
F(0, 0); G(2, 0); H(3, 1); I(3, 1)
3. (6, 3) 5. (4, 2)
7.
012345
y
x
100
200
300
400
500
012345
y
x
100
200
300
400
500
9. (a) Quadrant IV (b) Quadrants III or IV
11. (a)
(b)
(c) They are mirror images of each other, with the x-axis
being the mirror. In other words, they lie on the same
vertical line, on opposite sides of the x-axis, the same
distance from the axis.
13. 13;
2
1
, 1
15. 10;
2
3
,
7
2
17. 6 2
6
1.05;
2
2
3
,
3
2
19. 2
a b;
a
2
b
,
a
2
b
21. (3.5, 4.6) 23. 13 22
13
25. 15 square units
27. Hypotenuse from (1, 1) to (2, 2) has length 10; other
sides have lengths 2
and 8
. Since (2
)
2
(8
)
2
(10)
2
, this is a right triangle.
29. The distances are (1, 4) to (5, 2): 20; (5, 2) to (3, 2):
20; (3, 2) to (1, 4): 40. But (20)
2
(20)
2
40 (40)
2
, so the triangle is a right triangle with
hypotenuse 40.
31. (a) About 45 yd
(b)
34
1
6
, 27
1
2
33. (a)
(b) (2009, .85)
2002
Year
Billions of
passengers
1
x
y
2016
(2002, .65)
(2016, 1.05)
(–5, 4)
(–2, –3)
(3, –2)
(4, 1)
x
y
(–2, 3)
(–5, –4)
(4, –1)
(3, 2)
x
y
ANSWERS 969
(c) If linear growth is assumed, the midpoint suggests that
there will be 850 million passengers in 2009.
35. No 37. Yes 39. No
41. x 1, 5; y 5
43. x 5, 1; y 5
, 5
45. x 0, 10; y 0, 8
47. (a) Approximately 7.3 million.
(b) Approximately 2007; Approximately 7.8 million.
(c) 2000–2003, 2011–.
49. (a) Under 56; (b) Age 56;
(c) Retiring at age 60: about $30,000; retiring at age 65:
About $35,000.
51. B 53. C
55. (x 3)
2
(y 4)
2
4
57. x
2
y
2
3
59. (x 5)
2
(y 2)
2
5
61. (x 1)
2
(y 3)
2
9
63. Center: (4, 3); radius: 210
65. Center: (3, 2); radius: 27
67. Center:
2
2
5
, 5
; radius:
6
2
77
69. (a) On (b) Outside (c) Inside
(d) Outside (e) On
4
0
2
−4 −2
2
4
−2
(0, 3 + 2√2)
x
y
6
(0, 3 − 2√ 2)
(−1, 0)
−6
4
0
2
−4 −2
2
4
−2
−4
(4, 0) (6, 0)
x
y
6
71. (x 3)
2
(y 3)
2
18
73. (x 1)
2
( y 2)
2
8
75. (x 5)
2
( y 4)
2
16
77. x
2
y
2
4x 2y 0
79. (3, 4) and (2, 1)
81. Assume k d. The other two vertices of one possible
square are (c k d, d ), (c k d, k); those of another
square are (c (k d ), d ), (c (k d ), k); those of a
third square are
c
k
2
d
,
k
2
d
,
c
k
2
d
,
k
2
d
.
83. (0, 0), (6, 0)
85. (3, 5 11), (3, 5 11)
87. x 6
89. M has coordinates (s/2, r/2) by the midpoint formula.
Hence the distance from M to (0, 0) is
2
s
0
2
2
r
0
2
s
4
2
r
4
2
,
and the distance from M to (0, r) is the same:
2
s
0
2
2
r
r
2
2
s
2
2
r
2
s
4
2
r
4
2
as is the distance from M to (s, 0):
2
s
s
2
2
r
0
2
2
s
2
r
s
2
s
4
2
r
4
2
.
91. Place one vertex of the rectangle at the origin, with one side
on the positive x-axis and another on the positive y-axis. Let
(a, 0) be the coordinates of the vertex on the x-axis and
(0, b) the coordinates of the vertex on the y-axis. Then the
fourth vertex has coordinates (a, b) (draw a picture!). One
diagonal has endpoints (0, b) and (a, 0), so that its length
is
(0 a
)
2
(b
0)
2
a
2
b
2
. The other diagonal
has endpoints (0, 0) and (a, b) and hence has the same length:
(0 a
)
2
(0
b)
2
a
2
b
2
.
93. The circle (x k)
2
y
2
k
2
has center (k, 0) and radius k
(the distance from (k, 0) to (0, 0)). So the family consists of
every circle that is tangent to the y-axis and has center on
the x-axis.
95. The points are on opposite sides of the origin because one
first coordinate is positive and one is negative. They are
equidistant from the origin because the midpoint on the line
segment joining them is
c
2
(c)
,
d
2
(d)
(0, 0).
970 ANSWERS
Section 1.4, page 64
1. (a) C (b) B (c) B (d) D
3.
5
2
5. 4 7. t 22 9. t
1
5
2
11.
13. (d) 15. (a)
17. y 4x 5 19. y 2.3x 1.5
21. y
2
3
x 2 23. y
3
4
x 3
25. Slope 2; y-intercept 5
27. Slope
7
3
; y-intercept
7
11
29. y x 3 31. y x 8
33. y x 5 35. y
5
12
x
8
2
7
5
37.
39.
y
x
x 2
y 1
y
x
x 5
y 3
(1, 0)
Slope of L ==C
C − 0
1 − 0
P(1, C)
L
(0, 0)
x
y
41.
43.
Perpendicular 45. Parallel
47. Parallel 49. Perpendicular
51. Yes 53. y
1
2
x 6
55. y
5
1
x
1
5
3
57. y 3x 7
59. y
3
2
x 61. y x 5
63. y x 2 65. k
3
11
67. y
4
3
x
2
4
5
69. y
2
x
6
71. Both have slope
B
A
.
73. (a) (0, 60), (5, 66) (b) y 1.2x 60
(c) 64,800,000 (d) 2010
75. (a) y .03x (b) About 158.33 ft
77. (a) y
5
5
1
0
x 212
(b) 211° (c) 209.6° (d) 206.3° (e) 199.5°
79. (a) y .23x 3.2 (b) 7,800,000 (c) 2020
81. (a) y
2.75x
x
26,000
(b) 104,000
83. (a) r 1.4x (b) 24,167 items
85. (a) y 60 2x (b) y 80 2x (c) y 160 4x
87. (a) F
9
5
C 32 (b) C
5
9
(F 32) (c) 40°
89. (a) 12.5 gpm; 8.33 gpm; 25 gpm. (b) y 10x 75
0
24
Minutes
Gallons remaining
681357
10
20
30
40
50
60
70
80
y
x
x 0
y 0
ANSWERS 971
91. (a) y 1360x 15,350 (b) $1360 per year
(c) $7190
93. Let y mx b and y mx c be equations of lines with
same slope m, and b c. Suppose (x
1
, y
1
) is an arbitrary point
lying on both lines. Then, y
1
mx
1
b and y
1
mx
1
c.
So, mx
1
b mx
1
c and, b c, a contradiction. Thus,
the lines share no point in common so must be parallel.
95.
Equation of D
1
; y x, with slope 1
Equation of D
2
; y x a, with slope 1
(slope D
1
)(slope D
2
) (1)(1) 1
Chapter 1 Review, page 70
1. (a) (b) (c) (d) (e)
3. (a) 10 y 0
(b) 0 x 10
5. (a) (8, )
(b) (, 5]
7. (a) 1.232 10
16
(b) 7.89 10
11
9. (a) x 7 3
(b) y x 3
11. x 2 or 8 13. x
2
1
or
2
11
15. x 4 or x 0
17. (a) 7 p
(b) 23
3
19. 28/99 21. x 44/7
23. No real solutions 25. z
3
5
2 11
27. 2.25 times as large 29. 2
31. x 3 or 1 33. 6
35. 1
37. x 3 or 3 or 2
or 2
39. x 1 or 53
41. 58 43. c
2
d
2
45.
d,
c
2
2d
47. (a) 17
(b) (x 2)
2
(y 3)
2
17
(a, 0)
(a, a)
D
2
D
1
(0, 0)
(0, a)
x
y
49.
51. b and d 53. c
55.
57. (a) 1 (b) 45 59. y 3x 7
61. y 2x 1 63. x 5y 29
65. 25,000 ft 67. False 69. False 71. False
73. False 75. (d) 77. (e) 79. (c)
81. (a) y .155x 74.7 (b) About 74.5 yrs
(c) 2019
83. (c) 85. (d)
Chapter 1 Test, page 73
1. (a) (b) (, 2]
2. 49
3. (a) 740.76 pounds per square ft (b) 14,112 ft
4. (a) $5.622 10
12
(b) 2.811 10
8
(c) $20,000
5. t 12 6. x 7 or 12
7. 1499
8. b
2E
h
ch
9. x
3
4
29
10. (a) All numbers that are less than 6 units from 16 on the
number line.
(b) c 16 5
11. k 8 or 8
12. (a) (y 1)
2
(b) x
2
1
13. x 74 or 174 14. (x 1)
2
(y 6)
2
37
0−32
2016 24 3228
2
1
4
3
6
5
8
7
9
x
y
2016 24 3228
2
1
4
3
6
5
8
7
9
x
y
y
x
5
3
972 ANSWERS
15. Neither. Reason: one line has slope 4 and the other has
slope 2.
16. (a) (12, 1) (b) 5
17.
5
2
5
18. y-intercept is 2; x-intercepts are 1 and 23.
19. (a) $425/year (b) $17,609 (c) $21,859
20. (a) (1, 1) (b) y
1
2
x 4
21. Center (1, 4); radius 3
22.
23.
12 28
24. (a) y 7.82x 412.1
(b) About 247.88 per 100,000
(c) About 185.32 per 100,000
(d) No; it predicts a negative death rate in 2033.
25. r 94 26. Neither
Chapter 2
Section 2.1, page 89
1.
3.
−1
8
−77
−6
6
−10 10
5
−2
x
y
5.
7.
9.
11.
13.
(a)
(b) Yes. For x larger than about 2.2 in absolute value, the
value of the function is less than the height of one
pixel.
−10
10
−10 10
−10
10
−10 10
−10
10
−10 10
−10
10
−10 10
−10
10
−10 10
ANSWERS 973
(c)
There are many possible square windows for Exercises 15–24;
these were made on a TI-84. To have a square window with
the same y-axis shown here on wider screen calculators (such
as TI-86/89), the x-axis should be longer than the one shown
here. Because of the calculator’s limited resolution, some
graphs that should be connected (such as circles) may appear
to have breaks in them.
15.
17.
19.
−8
4
−513
−5.3
5.3
−88
−4
4
−66
−0.1
1.1
−10 10
21.
23.
25. Answers vary, but most calculators are correct to 6–8 deci-
mal places.
27. Highest at about (.7612, 3.9358); lowest at about
(4.0812, 3.3356)
29. in late 2003 (x 3.9); 5.17%
31. (c) 33. (d)
35. (e)
37. 5 x 5 and 100 y 100
39. 10 x 10 and 2 y 20
41. 6 x 12 and 100 y 250
43.
200
0
010
200
0
0
10
−3.1
3.1
−4.7 4.7
−8
8
−12 12
974 ANSWERS
45.
47. (a)
(b)
(c) Because the vertical distance between points on the
graph is very small at the top of the graph, the graphed
pixels make a solid line on the screen. When the vertical
distance between adjacent pixels is larger, more “space”
shows and the individual points appear isolated.
49. No
−10
10
−10 10
10
10
10 10
10
10
10
10
650
300
0
10
650
0
300
10
51. Yes
53. No
55. (a)
(b)
57. Possibly true
59. (c)
61. An appropriate viewing window should be large enough
to contain the points (0, 4000), (20, 104000), and
(25.080869, 0), and should not show negative values.
100
10
100
10
−50
50
−10 10
−10
10
−10 10
−10
10
−10 10
−10
10
−10 10
ANSWERS 975
63. x should range from 0 to 48, and y should range from 0 to at
least 31.5.
65. (a) 1917; 1929 (b) 1925; 104,000
67. (a) After about 6.3 hours (b) After about 18.7 hours
69. All four graphs have the same shape. Graph (b) is graph (a)
shifted 5 units vertically upward, graph (c) is graph (a)
shifted 5 units vertically downward, and graph (d) is graph
(a) shifted 2 units vertically downward.
71. Graph (b) is graph (a) stretched by a factor of 2 vertically
away from the x-axis; similarly for graph (c), except the
stretching factor is 3. Graph (d) is graph (a) shrunk verti-
cally toward the x-axis by a factor of
1
2
.
73. The graphs are “mirror images” of each other, with the
straight line y x being the mirror.
75. Same answer as Exercise 73
Section 2.2, page 100
1. 3 3. 3 5. 2
7. x 2.42645 9. 1.4526 11. 1.4751
13. 1.1921235 15. 1.379414 17. 1.3289
19. 2.1149 21. x 2.1017 23. x 1.7521
25. x .9505 27. x 0 or 2.2074
29. x 2.3901454
31. x .6513878188 or 1.151387819
33. x 7.033393042 35. x 2/3
37. x 1/12 39. x 3
41. x 1.4528 or 112.00 43. 3.00242
45. 1.7388 47. 2009
49. (a) late 2003
(b) mid 2010
51. about 96 ft 53. late 2012
Section 2.3, page 111
1. (a) Let x represent the student’s score on the fourth exam.
(b) x, 88 62 79 x, (88 62 79 x)/4
3. English Language Mathematical Language
Length of rectangle x
Width of rectangle y
Perimeter is 45 x y x y 45 or
2x 2y 45
Area is 112.5 xy 112.5
5. x .08x 2619
7. The circle has radius r 16/2 8, so its area is
pr
2
p
8
2
64p. Let x be the amount by which the
radius is to be reduced. Then r 8 x and the new area is
p(8 x)
2
, which must be 48p less than the original area, that is,
p(8 x)
2
64p 48p, or equivalently, p(8 x)
2
16p.