1 Numbers, Data, and Problem Solving
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1.1 Numbers, Data, and Problem Solving
EXAMPLE 1
3
Classifying numbers
Classify each real number as one or more of the following: natural number, integer, rational
number, or irrational number.
5, -1.2,
13
, - 27, -12, 216
7
SOLUTION
5: natural number, integer, and rational number
-1.2: rational number
13
: rational number
7
- 27: irrational number
-12: integer and rational number
216 = 4: natural number, integer, and rational number
Now Try Exercise 7
᭣
Order of Operations
6Ϫ3ء2
–52
Figure 1.1
0
–25
Does 6 - 3 # 2 equal 0 or 6? Does - 52 equal 25 or - 25? Figure 1.1 correctly shows that
6 - 3 # 2 = 0 and that - 52 = - 25. Because multiplication is performed before subtraction, 6 - 3 # 2 = 0. Similarly, because exponents are evaluated before performing negation, -52 = - 25. It is essential that algebraic expressions be evaluated consistently, so the
following rules have been established.
Order of Operations
Using the following order of operations, first perform all calculations within parentheses, square roots, and absolute value bars and above and below fraction bars. Then use
the same order of operations to perform any remaining calculations.
1. Evaluate all exponents. Then do any negation after evaluating exponents.
2. Do all multiplication and division from left to right.
3. Do all addition and subtraction from left to right.
EXAMPLE 2
Evaluating arithmetic expressions
Evaluate each expression by hand.
10 - 6
- 4 - ƒ7 - 2ƒ
(a) 3(1 - 5)2 - 42
(b)
5 - 3
SOLUTION
(a) 3(1 - 5)2 - 42 = 3( -4)2 - 42
NOTE
(b)
10 - 6
4
- 4 - ƒ7 - 2ƒ = - 4 - ƒ5ƒ
5 - 3
2
= 3(16) - 16
= 2 - 4 - 5
= 48 - 16
= -2 - 5
= 32
= -7
(-4)2 = ( -4)( -4) = 16 and -42 = - (4)(4) = - 16.
Now Try Exercises 19 and 21 ᭣
4
CHAPTER 1 Introduction to Functions and Graphs
Scientific Notation
Numbers that are large or small in absolute value are often expressed in scientific notation. Table 1.1 lists examples of numbers in standard (decimal) form and in scientific
notation.
Table 1.1
Standard Form
Scientific Notation
93,000,000 mi
9.3 * 10 mi
Distance to the sun
13,517
1.3517 * 10
Radio stations in 2005
9,000,000,000
9 * 10
Estimated world population in 2050
7
4
9
-6
0.00000538 sec
5.38 * 10
0.000005 cm
5 * 10-6 cm
sec
Application
Time for light to travel 1 mile
Size of a typical virus
To write 0.00000538 in scientific notation, start by moving the decimal point to the
right of the first nonzero digit, 5, to obtain 5.38. Since the decimal point was moved six
places to the right, the exponent of 10 is -6. Thus, 0.00000538 = 5.38 * 10-6. When the
decimal point is moved to the left, the exponent of 10 is positive, rather than negative. Here
is a formal definition of scientific notation.
Calculator Help
To display numbers in scientific
notation, see Appendix A
(page AP-2).
Scientific Notation
A real number r is in scientific notation when r is written as c * 10n, where
1 … ƒ c ƒ 6 10 and n is an integer.
An Application Nanotechnology involves extremely small electrical circuits. Someday
this technology may use the movement of the human body to power tiny devices such as
pacemakers. The next example demonstrates how scientific notation appears in the
description of this new technology.
EXAMPLE 3
Analyzing the energy produced by your body
Nanotechnology is a technology of the very small: on the order of one billionth of a meter.
Researchers are looking to power tiny devices with energy generated by the human body.
(Source: Z. Wang, “Self-Powered Nanotech,” Scientific American, January 2008.)
(a) Write one billionth in scientific notation.
(b) While typing, a person’s fingers generate about 2.2 * 10-3 watt of electrical energy.
Write this number in standard (decimal) form.
SOLUTION
1
(a) One billionth can be written as 1,000,000,000
= 101 9 = 1 * 10-9.
(b) Move the decimal point in 2.2 three places to the left: 2.2 * 10-3 = 0.0022.
Now Try Exercise 83 ᭣
The next two examples illustrate how to evaluate expressions involving scientific
notation.
1.1 Numbers, Data, and Problem Solving
EXAMPLE 4
5
Evaluating expressions by hand
Evaluate each expression. Write your result in scientific notation and standard form.
4.6 * 10-1
(a) (3 * 103)(2 * 104) (b) (5 * 10-3)(6 * 105) (c)
2 * 102
SOLUTION
(3 * 103)(2 * 104) =
=
=
=
(b) (5 * 10-3)(6 * 105) =
=
=
=
4.6 * 10-1
(c)
=
2 * 102
=
=
=
(a)
Algebra Review
To review exponents, see Chapter R
(page R-7).
3 * 2 * 103 * 104
6 * 103 + 4
6 * 107
60,000,000
5 * 6 * 10-3 * 105
30 * 102
3 * 103
3000
4.6
10-1
*
2
102
2.3 * 10-1 - 2
2.3 * 10-3
0.0023
Commutative property
Add exponents.
Scientific notation
Standard form
Commutative property
Add exponents.
Scientific notation
Standard form
Multiplication of fractions
Subtract exponents.
Scientific notation
Standard form
Now Try Exercises 53, 55, and 57 ᭣
Calculators Calculators often use E to express powers of 10. For example, 4.2 * 10-3
might be displayed as 4.2E–3. On some calculators, numbers can be entered in scientific
notation with the EE key, which you can find by pressing 2nd , .
EXAMPLE 5
Computing in scientific notation with a calculator
Approximate each expression. Write your answer in scientific notation.
6 * 103
103 + 450 3
2
(a) a
(b) 24500pa
b
6 b(1.2 * 10 )
4 * 10
0.233
SOLUTION
(a) The given expression is entered in two ways in Figure 1.2. Note that in both cases
a
6 * 103
b(1.2 * 102) = 0.18 = 1.8 * 10-1.
4 * 106
(b) Be sure to insert parentheses around 4500p and around the numerator, 103 + 450, in
the ratio. From Figure 1.3 we can see that the result is approximately 1.59 * 1012.
Calculator Help
To enter numbers in scientific
notation, see Appendix A
(page AP-2).
(6ء10^3)/(4ء10^6
)(ء1.2ء10^2)
.18
( 6 E3 ) / ( 4 E6 ) ( ء1 . 2
E2 )
.18
Figure 1.2
√ (4500) (( ء103ϩ4
50) / .233)^3
1.58960355 E 12
Figure 1.3
Now Try Exercises 61 and 63 ᭣
6
CHAPTER 1 Introduction to Functions and Graphs
EXAMPLE 6
Computing with a calculator
Use a calculator to evaluate each expression. Round answers to the nearest thousandth.
1 + 22
(a) 2
3 131 (b) p3 + 1.22 (c)
(d) ƒ 23 - 6 ƒ
3.7 + 9.8
Algebra Review
To review cube roots, see Chapter R
(page R-40).
SOLUTION
(a) On some calculators the cube root can be found by using the MATH menu. If your calculator does not have a cube root key, enter 131^(1/3). From the first two lines in
Figure 1.4, we see that 2
3 131 L 5.079.
(b) Do not use 3.14 for the value of p. Instead, use the built-in key to obtain a more accurate value of p. From the bottom two lines in Figure 1.4, p3 + 1.22 L 32.446.
(c) When evaluating this expression be sure to include parentheses around the numerator
and around the denominator. Most calculators have a special square root key that can
1 + 22
be used to evaluate 22. From the first three lines in Figure 1.5, 3.7
+ 9.8 L 0.179.
(d) The absolute value can be found on some calculators by using the MATH NUM
menus. From the bottom two lines in Figure 1.5, ƒ 23 - 6 ƒ L 4.268.
Calculator Help
3 √ (131)
5.078753078
^3ϩ1.2 2
32.44627668
To enter expressions such as
3
2131 , 22, p, and ƒ 23 - 6 ƒ ,
see Appendix A (page AP-2).
Figure 1.4
(1ϩ √ (2)) / (3.7ϩ9.
8)
.1 7 8 8 3 0 6 3 4 2
abs(√ (3)Ϫ6)
4.267949192
Figure 1.5
Now Try Exercises 67, 69, 71, and 73 ᭣
Problem Solving
Many problem-solving strategies are used in algebra. However, in this subsection we focus
on two important strategies that are used frequently: making a sketch and applying one or
more formulas. These strategies are illustrated in the next three examples.
EXAMPLE 7
Finding the speed of Earth
Earth travels around the sun in an approximately circular orbit with an average radius of
93 million miles. If Earth takes 1 year, or about 365 days, to complete one orbit, estimate
the orbital speed of Earth in miles per hour.
SOLUTION
D
Getting Started Speed S equals distance D divided by time T, S = T . We need to find the
Geometry Review
To find the circumference of a circle,
see Chapter R (page R-2).
number of miles Earth travels in 1 year and then divide it by the number of hours in
1 year. ᭤
Distance Traveled A sketch of Earth orbiting the sun is shown in Figure 1.6. In 1 year
Earth travels the circumference of a circle with a radius of 93 million miles. The
circumference of a circle is 2pr, where r is the radius, so the distance D is
D = 2pr = 2p(93,000,000) L 584,300,000 miles.
1.1 Numbers, Data, and Problem Solving
Hours in 1 Year
7
The number of hours H in 1 year, or 365 days, equals
H = 365 * 24 = 8760 hours.
93,000,000 mi
Earth
Speed of Earth
S =
D
H
=
584,300,000
8760
L 66,700 miles per hour.
Now Try Exercise 85 ᭣
Sun
(Not to scale)
Figure 1.6 Earth’s Orbit
Many times in geometry we evaluate formulas to determine quantities, such as
perimeter, area, and volume. In the next example we use a formula to determine the number of fluid ounces in a soda can.
EXAMPLE 8
Finding the volume of a soda can
The volume V of the cylindrical soda can in Figure 1.7 is given by V = pr2h, where r is
its radius and h is its height.
(a) If r = 1.4 inches and h = 5 inches, find the volume of the can in cubic inches.
(b) Could this can hold 16 fluid ounces? (Hint: 1 cubic inch equals 0.55 fluid ounce.)
r
SOLUTION
(a) V = pr 2h = p(1.4)2(5) = 9.8p L 30.8 cubic inches.
(b) To find the number of fluid ounces, multiply the number of cubic inches by 0.55.
30.8 * 0.55 = 16.94
h
Now Try Exercise 93 ᭣
Yes, the can could hold 16 fluid ounces.
Figure 1.7 A Soda Can
Measuring the thickness of a very thin layer of material can be difficult to do directly.
For example, it would be difficult to measure the thickness of a sheet of aluminum foil or
a coat of paint with a ruler. However, it can be done indirectly using the following formula.
Thickness =
Volume
Area
That is, the thickness of a thin layer equals the volume of the substance divided by the area
that it covers. For example, if a volume of 1 cubic inch of paint is spread over an area of
1
100 square inches, then the thickness of the paint equals 100
inch. This formula is illustrated
in the next example.
EXAMPLE 9
Calculating the thickness of aluminum foil
A rectangular sheet of aluminum foil is 15 centimeters by 35 centimeters and weighs
5.4 grams. If 1 cubic centimeter of aluminum weighs 2.7 grams, find the thickness of the
aluminum foil. (Source: U. Haber-Schaim, Introductory Physical Science.)
SOLUTION
Getting Started Start by making a sketch of a rectangular sheet of aluminum, as shown in
Figure 1.8. To complete this problem we need to find the volume V of the aluminum foil
and its area A. Then we can determine the thickness T by using the formula T = VA. ᭤
15 cm
T
35 cm
(Not to scale)
Figure 1.8 Aluminum Foil
8
CHAPTER 1 Introduction to Functions and Graphs
NOTE
For the rectangular box shape shown in Figure 1.8 on the previous page,
Volume = Length * Width * Thickness.
('')''*
Area
It follows that Thickness =
Geometry Review
To find the area of a rectangle, see
Chapter R (page R-1). To find the
volume of a box, see Chapter R
(page R-3).
Volume
Area .
Volume Because the aluminum foil weighs 5.4 grams and each 2.7 grams equals 1 cubic
centimeter, the volume of the aluminum foil is
5.4
= 2 cubic centimeters.
2.7
Area
Divide weight by density.
The aluminum foil is rectangular with an area of 15 * 35 = 525 square centimeters.
Thickness The thickness of 2 cubic centimeters of aluminum foil with an area of 525
square centimeters is
Thickness =
Volume
2
=
L 0.0038 centimeter.
Area
525
Now Try Exercise 89 ᭣
1.1 Putting
It All
Together
Concept
Numbers play a central role in our society. Without numbers, data could be
described qualitatively but not quantitatively. For example, we could say that
the day seems hot but would not be able to give an actual number for the temperature. Problem-solving strategies are used in almost every facet of our lives,
providing the procedures needed to systematically complete tasks and perform computations.
The following table summarizes some of the concepts in this section.
Comments
Examples
Natural numbers
Sometimes referred to as the counting numbers
1, 2, 3, 4, 5, . . .
Integers
Include the natural numbers, their opposites, and 0
. . . , -2, -1, 0, 1, 2, . . .
Rational numbers
Include integers; all fractions q, where p and q are integers
with q Z 0; all repeating and all terminating decimals
p
1
128
, -3,
, -0.335, 0,
2
6
1
1
0.25 = , 0.33 =
4
3
Irrational numbers
Can be written as nonrepeating, nonterminating decimals;
cannot be a rational number; if a square root of a positive integer is not an integer, it is an irrational number.
p, 22, - 25, 2
3 7, p4
1.1 Numbers, Data, and Problem Solving
Concept
Comments
Examples
Real numbers
Any number that can be expressed in
standard (decimal) form
Include the rational numbers and
irrational numbers
4
p, 27, - , 0, - 10, 1.237
7
2
0.6 = , 1000, 215, - 25
3
Order of
operations
Using the following order of operations,
first perform all calculations within parentheses, square roots, and absolute value
bars and above and below fraction bars.
Then perform any remaining calculations.
1. Evaluate all exponents. Then do any
negation after evaluating exponents.
2. Do all multiplication and division
from left to right.
3. Do all addition and subtraction from
left to right.
-42 - 12 , 2 - 2 =
=
=
=
2 + 42
=
3 - 3#5
Scientific notation
A number in the form c * 10n, where
1 … ƒ c ƒ 6 10 and n is an integer
3.12 * 104 = 31,200
- 1.4521 * 10-2 = - 0.014521
5 * 109 = 5,000,000,000
1.5987 * 10-6 = 0.0000015987
Used to represent numbers that are large
or small in absolute value
1.1
9
- 16 - 12 , 2 - 2
- 16 - 6 - 2
- 22 - 2
- 24
2 + 16
3 - 15
18
=
-12
3
= 2
Exercises
Classifying Numbers
Exercises 1–6: Classify the number as one or more of the following: natural number, integer, rational number, or real number.
1. 21
24 (Fraction of people in the United States completing at
least 4 years of high school)
2. 20,082 (Average cost in dollars of tuition and fees at a
private college in 2004)
3. 7.5 (Average number of gallons of water used each minute
while taking a shower)
4. 25.8 (Nielsen rating of the TV show Grey’s Anatomy the
week of February 12–18, 2007)
5. 9022 (Distance in feet from home plate to second base
on a baseball field)
6. -71 (Wind chill when the temperature is -30°F and the
wind speed is 40 mph)
Exercises 7–10: Classify each number as one or more of the
following: natural number, integer, rational number, or irrational number.
7. p, -3, 29, 29, 1.3, - 22
8. 31, - 58, 27, 0.45, 0, 5.6 * 103
9. 213, 13, 5.1 * 10-6, -2.33, 0.7, - 24
5.7 2
10. -103, 21
25 , 2100, - 10 , 9 , -1.457, 23
Exercises 11–16: For the measured quantity, state the set of
numbers that most appropriately describes it. Choose from
the natural numbers, integers, and rational numbers. Explain
your answer.
11. Shoe sizes
12. Populations of states
13. Gallons of gasoline
14. Speed limits
10
CHAPTER 1 Introduction to Functions and Graphs
15. Temperatures in a winter weather forecast in Montana
49. 0.045 * 105
50. -5.4 * 10-5
16. Numbers of compact disc sales
51. 67 * 103
52. 0.0032 * 10-1
Order of Operations
Exercises 53–60: Evaluate the expression by hand. Write
your result in scientific notation and standard form.
53. (4 * 103)(2 * 105)
54. (3 * 101)(3 * 104)
Exercises 17–28: Evaluate by hand.
17. ƒ 5 - 8 # 7 ƒ
18. -2(16 - 3 # 5) , 2
19. - 62 - 3(2 - 4)4
20. (4 - 5)2 - 32 - 329
55. (5 * 102)(7 * 10-4)
8 - 4
21. 29 - 5 4 - 2
6 - 42 , 23
22.
3 - 4
57.
6.3 * 10-2
3 * 101
58.
8.2 * 102
2 * 10-2
23. 213 - 12
13 - 29 + 16
24.
ƒ 5 - 7 ƒ2
59.
4 * 10-3
8 * 10-1
60.
2.4 * 10-5
4.8 * 10-7
2
25.
2
4 + 9
-32 # 3
2 + 3
5
27. - 52 - 20 , 4 - 2
26. 10 , 2 ,
5 + 10
5
28. 5 - (-4)3 - (4)3
56. (8 * 10-3)(7 * 101)
Exercises 61–66: Use a calculator to approximate the expression. Write your result in scientific notation.
8.947 * 107
(4.5 * 108)
61.
0.00095
Scientific Notation
62. (9.87 * 106)(34 * 1011)
Exercises 29–40: Write the number in scientific notation.
29. 184,800 (New lung cancer cases reported in 2005)
63. a
30. 29,285,000 (People worldwide living with HIV)
3 (2.5 * 10-8) + 10-7
64. 2
31. 0.04361 (Proportion of U.S. deaths attributed to accidents in 2004)
65. (8.5 * 10-5)(- 9.5 * 107)2
32. 0.62 (Number of miles in 1 kilometer)
33. 2450
34. 105.6
35. 0.56
36. -0.00456
37. - 0.0087
38. 1,250,000
39. 206.8
40. 0.00007
101 + 23 2
b + 23.4 * 10-2
0.42
66. 2p(4.56 * 104) + (3.1 * 10-2)
Exercises 67–76: Use a calculator to evaluate the expression. Round your result to the nearest thousandth.
3 192
67. 2
68. 2(32 + p3)
69. ƒ p - 3.2 ƒ
70.
1.72 - 5.98
35.6 + 1.02
Exercises 41–52: Write the number in standard form.
41. 1 * 10-6 (Wavelength in meters of visible light)
71.
0.3 + 1.5
5.5 - 1.2
72. 3.2(1.1)2 - 4(1.1) + 2
42. 9.11 * 10-31 (Weight in kilograms of an electron)
73.
1.53
12 + p - 5
74. 4.32 -
43. 2 * 108 (Years required for the sun to orbit our galaxy)
44. 9 * 10 (Federal debt in dollars in 2007)
12
75. 15 +
4 + 23
7
76.
5
17
5 + 25
2
45. 1.567 * 102
46. -5.68 * 10-1
Applications
47. 5 * 105
48. 3.5 * 103
Exercises 77–80: Percent Change If an amount changes
from A to a new amount B, then the percent change is
B - A
* 100.
A
1.1 Numbers, Data, and Problem Solving
Calculate the percent change for the given A and B. Round your
answer to the nearest tenth of a percent when appropriate.
77. A = $8, B = $13
78. A = $0.90, B = $13.47
79. A = 1.4, B = 0.85
80. A = 1256, B = 1195
81. Percent Change Suppose that tuition is initially $100
per credit and increases by 6% from the first year to the
second year. What is the cost of tuition the second year?
Now suppose that tuition decreases by 6% from the second to the third year. Is tuition equal to $100 per credit
the third year? Explain.
82. Tuition Increases From 1976 to 2004, average annual
tuition and fees at public colleges and universities
increased from $433 to $5132. Calculate the percent
change over this time period.
11
87. Federal Debt The amount of federal debt changed dramatically during the 30 years from 1970 to 2000.
(Sources: Department of the Treasury, Bureau of the Census.)
(a) In 1970 the population of the United States was
203,000,000 and the federal debt was $370 billion.
Find the debt per person.
(b) In 2000 the population of the United States was
approximately 281,000,000 and the federal debt was
$5.54 trillion. Find the debt per person.
88. Discharge of Water The Amazon River discharges
water into the Atlantic Ocean at an average rate of
4,200,000 cubic feet per second, the highest rate of any
river in the world. Is this more or less than 1 cubic mile
of water per day? Explain your calculations. (Source:
The Guinness Book of Records 1993.)
83. Nanotechnology (Refer to Example 3.) During inhalation, the typical body generates 0.14 watt of electrical
power, which could be used to power tiny electrical circuits. Write this number in scientific notation. (Source:
Scientific American, January 2008.)
84. Movement of the Pacific Plate The Pacific plate (the
floor of the Pacific Ocean) near Hawaii is moving at
about 0.000071 kilometer per year. This is about the
speed at which a fingernail grows. Use scientific notation to determine how many kilometers the Pacific plate
travels in one million years.
85. Orbital Speed (Refer to Example 7.) The planet Mars
travels around the sun in a nearly circular orbit with a
radius of 141 million miles. If it takes 1.88 years for
Mars to complete one orbit, estimate the orbital speed of
Mars in miles per hour.
Sun
Mars
141,000,000 mi
89. Thickness of an Oil Film (Refer to Example 9.) A drop
of oil measuring 0.12 cubic centimeter is spilled onto a
lake. The oil spreads out in a circular shape having a
diameter of 23 centimeters. Approximate the thickness
of the oil film.
90. Thickness of Gold Foil (Refer to Example 9.) A flat,
rectangular sheet of gold foil measures 20 centimeters
by 30 centimeters and has a mass of 23.16 grams. If
1 cubic centimeter of gold has a mass of 19.3 grams,
find the thickness of the gold foil. (Source: U. HaberSchaim, Introductory Physical Science.)
(Not to scale)
86. Size of the Milky Way The speed of light is about
186,000 miles per second. The Milky Way galaxy has an
approximate diameter of 6 * 1017 miles. Estimate, to
the nearest thousand, the number of years it takes for
light to travel across the Milky Way. (Source: C. Ronan,
The Natural History of the Universe.)
91. Analyzing Debt A 1-inch-high stack of $100 bills contains about 250 bills. In 2000 the federal debt was
approximately 5.54 trillion dollars.
(a) If the entire federal debt were converted into a stack
of $100 bills, how many feet high would it be?
(b) The distance between Los Angeles and New York is
approximately 2500 miles. Could this stack of $100
bills reach between these two cities?
12
CHAPTER 1 Introduction to Functions and Graphs
92. Volume of a Cone The volume V of a cone is given by
V = 13pr2h, where r is its radius and h is its height. Find
V when r = 4 inches and h = 1 foot. Round your
answer to the nearest hundredth.
95. Thickness of Cement (Refer to Example 9.) A 100foot-long sidewalk is 5 feet wide. If 125 cubic feet of
cement are evenly poured to form the sidewalk, find the
thickness of the sidewalk.
96. Depth of a Lake (Refer to Example 9.) A lake covers
2.5 * 107 square feet and contains 7.5 * 108 cubic feet
of water. Find the average depth of the lake.
r
h
Writing about Mathematics
97. Describe some basic sets of numbers that are used in
mathematics.
93. Size of a Soda Can (Refer to Example 8.) The volume
V of a cylindrical soda can is given by V = pr2h, where
r is its radius and h is its height.
(a) If r = 1.3 inches and h = 4.4 inches, find the volume of the can in cubic inches.
(b) Could this can hold 12 fluid ounces? (Hint: 1 cubic
inch equals about 0.55 fluid ounce.)
94. Volume of a Sphere The volume of a sphere is given by
V = 43pr3, where r is the radius of the sphere. Calculate
the volume if the radius is 3 feet. Approximate your
answer to the nearest tenth.
98. Suppose that a positive number a is written in scientific
notation as a = b * 10n, where n is an integer and
1 … b … 10. Explain what n indicates about the size of a.
EXTENDED AND DISCOVERY EXERCISE
1. If you have access to a scale that weighs in grams, find
the thickness of regular and heavy-duty aluminum foil.
Is heavy-duty foil worth the price difference? (Hint:
Each 2.7 grams of aluminum equals 1 cubic centimeter.)
1.2 Visualizing and Graphing Data
• Analyze one-variable data
• Find the domain and range
of a relation
• Graph in the xy-plane
• Calculate distance
• Find the midpoint
• Learn the standard
equation of a circle
• Learn to graph equations
with a calculator (optional)
Introduction
Technology is giving us access to huge amounts of data. For example, space telescopes,
such as the Hubble telescope, are providing a wealth of information about the universe.
The challenge is to convert the data into meaningful information that can be used to solve
important problems. Before conclusions can be drawn, data must be analyzed. A powerful
tool in this step is visualization, as pictures and graphs are often easier to understand than
words. This section discusses how different types of data can be visualized by using various mathematical techniques.
One-Variable Data
Data often occur in the form of a list. A list of test scores without names is an example;
the only variable is the score. Data of this type are referred to as one-variable data. If the
values in a list are unique, they can be represented visually on a number line.
Means and medians can be found for one-variable data sets. To calculate the mean (or
average) of a set of n numbers, we add the n numbers and then divide the sum by n. The
median is equal to the value that is located in the middle of a sorted list. If there is an odd
1.2 Visualizing and Graphing Data
13
number of data items, the median is the middle data item. If there is an even number of
data items, the median is the average of the two middle items.
EXAMPLE 1
Analyzing a list of data
Table 1.2 lists the monthly average temperatures in degrees Fahrenheit at Mould Bay,
Canada.
Table 1.2
Monthly Average Temperatures at Mould Bay, Canada
Temperature (°F)
- 27
- 31
- 26
-9
12
32
39
36
21
1
-17
-24
Source: A. Miller and J. Thompson, Elements of Meteorology.
(a)
(b)
(c)
(d)
Plot these temperatures on a number line.
Find the maximum and minimum temperatures.
Determine the mean of these 12 temperatures.
Find the median and interpret the result.
SOLUTION
(a) In Figure 1.9 the numbers in Table 1.2 are plotted on a number line.
–40
–30
–20
–10
0
10
20
30
40
Figure 1.9 Monthly Average Temperatures
CLASS DISCUSSION
In Example 1(c), the mean of the
temperatures is approximately
0.6°F. Interpret this temperature.
Explain your reasoning.
(b) The maximum temperature of 39°F is plotted farthest to the right in Figure 1.9.
Similarly, the minimum temperature of - 31°F is plotted farthest to the left.
(c) The sum of the 12 temperatures in Table 1.2 equals 7. The mean, or average, of these
7
temperatures is 12
L 0.6°F.
(d) Because there is an even number of data items, the median is the average of the middle two values. From the number line we see that the middle two values are -9°F and
1°F. Thus the median is - 9 2+ 1 = - 4°F. This result means that half the months have
an average temperature that is greater than -4°F and half the months have an average
Now Try Exercises 1 and 5 ᭣
temperature that is below -4°F.
Two-Variable Data
Sometimes a relationship exists between two lists of data. Table 1.3 lists the monthly average precipitation in inches for Portland, Oregon. In this table, 1 corresponds to January, 2
to February, and so on, until 12 represents December. We show the relationship between a
month and its average precipitation by combining the two lists so that corresponding
months and precipitations are visually paired.
Table 1.3 Average Precipitation for Portland, Oregon
Month
Precipitation (inches)
1
2
3
4
5
6
7
8
9
10
11
12
6.2
3.9
3.6
2.3
2.0
1.5
0.5
1.1
1.6
3.1
5.2
6.4