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Contemporary Mathematics

Chapter 10

Contemporary MathematicsChapter 10
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Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Your Turn

10.1
1.
\overrightarrow {BD} is the ray that starts at point B and extends infinitely in the direction of point D.
2.
\overline {AB} represents the line segment that starts at point A and ends at point B.
3.
\overleftarrow {BA} represents the ray that starts at point A and extends infinitely in the direction of point B.
4.
\overleftrightarrow {AD} represents a line that contains the points A and D. Notice the arrowheads on both ends of the line above AD, which means that the line continues infinitely in both directions.
10.2
1.
Answers will vary. One way {\overline {AE}} ,\,{\overline {EI}} ,\,{\overline {IE}} ,\,{\overline {EB}} ,\,{\overline {BC}} ,\,{\overline {CD}} ,\,{\overline {DG}} \,; Second way {\overline {AH}} ,\,{\overline {IE}} ,\,{\overline {EB}} ,\,{\overline {BC}} ,{\overline {CF}} ,\,{\overline {FG}} .
10.3
1.
\overleftrightarrow {RS}\parallel \overleftrightarrow {UV}
\overleftrightarrow {RS} \bot \overleftrightarrow {XY}
\overleftrightarrow {UV} \bot \overleftrightarrow {XY}
10.4
1.
{\overline {AB}} \cap {\overline {BC}} = B
2.
\overrightarrow {BC} \cup \overleftarrow {CA} = \overleftrightarrow {AC}
3.
{\overline {BC}} \cap {\overline {AC}} = {\overline {BC}}
10.5
1.
Point A is located at ( - 5, - 2); Point B is located at (- 3,4); Point C is located at (3, - 6); Point D is located at (5, - 2); Point E is located at (3,2); Point F is located at (5,5).
2.
Points A and B are on the straight line \overleftrightarrow {AB}.
3.
The line that begins at point E in the direction of point F is a ray, \overrightarrow {EF}.
4.
The line from point C to point D is a line segment \overline {CD} .
5.
Yes, this represents a plane. One reason is that the figure contains four points that are not on the line \overleftrightarrow {AB}.
10.6
1.
Plane MNOP intersects with plane OPQR, and plane OPQR intersects with plane QRST.
10.7
1.
Acute Angles Obtuse Angles Right Angles Straight Angles
\angle AOB
\angle AOC
\angle BOC
\angle BOD
\angle COD
\angle DOE
\angle FOE		 \angle AOE
\angle BOF
\angle COF 		\angle AOD
\angle BOE
\angle DOF 		\angle AOF
10.8
1.
\begin{array}{rcl}{5x}&{ =}&{{125^ \circ }}\\{2x + 5}&{=}&{{55^ \circ }}\end{array}
10.9
1.
m\measuredangle (6x) = {24^ \circ}, m\measuredangle (9x) = {36^ \circ}, m\measuredangle (7x + 2) = {30^ \circ}
10.10
1.
m\measuredangle 4 = {67^ \circ }, m\measuredangle 1 = {113^ \circ }, m\measuredangle 3 = {113^ \circ }
10.11
1.
{130^ \circ }
10.12
1.
m\measuredangle 5 = {120^ \circ}, m\measuredangle 4 = {120^ \circ}, m\measuredangle 8 = {120^ \circ}
10.13
1.
m\measuredangle 1 = {132^ \circ}, m\measuredangle 5 = {132^ \circ}
10.14
1.
m\measuredangle 5 = {56^ \circ }
m\measuredangle 6 = {62^ \circ }
m\measuredangle 7 = {118^ \circ }
m\measuredangle 8 = {62^ \circ }
m\measuredangle 9 = {118^ \circ }
10.15
1.
m\measuredangle x = {66^ \circ }, m\measuredangle (x + 1) = {67^ \circ }, m\measuredangle (x - 19) = {47^ \circ }
10.16
1.
m\measuredangle 1 = {35^ \circ }; m\measuredangle 2 = 2 = {85^ \circ }; m\measuredangle 3 = {60^ \circ }
10.17
1.
Triangle ABC is congruent to triangle DEF.
10.18
1.
SAS
10.19
1.
ASA
10.20
1.
The triangles are similar.
10.21
1.
x = 6, y = 24
10.22
1.
The tree is 86 feet high.
10.23
1.
60 ft
10.24
1.
rectangle
2.
pentagon
3.
heptagon
4.
parallelogram
10.25
1.
Shapes 1, 2, 4, and 6 are triangles; shape 3 is a pentagon; shape 5 is a parallelogram; and shape 7 is a rectangle.
10.26
1.
120 in
10.27
1.
22.4 in
10.28
1.
a = {108^ \circ }
The sum of the interior angles is {540^ \circ }.
10.29
1.
We have the sum of interior angles is {360^ \circ }. Then, x = - 9. The other angles measure {54^ \circ }, {111^ \circ }, {44^ \circ }.
10.30
1.
7(51.43) = {360^ \circ }
10.31
1.
C = 2(2.25)\pi = 14.14\text{ cm}
10.32
1.
r = 2.5\,\text{cm}
10.33
1.
You need to buy 15.74 feet of trim.
10.34
1.
The translated hexagon has labels A'', B'', C'', D'', E'', F''.
Three hexagons are graphed on a grid. Hexagon, A B C D E F is plotted. The bottom and top sides, A F and C D measure 3 units, each. The other sides, C B, B A, D E, and E F measure 2 units, each. The hexagon is translated 5 units to the right and 3 units up. The vertices of the translated hexagon are A prime, B prime, C prime, D prime, E prime, and F prime. The translated hexagon is again translated 7 units down. The vertices of the newly translated hexagon are A double prime, B double prime, C double prime, D double prime, E double prime, and F double prime.
10.35
1.
Rotate the triangle about the rotation point {90^ \circ } to the right three times.
10.36
1.
This tessellation could be produced with a reflection of the triangle vertically, then each triangle is rotated {180^ \circ } and translated to the right.
10.37
1.
From the first square on the left, rotate the square {30^ \circ } to the right, or A1 \to A2. Then, reflect the square over the horizontal, or A2 \to A3. Next, reflect all three squares over the vertical line. The lavender triangles comprise another pattern that tessellates, fits in with the squares, and fills the gaps.
10.38
1.
No
10.39
1.
Not by themselves, but by adding an equilateral triangle, the two regular polygons do tessellate the plane without gaps.
10.40
1.
We made a tessellation with a regular decagon (10 sides) and an irregular hexagon. We see that the regular decagon will not fill the plane by itself. The gap is filled, however, with an irregular hexagon. These two shapes together will fill the plane.
10.41
1.
8\,\text{cm}^2
10.42
1.
108 ft2
10.43
1.
45 boxes at a cost of $2,025.00
10.44
1.
270\,{\text{i}}{{\text{n}}^2}
10.45
1.
13,671 square feet; cost is $19,872.95.
10.46
1.
132\,{{\text{in}}^2}
10.47
1.
{d_2} =10
10.48
1.
40 in
10.49
1.
A = 96.25\,{\text{c}}{{\text{m}}^2}
10.50
1.
30\,{\text{y}}{{\text{d}}^2}
10.51
1.
28.3\,{\text{c}}{{\text{m}}^2}
10.52
1.
the 15-inch pizza
10.53
1.
A = 21.875 + 4.8 = 26.7\,{\text{f}}{{\text{t}}^2}
10.54
1.
21.46\,{\text{c}}{{\text{m}}^2}
10.55
1.
$300
10.56
1.
\begin{array}{rcl}{SA}&{ = }&{432\text{ cm}{^2}}\\{V}&{ = }&{540\text{ cm}{^3}}\\ \end{array}
10.57
1.
100\,\text{cm}{^2}
10.58
1.
\begin{array}{rcl} {SA}&{ = }&{336\text{ cm}{^2}}\\ {V}&{ = }&{480\text{ cm}{^3}}\\ \end{array}"
10.59
1.
1,192\text{ ft}{^2}
10.60
1.
\begin{array}{rcl} {SA}&{ = }&{527.78\text{ cm}{^2}}\\ {V}&{ = }&{769.69\text{ cm}{^3}}\\ \end{array}
10.61
1.
approximately 8 1/3 cans of soup
10.62
1.
6.25 ft by 6.25 ft
10.63
1.
1.26\,{\text{ft}}\,{\text{wide}}\, \times \,1.26\,{\text{ft}}\,{\text{long}}\, \times \,1.26\,{\text{ft}}\,{\text{high}}, $95.26
10.64
1.
a = 5
10.65
1.
1,140 ft
10.66
1.
The slanted distance will be 120.4 inches.
10.67
1.
The side lengths are 15,15\sqrt 3,30
10.68
1.
The ladder reaches 12 feet up the wall and sits 12\sqrt {3\,}{\text{ feet}} from the wall.
10.69
1.
Each side x equals x = 4\sqrt 2 = 5.66.
10.70
1.
\begin{array}{*{20}{l}} {r = 6.53} \\ {y = 4.2} \end{array}
10.71
1.
\alpha = {46.3^ \circ },\,\beta = {43.7^ \circ }, x = 5.73
10.72
1.
2,241 ft
10.73
1.
c = 8.06, one angle is {60^ \circ }, and the other angle is {30^ \circ }.
10.74
1.
{11.9^ \circ }
10.75
1.
46 ft

Check Your Understanding

1.
The line containing point D and point A is a line segment from point D to point A, \overline {DA} , or from point A to point D, \overline {AD} .
2.
The line containing points C and B is a straight line that extends infinitely in both directions and contains points C and B.
3.
This is a ray that begins at point E, although it does not contain point E, and extends in the direction of point F.
4.
{\overline {AB}} \cup {\overline {BD}} = \overline {AD}. The union of line segment \overline {AB} and the line segment {\overline {BD}} contains all points in each line segment combined.
5.
\overrightarrow {BD} \cap \overline {BC} = \overline {BC} . The intersection of the ray \overrightarrow {BD} and the line segment {\overline {BC}} contains only the points common to each set, \overline {BC} .
6.
\overleftarrow {BA} \cup \overrightarrow {BD} = \overleftrightarrow {AD}. The union of the ray starting at point B and extending infinitely in the direction of A and the ray starting at point B and extending infinitely in the direction of D is the straight line extending infinitely in both directions containing points A, B, C, and D.
7.
Two lines are parallel if the distance between the lines is constant implying that the lines cannot intersect.
8.
Perpendicular lines intersect forming a {90^ \circ } angle between them.
9.
Yes, because it contains a line and a point not on the line.
10.
straight
11.
obtuse
12.
right
13.
acute
14.
m\measuredangle 1 = {149^ \circ } by supplementary angles with \measuredangle {31^ \circ }.
15.
m\measuredangle 3 = {31^ \circ } by vertical angles with the angle measuring {31^ \circ }.
16.
m\measuredangle 5 = {149^ \circ } by corresponding angles with \measuredangle 1.
17.
x = {89^ \circ }
18.
x = {67^ \circ }
19.
x = {77^ \circ } and .
20.
These are similar triangles, so we can solve using proportions.
\begin{array}{*{20}{rcl}}{\frac{3}{2}}&{ = }&{\frac{{(3 + 4)}}{x}} \\ {3x}&{ = }&{14} \\ {x}&{ = }&{\frac{{14}}{3}} \\ \end{array} \begin{array}{*{20}{rcl}}{\frac{4}{2}}&{ = }&{\frac{{4 + y}}{{\frac{{14}}{3}}}} \\{\frac{{14}}{3}(4)}&{ = }&{2(4 + y)} \\{\frac{{56}}{3}}&{ = }&{8 + 2y} \\{56}&{ = }&{3(8 + 2y)} \\{56}&{ = }&{24 + 6y} \\{32}&{ = }&{6y} \\{\frac{{16}}{3}}&{ = }&{y}\end{array}
Then, x = \frac{{14}}{3} and y = \frac{{16}}{3}.
21.
Set up the proportions.
\begin{array}{*{20}{rcl}}{\frac{6}{a}}&{ = }&{\frac{{12}}{{14}}} \\{6(14)}&{ = }&{12a} \\{84}&{ = }&{12a} \\{7}&{ = }&{a}\end{array}
Thus, t = 20 and a = 7.
22.
pentagon
23.
octagon
24.
heptagon
25.
6(5) = 30\text{ cm}
26.
S = \left( {6 - 2} \right){180^ \circ } = {720^ \circ }
27.
{120^ \circ }
28.
\begin{array}{*{20}{c}}{360}&{=}&{152 + 9x + (5x + 1) + (x + 12)}\\{}&{=}&{15x + 165}\\{195}&{=}&{15x}\\{13}&{=}&{x}\\{x + 12}&{=}&{25^ \circ },\,9x = {117^ \circ },\,5x + 1 = {66^ \circ }\end{array}
29.
C = 2\pi (3) = 6\pi = 18.85\text{ cm}
30.
The patterns are repeated shapes that can be transformed in such a way as to fill the plane with no gaps or overlaps.
31.
Starting with the triangle with the point labeled , the triangle is translated point by point 3 units to the right and 3 units up to point A'. Then, the triangle labeled A' is translated 3 units to the right and 3 units up to point A''.
32.
The triangle is rotated about the rotation point 180^\circ to vertex B'.
33.
The dark triangle is reflected about the vertical line showing the light back, and then reflected about the horizontal line. The pattern is repeated leaving a white diamond between the shapes.
34.
3.3.3.3.3.3
35.
7.5 cm2
36.
25 ft
37.
168\,{\text{c}}{\text{m}^2}
38.
64.5\,{\text{c}}{{\text{m}}^2}
39.
201.1 in2
40.
116.38\,{\text{i}}{{\text{n}}^2}
41.
706.86\,{\text{i}}{{\text{n}}^2}
42.
386.6\,\text{cm}{^2}
43.
416\,\text{in}{^2}
44.
640\,\text{in}{^3}
45.
942.48\,\text{cm}{^2}
46.
1,570\,\text{cm}{^3}
47.
511.35\,\text{in}{^2}
48.
\begin{array}{*{20}{l}} {a = 2.5} \\ {b = 2.5\sqrt 3 } \end{array}
49.
\begin{array}{*{20}{l}} {b = 10} \\ {c = 10\sqrt 2 } \end{array}
50.
b = 6
51.
24.2 ft
52.
c = 11.7{\text{ cm}}
53.
x = 14.3{\text{ m}}
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