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 Contents  v

5 Analytic Summary, Review, and Test 795

Trigonometry 649 Review Exercises 798
Chapter 6 Test 800
5.1 Verifying Trigonometric Identities 650
Cumulative Review Exercises (Chapters P–6) 801
5.2 Sum and Difference Formulas 661
Appendix A: Where Did That Come From?
5.3 Double-Angle, Power-Reducing,
Selected Proofs 803
and Half-Angle Formulas 672
Mid-Chapter Check Point 683 Appendix B: The Transition From Precalculus to Calculus 807
5.4 Product-to-Sum and Sum-to-Product Formulas 684 Answers to Selected Exercises AA1
5.5 Trigonometric Equations 693 Subject Index I1
Summary, Review, and Test 706 Credits C1
Review Exercises 707
Chapter 5 Test 709
Cumulative Review Exercises (Chapters P–5) 709

6 Additional Topics
in Trigonometry 711
6.1 The Law of Sines 712
6.2 The Law of Cosines 724
6.3 Polar Coordinates 733
6.4 Graphs of Polar Equations 745
Mid-Chapter Check Point 756
6.5 Complex Numbers in Polar Form;
DeMoivre’s Theorem 757
6.6 Vectors 770
6.7 The Dot Product 785
 Preface
I’ve written Precalculus Essentials, Fifth Edition, to help • Precalculus develops trigonometry from the perspective
diverse students, with different backgrounds and future of the unit circle (4.2: Trigonometric Functions: The
goals, to succeed. The book has three fundamental goals: Unit Circle). In Algebra and Trigonometry, trigonometry
1. To help students acquire a solid foundation in algebra is developed using right triangles.
and trigonometry, preparing them for other courses such • Many of the liberal arts applications in Algebra and
as calculus, business calculus, and finite mathematics. Trigonometry are replaced by more scientific or higher
2. To show students how algebra and trigonometry can level applications in Precalculus. Some examples:
model and solve authentic real-world problems. • Black Holes in Space (P.2: Exponents and Scientific
3. To enable students to develop problem-solving skills, while Notation)
fostering critical thinking, within an interesting setting. • Average Velocity (1.5: More on Slope)
One major obstacle in the way of achieving these goals • Newton’s Law of Cooling (3.5: Exponential Growth
is the fact that very few students actually read their and Decay; Modeling Data)
textbook. This has been a regular source of frustration
for me and for my colleagues in the classroom. Anecdotal A Note on the Essentials Version of
evidence gathered over years highlights two basic reasons
that students do not take advantage of their textbook: Precalculus
Precalculus Essentials, Fifth Edition is a concise version
• “I’ll never use this information.”
of the Sixth Edition of Precalculus. The essential version
• “I can’t follow the explanations.”
differs from the Sixth Edition only in terms of length.
I’ve written every page of the Fifth Edition with the Chapter 7 (Systems of Equations and Inequalities),
intent of eliminating these two objections. The ideas and Chapter 8 (Matrices and Determinants), Chapter 9 (Conic
tools I’ve used to do so are described for the student in Sections and Analytic Geometry), Chapter 10 (Sequences,
“A Brief Guide to Getting the Most from This Book,” Induction, and Probability), and Chapter 11 (Introduction
which appears at the front of the book. to Calculus) have been eliminated. The essentials version
provides a lighter, less expensive alternative to the Sixth
How Does Precalculus Differ from Edition for instructors who do not cover the topics in
Algebra and Trigonometry? Chapters 7–11.
Precalculus is not simply a condensed version of my
Algebra and Trigonometry book. Precalculus students What’s New in the Fifth Edition?
are different from algebra and trigonometry students, • New Applications and Real-World Data. The Fifth
and this text reflects those differences. Here are a few Edition contains 53 worked-out examples and
examples: exercises based on new data sets, and 31 examples
• Algebra and Trigonometry devotes an entire chapter to and exercises based on data updated from
linear equations, rational equations, quadratic equations, the Fourth Edition. Many of the new
radical equations, linear inequalities, and developing applications involve topics relevant
models involving these equations and inequalities. to college students, including
Precalculus reviews these topics in three sections of the student-loan debt (Chapter P,
prerequisites chapter (P.7: Equations; P.8: Modeling with Mid-Chapter Check Point,
Equations; P.9: Linear Inequalities and Absolute Value Exercise 42), grade inflation
Inequalities). Functions, the core of any precalculus (Exercise Set P.7, Exercises
course, are then introduced in Chapter 1. 137–138), median earnings, by
• Precalculus contains a section on constructing functions final degree earned (Exercise
from verbal descriptions and formulas (1.10: Modeling Set P.8, Exercises 3–4), excuses
with Functions) that is not included in Algebra and for not meeting deadlines
Trigonometry. Modeling skills are applied to situations (Chapter P Summary, Exercise
that students are likely to see in calculus when solving 141), and political orientation
applied problems involving maximum or minimum of college freshmen (Chapter 1
values. Summary, Exercise 67).

vi vi
 Preface  vii

• Brief Reviews. Beginning with Chapter 1, the Brief What Content and Organizational
Review boxes that appear throughout the book
summarize mathematical skills, many of which are
Changes Have Been Made to the
course prerequisites, that students have learned, but Fifth Edition?
which many students need to review. This feature • Section P.1 (Algebraic Expressions, Mathematical
appears whenever a particular skill is first needed and Models, and Real Numbers) follows an example on
eliminates the need for you to reteach that skill. For the cost of attending college (Example 2) with a new
more detail, students are referred to the appropriate Blitzer Bonus, “Is College Worthwhile?”
section and objective in a previous chapter where the • Section P.6 (Rational Expressions) uses the least
topic is fully developed. common denominator to combine rational expressions
• Achieving Success. The Achieving Success boxes, with different denominators, including expressions
appearing at the end of many sections in Chapters P having no common factors in their denominators.
through 6, offer strategies for persistence and success in • Section P.8 (Modeling with Equations) has new
college mathematics courses. examples on modelling options for a toll (Example 3)
and dual investments (Example 5).
• Retaining the Concepts. Beginning with Chapter 1,
Section 1.2, each Exercise Set contains three or four • Section 1.1 (Graphing and Graphing Utilities)
review exercises under the header “Retaining the contains a new example of a graph with more than one
Concepts.” These exercises are intended for students to x-intercept (Example 5(d)).
review previously covered objectives in order to improve • Section 1.3 (More on Functions and Their Graphs)
their understanding of the topics and to help maintain contains a new discussion on graphs with three forms
their mastery of the material. If students are not certain of symmetry (Examples 2 and 3) before presenting
how to solve a review exercise, they can turn to the even and odd functions. A new example (Example 4)
section and worked example given in parentheses at the addresses identifying even or odd functions from graphs.
end of each exercise. The Fifth Edition contains 104 new • Section 1.4 (Linear Functions and Slope) includes a
exercises in the “Retaining the Concepts” category. new Blitzer Bonus, “Slope and Applauding Together.”
• New Blitzer Bonus Videos with Assessment. Many of • Section 1.8 (Inverse Functions) replaces an example on
the Blitzer Bonus features throughout the textbook 5
have been turned into animated videos that are finding the inverse of f(x) = + 4 with an example
x
built into the MyMathLab course. These videos help x + 2
students make visual connections to algebra and on finding the inverse of f(x) = (Example 4),
x - 3
trigonometry and the world around them. Assignable a function with two occurrences of x.
exercises have been created within the MyMathLab • Section 2.6 (Rational Functions and Their Graphs)
course to assess conceptual understanding and opens with a discussion of college students and video
mastery. These videos and exercises can be turned games. This is revisited in a new example (Example 9,
into a media assignment within the Blitzer “Putting the Video-Game Player Inside the Game”)
MyMathLab course. involving the Oculus Rift, a virtual reality headset that
• Updated Learning Guide. Organized by the textbook’s enables users to experience video games as immersive
learning objectives, this updated Learning Guide helps three-dimensional environments.
students make the most of their textbook for test • Section 4.1 (Angles and Radian Measure) has new
preparation. Projects are now included to give students examples involving radians expressed in decimal
an opportunity to discover and reinforce the concepts form, including converting 2.3 radians to degrees
in an active learning environment and are ideal for (Example 3(d)) and finding a coterminal angle for a
group work in class. -10.3 angle (Example 7(d)). Additional Great
• Updated Graphing Calculator Screens. All screens Question! features provide hints for locating terminal
have been updated using the TI-84 Plus C. sides of angles in standard position.
viii  Preface

• Section 4.2 (Trigonometric Functions: The Unit Circle) • Extensive and Varied Exercise Sets. An abundant
has a new Discovery feature on the use of parentheses collection of exercises is included in an Exercise Set
when evaluating trigonometric functions with a at the end of each section. Exercises are organized
graphing calculator, supported by new calculator screens within nine category types: Practice Exercises, Practice
throughout the section. Plus Exercises, Application Exercises, Explaining the
• Chapter 5 opens with a new discussion on trigonometric Concepts, Technology Exercises, Critical Thinking
functions and music. Exercises, Group Exercises, Retaining the Concepts,
and Preview Exercises. This format makes it easy to
What Familiar Features Have Been create well-rounded homework assignments. The order
of the Practice Exercises is exactly the same as the
Retained in the Fifth Edition? order of the section’s worked examples. This parallel
• Learning Objectives. Learning objectives, framed in the order enables students to refer to the titled examples
context of a student question (What am I supposed and their detailed explanations to achieve success
to learn?), are clearly stated at the beginning of each working the Practice Exercises.
section. These objectives help students recognize
• Practice Plus Problems. This category of exercises
and focus on the section’s most important ideas. The
contains more challenging practice problems that often
objectives are restated in the margin at their point
require students to combine several skills or concepts.
of use.
With an average of ten Practice Plus problems per
• Chapter-Opening and Section-Opening Scenarios. Exercise Set, instructors are provided with the option
Every chapter and every section open with a scenario of creating assignments that take Practice Exercises to
presenting a unique application of mathematics in a more challenging level.
students’ lives outside the classroom. These scenarios
are revisited in the course of the chapter or section in • Mid-Chapter Check Points. At approximately the
an example, discussion, or exercise. midway point in each chapter, an integrated set of
Review Exercises allows students to review and
• Innovative Applications. A wide variety of interesting assimilate the skills and concepts they learned
applications, supported by up-to-date, real-world data, separately over several sections.
are included in every section.
• Graphing and Functions. Graphing and functions are
• Detailed Worked-Out Examples. Each example introduced in Chapter 1, with an integrated graphing
is titled, making the purpose of the example clear. functional approach emphasized throughout the book.
Examples are clearly written and provide students with Graphs and functions that model data appear in nearly
detailed step-by-step solutions. No steps are omitted every section and Exercise Set. Examples and exercises
and each step is thoroughly explained to the right of use graphs of functions to explore relationships
the mathematics. between data and to provide ways of visualizing a
• Explanatory Voice Balloons. Voice balloons are problem’s solution. Because functions are the core
used in a variety of ways to demystify mathematics. of this course, students are repeatedly shown how
They translate algebraic and trigonometric ideas functions relate to equations and graphs.
into everyday English, help clarify problem-solving • Integration of Technology Using Graphic and
procedures, present alternative ways of understanding Numerical Approaches to Problems. Side-by-side
concepts, and connect problem solving to concepts features in the technology boxes connect algebraic
students have already learned. and trigonometric solutions to graphic and numerical
• Check Point Examples. Each example is followed approaches to problems. Although the use of graphing
by a similar matched problem, called a Check Point, utilities is optional, students can use the explanatory
offering students the opportunity to test their voice balloons to understand different approaches to
understanding of the example by working a similar problems even if they are not using a graphing utility in
exercise. The answers to the Check Points are the course.
provided in the answer section. • Great Question! This feature presents a variety
• Concept and Vocabulary Checks. This feature offers of study tips in the context of students’ questions.
short-answer exercises, mainly fill-in-the-blank and Answers to questions offer suggestions for problem
true/false items, that assess students’ understanding solving, point out common errors to avoid, and provide
of the definitions and concepts presented in each informal hints and suggestions. As a secondary benefit,
section. The Concept and Vocabulary Checks appear as this feature should help students not to feel anxious or
separate features preceding the Exercise Sets. threatened when asking questions in class.
 Preface  ix

• Chapter Summaries. Each chapter contains a review • Discovery. Discovery boxes, found throughout the
chart that summarizes the definitions and concepts in text, encourage students to further explore algebraic
every section of the chapter. Examples that illustrate and trigonometric concepts. These explorations are
these key concepts are also referenced in the chart. optional and their omission does not interfere with the
• End-of-Chapter Materials. A comprehensive collection continuity of the topic under consideration.
of Review Exercises for each of the chapter’s sections I hope that my passion for teaching, as well as my respect
follows the Summary. This is followed by a Chapter Test for the diversity of students I have taught and learned
that enables students to test their understanding of from over the years, is apparent throughout this new
the material covered in the chapter. Beginning with edition. By connecting algebra and trigonometry to
Chapter 2, each chapter concludes with a comprehensive the whole spectrum of learning, it is my intent to show
collection of mixed Cumulative Review Exercises. students that their world is profoundly mathematical, and
• Blitzer Bonuses. These enrichment essays provide indeed, p is in the sky.
historical, interdisciplinary, and otherwise interesting
connections to the algebra and trigonometry under Robert Blitzer
study, showing students that math is an interesting and
dynamic discipline.
x  Preface

Acknowledgments Angela Heiden, St. Clair Community College

An enormous benefit of authoring a successful series is Donna Helgeson, Johnson County Community College
the broad-based feedback I receive from the students, Celeste Hernandez, Richland College
dedicated users, and reviewers. Every change to this Gregory J. Herring, Cameron University
edition is the result of their thoughtful comments and
suggestions. I would like to express my appreciation Alysmarie Hodges, Eastfield College
to all the reviewers, whose collective insights form the Amanda Hood, Copiah-Lincoln Community College
backbone of this revision. In particular, I would like to Jo Beth Horney, South Plains College
thank the following people for reviewing College Algebra, Heidi Howard, Florida State College at Jacksonville-South
Algebra and Trigonometry, Precalculus, and Trigonometry. Campus
Winfield A. Ihlow, SUNY College at Oswego
Karol Albus, South Plains College
Nancy Raye Johnson, Manatee Community College
Kayoko Yates Barnhill, Clark College
Daniel Kleinfelter, College of the Desert
Timothy Beaver, Isothermal Community College
Sarah Kovacs, Yuba College
Jaromir Becan, University of Texas-San Antonio
Dennine Larue, Fairmont State University
Imad Benjelloun, Delaware Valley College
Mary Leesburg, Manatee Community College
Lloyd Best, Pacific Union College
Christine Heinecke Lehman, Purdue University North Central
David Bramlett, Jackson State University
Alexander Levichev, Boston University
Natasha Brewley-Corbin, Georgia Gwinnett College
Zongzhu Lin, Kansas State University
Denise Brown, Collin College-Spring Creek Campus
Benjamin Marlin, Northwestern Oklahoma State University
David Britz, Raritan Valley Community College
Marilyn Massey, Collin County Community College
Mariana Bujac-Leisz, Cameron University
Yvelyne McCarthy-Germaine, University of New Orleans
Bill Burgin, Gaston College
David McMann, Eastfield College
Jennifer Cabaniss, Central Texas College
Owen Mertens, Missouri State University-Springfield
Jimmy Chang, St. Petersburg College
James Miller, West Virginia University
Teresa Chasing Hawk, University of South Dakota
Martha Nega, Georgia Perimeter College-Decatur
Diana Colt, University of Minnesota-Duluth
Priti Patel, Tarrant County College
Shannon Cornell, Amarillo College
Shahla Peterman, University of Missouri-St. Louis
Wendy Davidson, Georgia Perimeter College-Newton
Debra A. Pharo, Northwestern Michigan College
Donna Densmore, Bossier Parish Community College
Gloria Phoenix, North Carolina Agricultural and Technical
Disa Enegren, Rose State College State University
Keith A. Erickson, Georgia Gwinnett College Katherine Pinzon, Georgia Gwinnett College
Nancy Fisher, University of Alabama David Platt, Front Range Community College
Donna Gerken, Miami Dade College Juha Pohjanpelto, Oregon State University
Cynthia Glickman, Community College of Brooke Quinlan, Hillsborough Community College
Southern Nevada
Janice Rech, University of Nebraska at Omaha
Sudhir Kumar Goel, Valdosta State University
Gary E. Risenhoover, Tarrant County College
Donald Gordon, Manatee Community College
Joseph W. Rody, Arizona State University
David L. Gross, University of Connecticut
Behnaz Rouhani, Georgia Perimeter College-Dunwoody
Jason W. Groves, South Plains College
Judith Salmon, Fitchburg State University
Joel K. Haack, University of Northern Iowa
Michael Schramm, Indian River State College
Jeremy Haefner, University of Colorado
Cynthia Schultz, Illinois Valley Community College
Joyce Hague, University of Wisconsin at River Falls
Pat Shelton, North Carolina Agricultural and Technical
Mike Hall, University of Mississippi State University
Mahshid Hassani, Hillsborough Community College Jed Soifer, Atlantic Cape Community College
Tom Hayes, Montana State University Caroline Spillman, Georgia Perimeter College-Clarkston
Christopher N. Hay-Jahans, University of South Dakota Jonathan Stadler, Capital University
 Preface  xi

Franotis R. Stallworth, Gwinnett Technical College Additional acknowledgments are extended to Dan Miller
John David Stark, Central Alabama Community College and Kelly Barber for preparing the solutions manuals;
Brad Davis for preparing the answer section, serving
Charles Sterner, College of Coastal Georgia
as accuracy checker, and writing the new learning
Chris Stump, Bethel College guide; the codeMantra formatting team for the book’s
Scott Sykes, University of West Georgia brilliant paging; Brian Morris and Kevin Morris at
Richard Townsend, North Carolina Central University Scientific Illustrators for superbly illustrating the book;
Francesca Monaco, project manager; and Kathleen
Pamela Trim, Southwest Tennessee Community College
Manley, production editor, whose collective talents kept
Chris Turner, Arkansas State University every aspect of this complex project moving through its
Richard E. Van Lommel, California State many stages.
University-Sacramento
I would like to thank my editor at Pearson, Dawn Murrin,
Dan Van Peursem, University of South Dakota who, with the assistance of Joseph Colella, guided
Philip Van Veldhuizen, University of Nevada at Reno and coordinated the book from manuscript through
Philip Veer, Johnson County Community College production. Finally, thanks to Peggy Lucas and Claire
Kozar for their innovative marketing efforts and to
Jeffrey Weaver, Baton Rouge Community College
the entire Pearson sales force for their confidence and
Amanda Wheeler, Amarillo College enthusiasm about the book.
David White, The Victoria College
Tracy Wienckowski, University of Buffalo Robert Blitzer
 Get the Most Out of
MyMathLab
®

MyMathLab is the leading online homework, tutorial, and assessment program for
teaching and learning mathematics, built around Pearson’s best-selling content.
MyMathLab helps students and instructors improve results; it provides engaging
experiences and personalized learning for each student so learning can happen in any
environment. Plus, it offers flexible and time-saving course management features to
allow instructors to easily manage their classes while remaining in complete control,
regardless of course format.

Preparedness
MyMathLab course solutions offer a complete College Algebra, Algebra & Trigonometry,
or Precalculus course with integrated review of select topics from developmental
algebra. These courses help remediate students “just-in-time” and help with student
retention of important concepts, ultimately boosting student success.

• Students begin each chapter by completing a Skills Check assignment to pinpoint

which developmental topics, if any, they need to review.
• Students who demonstrate mastery of the review topics will move straight into the
Algebra & Trigonometry content.
• A personalized review homework assignment will provide extra support for the
students who need it.
• Additional review materials (worksheets, videos, and more) are available in an
Integrated Review section at the start of each chapter in MyMathLab.

MyMathLab with Integrated Review are appropriate for students who struggle with
pre-requisite skills and for co-requisite course models. These Integrated Review
MyMathLab courses are available for a variety of College Algebra, Algebra &
Trigonometry, and Precalculus programs, as well as a variety of other disciplines.

Used by more than 37 million students worldwide, MyMathLab delivers

consistent, measurable gains in student learning outcomes, retention, and
subsequent course success.

www.mymathlab.com
xii
 Get the Most Out of
MyMathLab Online Course for®
MyMathLab
Precalculus Essentials
by Robert Blitzer
(access code required)

NEW! Video Program

A fresh, and all new, video program walks through
the concepts from every objective of the text. Many
videos provide an active learning environment
where students try out their newly learned skill.

NEW! Guided Visualizations

These HTML-based, interactive figures help students
visualize the concepts through directed explorations
and purposeful manipulation. They encourage active
learning, critical thinking, and conceptual learning.
They are compatible with iPad and tablet devices.
The Guided Visualizations are located in the
Multimedia Library and can be assigned as
homework with correlating assessment exercises.
Additional Exploratory Exercises are available to
help students think more conceptually about the
figures and provide an excellent framework for group
projects or lecture discussion.

NEW! Workspace Assignments

Students can now show their work like never before!
Workspace Assignments allow students to work through
an exercise step-by-step, and show their mathematical
reasoning as they progress. Students receive immediate
feedback after they complete each step, and helpful
hints and videos offer guidance when they need it. When
accessed via a mobile device, Workspace exercises use
handwriting recognition software that allows students
to naturally write out their answers. Each student’s
work is automatically graded and captured in the
MyMathLab gradebook so instructors can easily pinpoint
exactly where they need to focus their instruction.

www.mymathlab.com
xiii
 Resources for Success
Instructor Resources Student Resources
Additional resources can be downloaded from Additional resources to help student success are
www.mymathlab.com or www.pearsonhighered.com available to be packaged with the Blitzer textbook
or hardcopy resources can be ordered from your sales and MyMathLab access code.
representative.
Objective Level Videos
Annotated Instructor’s Edition An all new video program covers every objective
Shorter answers are on the page beside the of the text and is assignable in MyMathLab. Many
exercises. Longer answers are in the back of the text. videos provide an active learning environment where
students try out their newly learned skill.
Instructor’s Solutions Manual
Fully worked solutions to all textbook exercises. Chapter Test Prep Videos
Students can watch instructors work
PowerPoint® Lecture Slides through step-by-step solutions to
Fully editable lecture slides that correlate to the all the Chapter Test exercises from
textbook. the textbook. These are available in
MyMathLab and on YouTube.
Mini Lecture Notes
Additional examples and helpful teaching tips for Student Solutions Manual
each section. Fully worked solutions to odd-numbered exercises
and available to be packaged with the textbook.
TestGen®
Enables instructors to build, edit, print, and Learning Guide
administer tests using a computerized bank of The note-taking guide begins each chapter with
algorithmic questions developed to cover all the an engaging application, and provides additional
objectives of the text. examples and exercises for students to work through
for a greater conceptual understanding and mastery
of topics.
New to this edition: classroom projects are included
for each chapter providing the opportunity for
collaborative work. The Learning Guide is available
in PDF and customizable Word file formats in
MyMathLab. It can also be packaged with the
textbook and MyMathLab access code.

MathTalk Videos
Engaging videos connect mathematics to
real-life events and interesting applications. These
fun, instructional videos show students that math
is relevant to their daily lives and are assignable
in MyMathLab. Assignable exercises are available in
MyMathLab for these videos to help students apply
valuable information presented in the videos.

www.mymathlab.com
xiv
 To the Student
The bar graph shows some of the qualities that
students say make a great teacher. It was my goal to
incorporate each of these qualities throughout the
pages of this book.
Explains Things Clearly
I understand that your primary purpose in reading
Precalculus Essentials is to acquire a solid understanding
of the required topics in your precalculus course. In
order to achieve this goal, I’ve carefully explained each
topic. Important definitions and procedures are set off in
boxes, and worked-out examples that present solutions
in a step-by-step manner appear in every section. Each
example is followed by a similar matched problem,
called a Check Point, for you to try so that you can
actively participate in the learning process as you read
the book. (Answers to all Check Points appear in the
back of the book.)
Funny & Entertaining
Who says that a precalculus textbook can’t be entertaining? From our unusual cover to the photos in the
chapter and section openers, prepare to expect the unexpected. I hope some of the book’s enrichment essays,
called Blitzer Bonuses, will put a smile on your face from time to time.
Helpful
I designed the book’s features to help you acquire knowledge of algebra and trigonometry, as well as to show you
how algebra and trigonometry can solve authentic problems that apply to your life. These helpful features include:
• Explanatory Voice Balloons: Voice balloons are used in a variety of ways to make math less intimidating.
They translate algebraic and trigonometric language into everyday English, help clarify problem-solving
procedures, present alternative ways of understanding concepts, and connect new concepts to concepts
you have already learned.
• Great Question!: The book’s Great Question! boxes are based on questions students ask in class. The
answers to these questions give suggestions for problem solving, point out common errors to avoid, and
provide informal hints and suggestions.
• Achieving Success: The book’s Achieving Success boxes give you helpful strategies for success in
learning algebra and trigonometry, as well as suggestions that can be applied for achieving your full
academic potential in future college coursework.
• Chapter Summaries: Each chapter contains a review chart that summarizes the definitions and concepts in
every section of the chapter. Examples from the chapter that illustrate these key concepts are also referenced
in the chart. Review these summaries and you’ll know the most important material in the chapter!
Passionate about the Subject
I passionately believe that no other discipline comes close to math in offering a more extensive set of tools for
application and development of your mind. I wrote the book in Point Reyes National Seashore, 40 miles north of
San Francisco. The park consists of 75,000 acres with miles of pristine surf-washed beaches, forested ridges, and
bays bordered by white cliffs. It was my hope to convey the beauty and excitement of mathematics using nature’s
unspoiled beauty as a source of inspiration and creativity. Enjoy the pages that follow as you empower yourself
with the algebra and trigonometry needed to succeed in college, your career, and your life.
Regards,

Bob
Robert Blitzer
xv
 About the Author

Bob Blitzer is a native of Manhattan and received a Bachelor

of Arts degree with dual majors in mathematics and psychology
(minor: English literature) from the City College of New York.
His unusual combination of academic interests led him toward a
Master of Arts in mathematics from the University of Miami and
a doctorate in behavioral sciences from Nova University. Bob’s
love for teaching mathematics was nourished for nearly 30 years
at Miami Dade College, where he received numerous teaching
awards, including Innovator of the Year from the League for
Innovations in the Community College and an endowed chair
based on excellence in the classroom. In addition to Precalculus
Essentials, Bob has written textbooks covering developmental
mathematics, introductory algebra, intermediate algebra,
college algebra, algebra and trigonometry, trigonometry, and
liberal arts mathematics, all published by Pearson. When not secluded in his Northern California writer’s
cabin, Bob can be found hiking the beaches and trails of Point Reyes National Seashore and tending to
the chores required by his beloved entourage of horses, chickens, and irritable roosters.

xvi
 APPLICATIONS INDEX
A Angular speed to fire from two fire stations, 719, 721
Acid rain, 491 of audio records, 531 of jet from control tower, 639
Advertising, sales and price and, 421–422 of carousel, 530 true, of plane, 783–784
African Americans, percentage of, with high of hard drive in computer, 530 between two cities, 646
school diploma, 514 of propeller on wind generator, 644 Beauty, symmetry and, 184
Age(s) Antenna, height on top of building, 646 Bias, Implicit Association Test for, 48, 56–57
arrests and drunk driving as function of, 402 Applause, decibel level of, 203 Bicycles
calories needed to maintain energy by, 86 Archer’s arrow, path of, 326 banking angle, 424
chances of surviving to various, 179 Architecture, angles in, 518 manufacturing, 179, 400
of driver, and rate of fatal crashes, 107–108 Area, of a football field, 116–117 Biorhythms, 517, 535, 549, 597–599
height as function of, 221, 224, 243 Arrests, drunk driving, 402 Bird species population decline, 506
marriage and, 156–158, 302 Aspirin, half-life of, 506 Birth(s)
perceived length of time period and, 425 Atmospheric pressure and altitude, 513 out-of-wedlock, 291
percentage of U.S. population never Attitudes, of college freshmen, 112–113 in U.S. from 2000 through 2009, 246,
married, ages 25-29, 214, 216 Attractiveness, and success, 147 251–252
percent body fat in adults by, 199 Audio records, angular speed and linear speed Birthday, probability of sharing same, 271, 684
preferred age in a mate, 270–271 of, 531 Birthday cake, 48
racial prejudice and, 56–57 Automobiles Black holes, 28–29
and sleep, 160 computing work of pushing, 791, 793 Blood-alcohol concentration, 15–16, 19,
Aging rate, space travel and, 32, 44, 47 depreciation, 179 485–486, 491
AIDS/HIV drunk driving arrests as function of age, 402 Blood volume and body weight, 417–418
cases diagnosed (U.S.), 335–337, 339 fatal crashes, age of driver and, 107–108 Boats/ships
number of Americans living with, 335 leaving city at same time, distance between, bearing of, 633, 639
T cell count and, 162, 171–172 798 changing, 731
Aircraft/airplanes prices of, and average age, 122–123 distance traveled at certain, 639
approaching runway, vector describing, 782 rentals, 126, 129–130, 135–136, 402, 744 to sail into harbor, 639
direction angle of, given speed, 784 required stopping distance, 403, 413–414 direction angle of, 800
distance and angle of elevation of, 611 stopping distances, 403, 413–414 distance from lighthouse, 646, 723
distance flown by, 562 Average cost function, 395–396, 400 distance from pier, 731
ground speed of, 784 Average rate of change, 221–222, 243 ground speed, 800
height of, 578 leaving harbor at same time, distance
leaving airport at same time, distance B between after three hours, 730
between, 723, 727–728, 798 Ball on tilted ramp, vector components of force
linear speed of propeller, 644 angle of elevation and throwing distance on, 789, 793
Mach speed of, 682 of, 709 velocity of, 800
true bearing of, 783–784 thrown upward and outward, 331 velocity vector of, 780
vector describing flight of, 782 Ball (attached to spring) Body fat in adults by age and gender, percent,
velocity vector of, 780 finding amplitude and period of motion 199
wind speed and direction angle exerted on, of, 670 Body-mass index, 424
783–784 simple harmonic motion of, 633–634, Body temperature, variation in, 645
Airports, distance between, 730 705, 709 Box dimensions, 364
Alcohol use Ball (height above ground) Brain
drunk driving arrests, 402 baseball, 516 exercising your, 175
by high school seniors, 159–160 bounce height, 424 growth of the human, 501
moderate wine consumption and heart football, 17, 324–325 modeling brain activity, 611
disease, 215–216 maximum height, 801 Break-even analysis, 648. See also Cost and
number of moderate users in U.S., 514 when thrown from rooftop, 413 revenue functions/break-even points
and risk of accident, 485–486, 491 when thrown from top of Leaning Tower of Breathing cycle, 578
Alligator, tail and body length of, 424 Pisa, 411 modeling, 592–593
Altitude Banking angle and turning radius of velocity of air flow in, 705
atmospheric pressure and, 513 bicycle, 424 Bribery (Corruption Perceptions Index), 178
gained by hiker climbing incline, 645 Baseball Bridge coupon book/toll passes, 260, 612
increase on inclined road of, 562 angle of elevation and throwing distance Budgeting, groceries vs. health care, 226
American Idol, ratings of, 333 of, 705 Building
Amusia (tone deafness), sound quality and, height of ball above ground, 516 height of, 557–558, 638, 639, 646, 647, 723
661, 663 pitcher’s angle of turn to throw ball, 732 shadow cast by, 705
Angle(s) Baseball diamond, distance from pitcher’s Butterflies, symmetry of, 733
in architecture, 518 mound to bases on, 731
clock hands forming, 518, 519 Bearings, 632–633, 647 C
of elevation, 557–559, 561–563, 578, 611, 638, of boat, 633, 639, 731 Cable car, distance covered by, 722–723
639, 645, 647, 648, 721–722 distance at certain, 639, 646 Call of Duty video game, retail sales of, 492

xvii
xviii  Applications Index

Calories, needed to maintain energy balance, 86 Concentration of mixture. See Mixture of ship from lighthouse, 646
Camera problems of stolen car from point directly below
price reductions, 114–115 Cone volume, 423 helicopter, 638
viewing angle for, 627 Continuously compounded interest, 477, 486, traveled by plane, 562
Car(s). See Automobiles 490, 514, 515, 709 between two points on Earth, 533
Carbon-14 dating, 496, 505, 506 Coronary heart disease, 507 between two points on opposite banks of
Carbon dioxide, atmospheric global warming Corruption Perceptions Index, 178 river, 721
and, 149, 210–212, 599, 612 Cost(s). See also Manufacturing costs between two trains leaving station at same
Carousel, linear speed and angular speed of of college education, 2, 4–5, 19 time, 756
animals on, 530, 534 of text messaging plans, 283–285 Distance traveled, combined walking and bus
Cave paintings, prehistoric, 506 of yacht, 120–121 travel, 19
Celebrity earnings, 162–165 Cost and revenue functions/break-even points Diver’s height above water, 413
Cell phones, 180 average, 395–396, 400 Diversity index, 142
Celsius/Fahrenheit temperature bike manufacturing, 400 Diving board motion, modeling, 611
interconversions, 17 customers and revenue, 285–286 Divorce, marriage age and probability of,
Centrifugal force, 422–423 radio manufacturing, 259 156–158
Chaos, patterns of, 711, 757 running shoe manufacturing, 400 DNA, structure of, 550
Chernobyl nuclear power plant accident, 449 virtual reality headset manufacturing, Doctor, visits to, 200
Child mortality, literacy and, 201, 215 395–396 “Don’t ask, don’t tell” policy, 226–227
Children, modeling height of, 459, 465, 466, 487 wheelchair manufacturing, 396 Drivers, age of. See under Age(s)
Cigarette consumption. See Smoking Cove, distance across, 731 Driving rate and time for trip, 419
Circle, finding length of arc on, 646, 691 Crane lifting boulder, computing work of, 793 Drug concentration, 223, 400
Cliff, distance of ship from base of, 638 Crate, computing work of dragging, 800 Drug dosage, child vs. adult, 709
Clock(s) Crime, decrease in violent, 216 Drug use among teenagers, 508
angles formed by hands of, 518, 519 Cycles, 564 Drunk driving, and age as function of
degrees moved by minute hand on, 533 arrests, 402
distance between tip of hour hand and D Dual investments, 19, 115–116, 200
ceiling, 597 Daylight, number of hours of, 535, 549, 595,
distance between tips of hands at 10:00, 732 597, 611, 705 E
minute hand movement in terms of p, 533 Dead Sea Scrolls, carbon-14 dating of, 496 Eagle, height and time in flight, 302
Coffee consumption, sleep and, 516 Death penalty, sentences rendered by Earnings. See Salary(-ies)
College education U.S. juries, 352 Earth
cost of, 2, 4–5, 19 Deaths age of, 24
and success, 121 from 2000 through 2009, 246, 251–252 angular velocity of point on, 534
women vs. men, 146 by snakes, mosquitoes, and snails, 180 distance between two points on, 533
College graduates Debt finding radius of, 640
among people ages 25 and older, in U.S., 450 national, 20, 27–28, 31, 32, 142 motion of Moon relative to, 550
salaries of, in first job after college, 300 student loan, 71 Earthquake
Colleges, percentage of U.S. high school Decay model for carbon-14, 505, 506 epicenter, 281
seniors applying to more than Decibels. See Sound intensity intensity, 512
three, 450 Defense budget, 121 intensity of, 452, 460
College students Deforestation, Amazon, 430 simple harmonic motion from, 636
excuses of, for not meeting assignment Delicate Arch, angle of elevation to determine Eating, lifetime hours spent, 122
deadlines, 144 height of, 563 Education. See also College education
freshmen Depreciation, 179 level of, and earnings, 110–111, 122
attitudes of, 112–113 Depression, exercise and, 229 percentage of U.S. adults completing high
claiming no religious affiliation, Digital camera, price reduction for, 114–115 school, 514
163–164, 166 Dinosaur bones and potassium-40 dating, 506 Electrical resistance, 86, 425
political orientation, 302 Dinosaur footprints, pace angle and stride Elephant’s weight, 491
grade inflation among, 107 indicated by, 724, 730 Elevation, angle of, 557–559, 561–563, 578, 611,
loan debt, 71 Direction, 770–771 638–639, 645, 647, 648, 721–722
College tuition, and student loan debt, 71 Distance Elk population, 516
Comedians, net worth of, 466 across cove, 731 Endangered species, 506
Commuters, toll discount passes, 260, 612 across lake, 558, 561, 645, 730, 731 Equator, linear velocity of point on, 533
Compound interest, 445–449, 451, 486, 489–491 from base to top of Leaning Tower of Equilibrium, forces in, 783
continuously compounded, 477, 486, 490, Pisa, 721 Ethnic diversity, 142
514, 515, 709 between cars leaving city at same time, 798 Exercise
investments, 512, 640 of forest ranger from fire, 639 depression and, 229
Computer(s) of island from coast, 638 heart rate and, 3
angular speed of hard drive in, 530 of marching band from person filming target heart rate ranges for, 18
computer-generated animation, 229 it, 611 of your brain, 175
discounts, 252–253, 260 of oil platform from ends of beach, 721 Exponential decay model, 506, 514, 516
PC vs. tablet sales, 513 between pairs of cities, 281 Exponential growth, 648
price before reduction, 509 of rotating beam of light from point, Expressway speeds and safe distances, 142
prices, 262 610, 611
ratio of students to computers in U.S. public safe, expressway speed and, 142 F
schools, 339 of ship from base of cliff, 638 Fahrenheit/Celsius temperature
sale price, 69 of ship from base of Statue of Liberty, 638 interconversions, 17
 Applications Index   xix

Federal budget Ground speed, 784 Housework, 465

deficit, 142 (See also National debt) Gutter cross-sectional area, 332 Human resources, federal budget expenditures
expenditures on human resources, 401 Guy wire attached to pole, angle made with on, 401
Federal income tax, 180 ground and, 631 Hurricanes, barometric air pressure and, 491
Fencing, maximum area inside, 328, 331–332 Hydrogen ion concentration, 491
Ferris wheel, 281 H
height above ground when riding, 550 Half-life, for radioactive elements, 506, 514 I
linear speed of, 534 Hamachiphobia, 507 Illumination intensity, 424, 425
Films. See Movies Happiness Implicit Association Test, 48, 56–57
Fire average level of, at different times of Income, of highest paid TV celebrities, 162–165
distance of forest ranger from, 639 day, 271 Income taxes, federal, 180
locating, from tower, 712, 719, 721, 756, 800 per capita income and national, 215 Inflation
Flagpole Harmonic motion, simple. See Simple grade, 107
height of, finding, 709 harmonic motion rate of, 450
leaning, angle made with ground, 723 Headset manufacturing costs, 382, 395–396 Influenza. See Flu
on top of building, height of, 639 Health care Inoculation costs for flu, 86
Flu budgeting for, 226 Insulation, rate of heat lost through, 648
epidemic, 497–498 gross domestic product (GDP) spent on, 490 Insurance, pet, 199
inoculation costs, 86 savings needed for expenses during Intelligence quotient (IQ) and mental/
time-temperature scenario, 181–182 retirement, 507 chronological age, 424
vaccine mixture, 179 Heart beats over lifetime, 32 Interfaith marriages, 139
Focal length, 93 Heart disease Investment(s)
Food cost per item, 226 coronary, 507 accumulated value of, 446, 449, 486
Football moderate wine consumption and, 215–216 choosing between, 447
area of field, 116–117 smoking and, 402 compound interest, 445–449, 451, 477, 486,
height above ground, 17, 324–325 Heart rate 489–491, 512, 514, 515, 640, 709
vector describing thrown, 782 exercise and, 3, 18 dual, 19, 115–116, 200
Force(s) of mammals, and life span, 433 and interest rates, 19
on body leaning against wall, 770, 773 before and during panic attack, 351 IQ (intelligence quotient) and mental/
in equilibrium, 783 Heat generated by stove, 425 chronological age, 424
pulling cart up incline, 770 Heat loss of a glass window, 425 IRA. See Individual Retirement Account
resultant, 783, 799, 800 Height Island, distance from coast of, 638
FoxTrot comic strip, 46 of antenna on top of building, 646
Freedom 7 spacecraft flight, 272 of ball above ground (See Ball [height above J
Free-falling object’s position, 410–411, 413 ground]) Jeans, price of, 260
Frequency, length of violin string and, 421 of building, 557–558, 638, 639, 646, 647, 723 Jobs, in U.S. labor force, by gender, 108
Freshmen. See under College students child’s height modeled, 459, 465, 466, 487
Fuel efficiency, 181 diver’s height above water, 413 K
of eagle, in terms of time in flight, 302 Kinetic energy, 425
G on Ferris wheel while riding, 550 Kite, angle made with ground of flying, 631
Garbage, daily per-pound production of, 71 of flagpole, 639, 709
Gasoline as function of age, 221, 224, 243 L
average U.S. price, 351 of leaning wall, finding, 722 Labrador retrievers, color of, 55
gallons of premium sold, 793 maximum, 801 Lake, distance across, 558, 561, 645, 730, 731
gallons of regular sold, 793 of Mt. Rushmore sculpture, 632 Land
Gas pressure, in can, 420 percentage of adult height attained by girl fencing for (See Fencing)
Gay marriage, U.S. public opinion on, 507 of given age, 465, 487 triangular plot, 732, 798
Gay service members discharged from of plane, 578 Landscape design, 117–118
military, 226–227 of tower, finding, 630, 638, 769 Leaning Tower of Pisa, distance from base to
Gender of tree, finding, 756 top of, 721
bachelor’s degrees awarded and, 146 weight and height recommendations/ Leaning wall, finding height of, 722
calories needed to maintain energy by, 86 calculations, 424 Learning rate and amount learned, measuring,
educational level and earnings by, 110–111 High school education, percentage of U.S. 801
first-year U.S. college students claiming no adults completing, 514 Learning theory project, 498
religious affiliation by, 163–164, 166 High school seniors, marijuana/alcohol use by, Length of violin string and frequency, 421
housework and, 465 159–160 Lens, focal length of a, 93
jobs in U.S. labor force by, 108 Hiking trails, finding bearings on, 633 Life expectancy, 214, 433
life expectancy by year of birth and, 214 Hill, magnitude of force required to keep car Light intensity of sunlight beneath ocean’s
percentage of United States population from sliding down, 783 surface, 489
never married, ages 25-29 and, 214, 216 Hispanic Americans, population growth of, 514 Light waves, modeling, 645
percent body fat in adults by, 199 HIV/AIDS Linear speed, 534
wage gap by, 179 cases diagnosed (U.S.), 335–337, 339 of airplane propeller, 644
Global warming, 149, 210–212 number of Americans living with, 335 of animals on carousel, 530, 534
Golden rectangles, 47 T cell count and, 162, 171–172 of wind machine propeller, 531
Grade inflation, 107 Hot-air balloon, distance traveled by Literacy, child mortality and, 201, 215
Gravitational force, 422 ascending, 631, 639, 671 Living alone, number of Americans,
Gravity model, 425 House sales prices, 180 217, 220–221, 306
Groceries, budgeting for, 226 House value, inflation rate and, 450 Loudness, 203, 425, 465, 476, 507, 515
xx  Applications Index

Love, components of, 138 O Uganda, 509

Luggage, volume of carry-on, 378–379 Oculus Rift headset manufacturing costs, 382, United States, 494–495
395–396 age 65 and older, 508–509, 550
M One-person households. See Living alone, by gender, 259
Mach speed of aircraft, 682 number of Americans percentage never married, ages 25–29,
Mailing costs, 199 Oscar-winning films, top ten, 261 214, 216
Mall browsing time and average amount spent, total tax collections and, 31
438, 439 P and walking speed, 501
Mammals, heart rate of, 433 Package, forces exerted on held, 779 world, 146, 260, 492, 502–504, 506, 507, 515
Mandelbrot set, 757, 766, 768, 769 Panic attack, heart rate before and during, 351 Population projections, 46–47, 505
Manufacturing costs. See also Cost and Parthenon at Athens, as golden rectangle, 47 Price(s)
revenue functions/breakeven points Pay. See Salary(-ies) advertising and, 421–422
bicycles, 179 PC (personal computer) sales, 513 computer, 262, 509
virtual reality headsets, 382, 395–396 Per capita income and national happiness, 215 gasoline, 351
wheelchair, 396 Perceived length of time period and age, 425 of a house, 180
Maps, making, 559 Periodic rhythms, 693 jeans, 260
Marijuana use, by high school seniors, Personal computer (PC) sales, 513 ticket, 144
159–160 Pets, insurance for, 199 Price reductions, 114–115, 262
Marital status, and unmarried Americans pH Problem solving, time for, 422
(ages 25-29), 214, 216 of human mouth after eating sugar, 400 Profits
Marriage age pH scale, 490–491 department store branches, 260
of men, 302 Phone calls between cities, 416, 425 maximizing, 381
preferred age in a mate, 270–271 Phonograph records, angular speed and linear Projectiles, paths of, 316. See also Ball (height
and probability of divorce, 156–158 speed of, 531 above ground); Free-falling object’s
Marriage equality, U.S. public opinion on, 507 Physician visits, 200 position
Marriages, interfaith, 139 Pier, finding length of, 722 Propeller
Mass attached to spring, simple harmonic Planes. See Aircraft/airplanes of airplane, linear speed of, 644
motion of, 635–636 Playground, dimensions of, 332 on wind generator, angular speed of, 644
Maximum area, 328, 331–332, 381 Play production, break-even analysis of, 648
Maximum height, 801 Pole, angle made by rope anchoring circus tent R
Maximum product, 331, 381 and, 647 Racial diversity, 142
Maximum profit, 381 Police officers, average salary of, 28 Racial prejudice, Implicit Association Test for,
Median age. See under Age(s) Political identification 48, 56–57
Medication dosage, adult vs. child/infant, 709 college freshmen, 302 Radiation intensity and distance of radiation
Memory retention, 465, 466, 490, 513 Implicit Association Test scores, 57 machine, 424
Mental illness, number of U.S. adults with, 514 Pollution, removal costs for, 72 Radio stations, locating illegal, 721
Merry-go-round Population Radio waves, simple harmonic motion of, 639
linear speed of horse on, 578 Africa, 495 Railway crossing sign, length of arcs formed
polar coordinates of horses on, 743 Asia, 516 by cross on, 533
Miles per gallon, 181 Belgium, 123 Rain gutter cross-sectional area, 332
Military, gay service members discharged bird species in danger of extinction, 506 Ramp
from, 226–227 Bulgaria, 505 computing work of pulling box along, 793
Minimum product, 327 California, 489 force and weight of box being pulled
Miscarriages, by age, 507 Canada, 509 along, 783
Mixture problems, 179 Colombia, 505 magnitude of force required to keep object
Moon, weight of person on, 424 elk, 516 from sliding down, 783
Moth eggs, and abdominal width, 353, 364 exponential growth modeling, 505, 506 vector components of force on boat on
Mt. Rushmore sculpture, height of, 632 Germany, 505, 515 tilted, 789, 793
Mountain, measuring height of, 550, 559, gray wolf, 444–445 wheelchair, angle of elevation of, 639
722–723 Greece, 123 Rate of travel
Movies Hispanic, 514 average rate and time traveled, 179
ticket prices, 144 Hungary, 492 average rate on a round-trip commute, 86
top ten Oscar-winning, 261 India, 449, 505 and time for trip, 419
Movie theater, finding best viewing angle in, Iraq, 505 walking speed and city population, 501
613, 627, 628 Israel, 505 Records, angular speed and linear speed
Music Japan, 505 of, 531
amplitude and frequency of note’s sine Madagascar, 505 Rectangle
wave, 688 Mexico, 506 area of, 47
amusia and, 661, 663 New Zealand, 506 dimensions of, 245, 414
modeling musical sounds, 635, 640 Nigeria, 508 dimensions of, maximizing enclosed area, 328
Pakistan, 505 golden, 47
N Palestinian, 505 perimeter of, 47, 86
National debt, 20, 27–28, 31, 32, 142 Philippines, 505 Rectangular box dimensions, 364
National diversity index, 142 Russia, 505 Rectangular solid, volume of, 57
Navajo sand painting, 549 in scientific notation, 26 Redwood trees, finding height of, 722
Navigation, 550. See also Bearings single, 217–218, 220–221 Reflections, 234
Neurons, human brain vs. gorilla brain, 71 Texas, 489 Relativity theory, space exploration and,
Newton’s Law of Cooling, 509 tigers, worldwide, 350 32, 44, 47
 Applications Index   xxi

Religious affiliation, first-year U.S. college from touching buttons on touch-tone phone, Telephone(s)
students claiming no, 163–164, 166 684, 690 calls between cities, 416, 425
Rentals, car, 126, 129–130, 135–136, 402, 744 Sound intensity, 203, 425, 465, 476, 507, 515, 688 sound from touching buttons on,
Resistance, electrical, 86, 425 Sound quality, amusia and, 661, 663 684, 690
Resultant forces, 783, 784, 799, 800 Space flight/travel Telephone plans
Revenue functions. See Cost and revenue aging rate and, 32, 44, 47 cellular, 180
functions/break-even points Freedom 7 spacecraft, 272 per-minute costs, 191–192, 198
Reversibility of thought, 58 relativity theory and, 32, 44, 47 Telephone pole
Roof of A-frame cabin, finding length of, 798 Speed. See also Rate of travel angle between guy wire and, 562
Rotating beam of light, distance from point, angular, 530, 644 tilted, finding length of, 722
610, 611 linear, 534 Television
Runner, speed of, 291 of airplane propeller, 644 lifetime hours spent watching, 122
Runner’s pulse, 491 of animals on carousel, 530, 534 sale prices, 69
of wind machine propeller, 531 Temperature
S Mach speed of aircraft, 682 average monthly, 597, 598
Sailing angle to 10-knot wind, sailing speed Speed skating, winning time for, 306 body, variation in, 645
and, 743, 754 Spring(s) of cooling cup of coffee, 512
Salary(-ies) force required to stretch, 424 and depth of water, 424
earnings with overtime, 516 simple harmonic motion of object attached in enclosed vehicle, increase in, 461–462
educational level and, 110–111, 122 to, 633–636 Fahrenheit-Celsius interconversions, 17
first job after college, 300 ball, 633–634, 705, 709 global warming, 149, 210–212
increases in average, 304 distance from rest position, 637, 646 home temperature as function of
police officers, average, 28 frequency of, 637 time, 244
wage gap in, by gender, 179 maximum displacement of, 637 increase in an enclosed vehicle, 507
Sale prices, 69. See also Price reductions phase shift of motion, 637 as magnitude, 770
Sales volume/figures time required for one cycle, 637 Newton’s Law of Cooling, 509
PCs vs. tablets, 513 Statue of Liberty, distance of ship from base time-temperature flu scenario, 181–182
price/advertising and, 421–422 of, 638 Text messaging plans, modeling costs of,
video games, 492 Stomach acid, pH of, 491 283–285
Sand painting, 549 Stonehenge, raising stones of, 562 Throwing distance, 672, 682
Savings Stopping distances angle of elevation of, 705, 709
and compound interest, 489–490 for cars, 403, 413–414 maximum height of thrown ball, 801
needed for health-care expenses during for trucks, 414 shot put, 682, 722
retirement, 507 Stress levels, 329 Ticket prices/sales, 144
Shaded region areas, 57, 69 String length and frequency, 421 Tides, behavior of, 535, 544, 549
Shadow, length of, 782 Strontium-90, 497 modeling cycle of, 594
Shipping costs, 302. See also Mailing costs Student loan debt, 71 modeling water depth and, 597
Ships. See Boats/ships Success, attractiveness and, 147 Tigers, worldwide population, 350
Shot put Sun, finding angle of elevation of, 559, 561–562, Time, perceived length of, 425
angle and height of, 331 578, 639, 645 Time traveled, average rate and, 179
throwing distance, 682, 722 Sunlight, intensity beneath ocean’s Tobacco use. See Smoking
Shower, water used when taking, 418 surface, 489 Tolls, 113–114, 260, 612
Simple harmonic motion, 801 Sunscreen, exposure time without burning Touch-tone phone, sounds from touching
ball attached to spring, 633–634, 705, 709 and, 2 buttons on, 684, 690
earthquake, 636 Supply-side economics, 365 Tower
modeling, 633–636, 639, 646, 647 Surface sunlight, intensity beneath ocean’s angle of elevation between point on ground
radio waves, 639 surface, 489 and top of, 648, 769
tuning fork, 639 Surveying height of, finding, 630, 638, 639, 769
Skeletons, carbon-14 dating of, 506 bearings in, 632–633 length of two guy wires anchoring, 731
Skydiver’s fall, 418–419 to find distance between two points on Trains leaving station at same time, distance
Sled, pulling opposite banks of river, 721 between, 756
computing work of, 792 Sushi, population who won’t try, 507 Tree(s)
forces exerted, 782 Synthesizers, musical sounds modeled by, Amazon deforestation, 430
Sleep 629, 635 annual yield of fruit/nut, 306, 332, 430
and age, 160 finding height of, 756
coffee consumption and, 516 T Triangle
hours of, lifetime, 122 Tablet sales, 513 area of, 718, 728
Smoking Target heart rate for exercise, 18 oblique, 718
deaths and disease incidence ratios, Task mastery, 476, 513 Triangular piece of land
400–401 Tax code, U.S., increase in number of pages cost of, 732, 798
and heart disease, 402 in, 451 length of sides of, 798
Snow, water supply from melting, 432 Taxes Trucks, stopping distances required for, 414
Soda consumption, modeling, 282–283 federal tax rate schedule for tax owed, 199 Tugboats towing ship, resultant force of two,
Soft drink can, surface area of, 289–290 government spending and, 31 783, 784
Solar energy industry, number of U.S. jobs income, 180 Tuning fork
in, 515 tax rate percentage and revenue, 365 eardrum vibrations from, 671
Sound U.S. population and total tax collections, 31 simple harmonic motion on, 639
amplitude and frequency of, 688 Teenage drug use, 508 TV. See Television
xxii  Applications Index

V Water Wind force, 425

Vaccine, mixture for flu, 179 pressure and depth, 416–417 Wind generator
Velocity vector temperature and depth, 424 angular speed of propeller on, 644
of boat, 780 used in a shower, 418 linear speed of propeller of, 531
of plane, 780 Water pipe diameter, number of houses served Wind pressure, 425
of wind, 779, 780, 783–784 and size of, 424 Wine consumption, heart disease and, 215–216
Video games, retail sales of, 492 Water supply, from melting snow, 432 Wing span of jet fighter, finding, 723
Videos rented, number of one-day and Water wheel, linear speed of, 534 Women. See also Gender
three-day, 793 Weight average level of happiness at different times
Violent crime, decrease in, 216 blood volume and body, 417–418 of day, 271
Violin string, length and frequency of, 421 of elephant, 491 and housework, 465
Virtual reality headset manufacturing costs, of great white shark, cube of its length Work, 791–793
382, 395–396 and, 419 crane lifting boulder, 793
Volume and height recommendations/ dragging crate, 800
of carry-on luggage, 378–379 calculations, 424 pulling box up ramp, 793
of sound (See Sound intensity) of person on Moon, 424 pulling wagon, 791, 793, 800
Weightlifting, 508, 785, 794 pushing car, 791, 793
W Wheelchair business, manufacturing costs of weightlifter, 785, 794
Wage gap, 179 in, 396
Wages. See Salary(-ies) Wheelchair ramp, angle of elevation X
Wagon, computing work of pulling, 791, 793, 800 of, 639 Xanax, half-life of, 506
Walking speed and city population, 501 Wheel rotation, centimeters moved
Washington Monument, angle of elevation to with, 533 Y
top of, 562 Wind, velocity vector of, 779, 780, 783–784 Yacht, dividing cost of a, 120–121
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while decorations have been simply raining down on the Navy and
the Schutztruppe, Bwana Linda still walks among mortals without a
single order. He is, however, a philosopher as well as a hero.
The song runs as follows:—
Ulendo wa Linda (er); pa kwenda ku Masasi na gumiri chikuo: mkasálile
mbwana mkubwa ngondo jaiche nand autwiche lunga yangadye. Mkasálile akida
Matora: ngondo jaiche na gombel(r)e lilōmbe. Tukujir(l)a Masasi; Mwera kupita
mchikasa mpaka pe Lindi. Ne wapere rukhsa. Yendeye ku mangwenu; mkapānde
mapemba.
The translation is as follows:—
“The journey of Linder, when he went to Masasi, and I shouted
with a shouting.—‘Tell the Bwana Mkubwa, war has come, and I ran
away without looking back. Tell the akida Matora, (that) the war has
come, and I have beaten the great-drum.’ Then we went to Masasi,
the Wamwera are beaten and go as far as Lindi, and they get
permission. ‘Go to your homes, and plant Mapemba (sorghum).’”
This is delivered in very quick recitative, and relates in a few words
the history of the whole campaign, of course making the singer the
central point. Mr. Linder comes to Masasi in the course of one of his
official tours, his principal duty being to ascertain whether the local
headmen have cultivated the various crops prescribed by
government. There the loyal Likoswe of course hastens to him and
warns him of impending hostilities on the part of the Wamwera.
Linder in his turn sends word to the District Commissioner at Lindi,
and at the same time despatches Likoswe with an urgent message to
Matola’s. Likoswe, on arriving, beats the war-drum (lilombe),
Matola’s warriors immediately hasten to the spot, six hundred men
with guns and many more with spears, bows and arrows, and the
chief marches on Masasi, to take the Wamwera in the rear. It is
related as a fact that Seliman Mamba and his subordinates had each,
at the beginning of the rising when their hopes were highest and they
already saw the Germans driven into the sea, fixed on a house at
Lindi with all its contents as his own share of the spoil. Possibly, the
line about the enemy’s going back to Lindi refers to these unrealised
plans. Matola, I believe, lost about forty men in fighting the rebels,
but certainly did not drive them back to Lindi. The last sentence
relates to the conclusion of peace:—the vanquished are pardoned,
and directed to go home quietly and plant their gardens once more.

YAO DANCE AT CHINGULUNGULU

My cinematograph, too, has been several times in requisition

during my stay at Chingulungulu, as I have found opportunity to take
a whole series of dances of the Wayao and Makua. The latter, it is
well known, are the hunting-tribe par excellence of the east—indeed
professional hunters of any tribe are generally described as Makua.
They are, moreover, typical for all other tribes in their method of
hunting, and in all appliances and customs connected therewith. One
day, by Matola’s orders, a troop appeared at Chingulungulu to
perform, as they said, the makwaru—a dance entirely based on the
details of the hunter’s life. I had quickly got my apparatus arranged
in a suitable place, not an easy matter here in the loose alluvial soil,
as, if one presses too hard on the legs of the tripod, they are apt to
sink into the sand up to their whole height. Grown wise by
experience, I now take the precaution of driving a wooden wedge
obliquely from above under each leg before beginning operations. It
is more difficult to remedy the results of a mistaken economy. In
order to save the African Fund about twelve shillings and a quarter of
a carrier, I did not bring the heavy stand necessary for the Ernemann
cinematograph, thinking that I could use my ordinary camera-stand.
This, though excellent of its kind, is far too light to stand the
continual jerks of the cinematograph, and I have to balance matters
by hanging a heavy stone or one of my packing-cases under it. If
matters become very serious one of the carriers has to sacrifice
himself and do duty as a tripod-holder. Everything being now ready
for the makwaru, the same band which figured at Sulila’s and
Likoswe’s performances takes its place. It consists of six or seven
men and youths, squatting before their long white logs with their
drumsticks in hand. Suddenly, a fantastically decorated something
flashes into the circle, moving so rapidly that it is impossible to
distinguish whether it is a man or a woman. Being compelled to
pause for breath it is revealed as a middle-aged man in a kilt of long
green leaves resembling a ballet-dancer’s skirt. The man scarcely
stirs from the spot, but his skirt flies in the wind, and he works his
feet in quick, regular time, while at the same time his arms move in a
manner difficult to describe, as there is nothing in European dancing
which in the least degree corresponds to it; and both, arms and legs,
keep exact time with the band. Whether the rest of the body in its
incessant motion backwards and forwards also keeps time it is
impossible to decide, as the vibrations are too rapid to let the eye
make out the details. This stage lasts so long that I am tempted to
regret the waste of my precious film.
 “BUSH SCHOOL” IN THE PORI, NEAR CHINGULUNGULU

At last the hunter changes his tactics. The dancer is, in fact, a
hunter, and not only that, but a very successful elephant-hunter; and
having just killed a large elephant, he is celebrating this deed of
prowess before the assembled inhabitants of his native village, just as
he does after his return from the actual hunt. Here, too, the people
have collected from far and near to see this celebrity, and to admire
his skill in the dance. His performance becomes more and more
vivacious—he no longer remains on one spot but trips forward, first
in a straight line, then in a zig-zag. At last he revolves in a circle,
moving round with short, cautious jumps, and all the time keeping
up the movements of his arms and hips without a moment’s
intermission. After one more rapid trip round the circle and a frantic
vibration of the whole body, the dancer stands still, breathing deeply.
This kind of dance is too peculiar, too divergent from all European
standards for us to judge of it critically according to the rules of art. I
had expected a pantomimic representation of an elephant-hunt, or at
least of the stalking and killing of the game, and I must confess that I
can find nothing in the performance which seems to have any such
reference, and must confine myself to admiring the incredible
dexterity shown by this acrobat in setting all his muscles a-quiver. I
have no sooner got a fresh film ready, than a second dancer has
appeared on the scene, whose action is still more curious and
perplexing. At first one sees nothing but a confused mass of green
leaves rolling and writhing on the ground in convulsive motions.
After a while, this resolves itself into a man much like the previous
one, except that his costume is much more voluminous. He quivers
in a masterly manner and shows as much staying power as his
predecessor; but his chief strength lies in his legs, whose suppleness
and power of assuming the most grotesque attitudes are nothing
short of marvellous. When he has exhausted his repertoire and made
way for a third performer, we at last get the expected pantomime.
Stooping as if for a spring, the hunter creeps up, noiselessly, making
use of every bit of cover, to stalk the elephant, whose scent is
exceedingly keen. At last the goal is reached—swiftly, but as
noiselessly as the hunter, the quarry, represented by another man,
has slipped into the arena, and squatted down, and the hunter circles
round him in diminishing spirals. We expect the deadly shot, but it
does not come off, and the third dancer, quite regardless of the
elephant he is supposed to represent, begins to “triumph” in
precisely the same way as the two others, practising highly artistic
short steps, swaying his hips and flourishing his arms. “Bassi”—
(finished,) I exclaim, as the last of my three films whizzes off the reel.
Quite in contrast to these are the typical unyago dances of the
Wayao. There seems to be a great variety of these; but so far I have
only seen two at Chingulungulu, a masewe, so called from the rattles
worn, as already mentioned, on the legs and feet, and a luwanja.
Both are essentially the same in character. The primitive xylophone
of the Makua hunting-dance is here replaced by a complete band of
drums, of the most various shapes and sizes. A certain musical
faculty inherent in the race is evidenced by the fact that the
musicians take care to tune up before the dance begins. Each beats
his own drum, listening carefully to hear whether it is in tune with
the rest, and if not, hurries away to the nearest hut and comes back
with a brand from the hearth and a large bundle of dry grass. The
grass is heaped on the ground and set on fire, and then every drum is
held with the open end over it, for a longer or shorter time—some for
a few seconds only, some for half a minute or more—the pitch being
tested by striking from time to time. At last all the skins are
sufficiently tense and the drumming begins.
 A YAO DRESSED FOR
THE MASEWE DANCE

MASEWE DANCE OF THE YAOS AT MTUA

At the same moment a dense cloud of dust is seen approaching

with lightning speed, and discloses a seemingly endless procession of
men, youths and boys, all decked in bundles of masewe at the ankle
and above the knee, and a kilt of leaves and strips of skin round the
waist. They take their places in the arena in front of the band, and
immediately fall into position and trot along in Indian file, till the
line closes up into a circle and moves round to the left, then round to
the right, and so on. It is astonishing how uniformly and accurately
the movements are executed by every individual performer, even the
youngest boys. There is nothing very exciting about this dance; in
fact, I find all native dances monotonous, perhaps owing to the
prevailing character of the continent, which is very uninteresting,
except in a few favoured spots. Perhaps a native critic, however,
might object that there is no great variety in our waltzes or polkas.
Just as these reflections were passing through my mind, the scene
changed, somewhat to its advantage, and the circle broke up into
groups which vied with each other in the most remarkable leg-
movements. These, in fact, seem to be the strong point of all these
dancers. One group floated along on tip-toe, another imitated the
dignified gait of some kind of wading-bird, yet another swayed
merrily in and out between the rest, and a fourth stalked along with
legs held perfectly stiff. Long after my last film was finished the
company were still disporting themselves, unable to leave off, but at
last this “turn,” too, came to an end; the band produced only horrible
discords; I was tired out with standing; Knudsen complained of the
first symptoms of fever, and the function was over.
The performance of dances like the one just described, which is
connected with the circumcision rite, have naturally increased my
interest in this tribal festival, and my desire to see and study it as
closely as possible.
My curiosity was increased by the two following incidents. One
afternoon I was strolling through the bush in the neighbourhood of
Chingulungulu; we had already obtained some interesting
photographs of graves, had studied the exterior and interior of some
outlying homesteads, and were about to take some views of the pori
showing the character of the vegetation. After straggling in Indian
file through the high grass and the underwood, which was here
exceptionally dense, we came to a little circular clearing, perhaps
from fifteen to twenty yards in diameter, and studded with a few
scattered bushes. The unique feature of the place was two concentric
circles of stumps having another stump in the centre. These stumps
were about a foot high, cut off with a perfectly smooth horizontal
surface, and excellently well adapted for seats. I took a photograph of
this remarkable object without loss of time, and, on my return to
camp, made inquiries of Matola and others as to its meaning. I found
that the stumps were seats for the wari, as the boys under initiation
are called after a certain point in the ceremony, and the seat in the
middle was that reserved for the instructor who has charge of the
boys during the months which they have to spend in a hut built for
the purpose in the bush. My informants added that the hut had stood
close to the circle, but was no longer in existence, as the unyago for
which it had been built had taken place some years ago.
Some days later, Knudsen and I were sitting under our baraza in
the early part of the afternoon, pressing our hands to our temples. It
was no wonder that every day about this time we both suffered from
excruciating headaches, for the temperature had been steadily rising
during the last few weeks, and on this particular afternoon the
thermometer stood at 93·36°F. We had given vent to our disgust at
the Dark Continent in the strongest of language, and I was just about
to soothe our ruffled feelings with a cigar apiece, when we saw two
black figures approaching. These proved to be Akundonde, the wise
old Yao chief, and his councillor, Akumapanje. We had sent to ask
Akundonde to find us some men capable of giving accurate
information, and now he came himself, though far from well. He was
suffering from the usual neglected ulcer on the leg, and could only
limp along painfully with the help of his staff, so that his taking a
four hours’ walk to oblige us shows a degree of goodwill deserving
the amplest recognition.
Akundonde being established in Knudsen’s long chair, while his
companion took a seat on a packing-case, I made an effort to divert
the conversation from the trifles which at first threatened to engross
it to the subjects which chiefly interest me, and succeeded, more by
luck than good guidance. As usually happens, we were soon
discussing the most recondite matters, such as the attitude of the
natives towards eclipses, the fall of meteorites, and the moon.
Meteorites are considered by the Yaos as of evil omen. When they are
heard to explode, people say, “Either a great chief will die this year,
or a great multitude of the people will perish.” An eclipse of the
moon is thought, as among all primitive people, to be a personal
encounter between two foes. The enemy of the moon is, of course,
the sun; they seize each other fiercely and wrestle together. As both
are equally matched, the battle remains undecided, which forces
mankind to interfere. The Wayao run in haste to fetch hoes and axes,
and strike them against each other, looking up at the scene of strife
and calling out:—
“Mlekangane, mlekangane, mwesi na lyuwa, mkamulene,
Mlekangane, mlekangane sambano.”
“Go asunder, go asunder, sun and moon, you have seized one
another. Go asunder, go asunder now.”
The same custom is observed in eclipses of the sun, as is only
logical.
The full moon with her pale light exercises the same magical
influence on the native mind as on the feelings of every other mortal,
except that our black brother is not like us filled with emotional
enthusiasm, but, quite in conformity with his views on other matters,
makes use of this favourable opportunity for heightening the virtue
of his medicines and charms. When the moon is at the full, the native
goes to the nearest cross-roads, or to a place where two paths meet,
carrying with him a sufficient quantity of a certain gum called ubani.
In perfect silence he then kindles a fire by means of the primitive
appliance of the drill (to be described later on). The dust produced by
boring catches fire, but the glimmer is at first so faint that it is
scarcely perceptible even to the keen eyesight of the savage. Very
carefully he blows on the tiny spark—it grows, catches the bunch of
dry grass and then the sticks, and when the flame leaps up, he drops
his powder into it. The flame now burns dimly, a thick smoke rises,
and the man takes the amulets he is accustomed to wear round his
neck, arms and waist, and holding them in the smoke, says: “You
moon, a little while ago you were not there, and the sky was dark.
Now you are there and shine down brightly. All beasts and plants are
glad and have new strength, so let my medicine also have new
strength.” Then he prays thus: “Let the medicine protect my body
against lions and serpents, against witchcraft and everything that
may hurt me, and let my body have new strength.” Once more he
swings his charms through the smoke, as it becomes thinner and
more transparent; the fire dies down, and as noiselessly as he came
the man creeps back to his hut.

FRESCO ON THE WALL OF A HUT AT AKUNDONDE’S,

REPRESENTING TWO EUROPEANS WITH THEIR ESCORT:
THE WORK OF A YAO BOY

Being now on the subject of magic, the three ethnographic

specialists, Knudsen, Akundonde and Akumapanje, keep to it, and
speak of the tying of knots. Akundonde relates how a man in this
country, if he has designs on any particular girl, takes a strip of bark,
makes a knot in it, without drawing it tight, and says to it, “You tree,
your name is sangalasa (joy)—you are to fetch me that girl, and as a
sign that it shall come to pass, I shut my words up in you.” He then
holds the open knot in front of his mouth, puts his tongue through it
and draws it tight. He afterwards wears the knotted piece of bark-
string tied round his wrist. This proceeding, though simple enough,
is connected with a long and important chapter in racial psychology.
The tying of a knot in fact, in many strata of mankind, has an occult
meaning; the binding power of the knot is supposed to be transferred
to certain persons, and, so long as the knot itself cannot be untied,
those persons are indissolubly attached to him or her who has tied it
according to certain rules and with the proper ceremonies.
Interesting as these matters were, and glad as I should have been
to know more of them, I was just now still more eager to hear about
the much-discussed unyago. I brought up the subject, but both
natives cleverly evaded it. After a while, I noticed the old chief’s eye
roaming wistfully about our study, saw that he was tired and thirsty,
and remembered that Daudi, the native clergyman, had sent us a
large pot of pombe whose quality precluded our drinking it
ourselves. “I suppose it will be quite good enough for these two old
sinners,” I remarked to Knudsen, who must have been revolving
similar cogitations; for he at once seized the import of my words,
fetched a huge tin mug from his tent, filled it with the yellow,
fermenting liquor, and handed it to Akundonde. The latter took it,
but did not drink, handing it to his companion instead. “There’s a
polite chief for you!” I thought to myself—but, seeing how very
cautiously Akumapanje touched the beer with his lips, it became
clear to me that I was witnessing an ancient traditional custom,
arising from the innate suspiciousness of the negro, who scents—not
indeed poison, but certainly witchcraft—everywhere, and dreads it
accordingly. The precaution is intended to divert the risk from the
superior to the subordinate.
Akumapanje, after tasting, handed the cup back to Akundonde,
who thereupon emptied it at a draught. A few seconds later it was
again at the lips of the prime minister, who faithfully copied his
master. Drink and counter-drink succeeded each other at the same
rapid rate, and we Europeans looked on with mixed feelings of envy
and admiration. This did not prevent me from remembering our
ethnographical purpose, and I found that what had previously
seemed impossible was now child’s play. The two old men, by turns
completing each other’s statements, gave a fluent description of the
general features of the boys’ unyago: the arrangement for holding
the festival at different villages every year (which was not new to
me); the introductory ceremony, held in an open square surrounded
by the huts erected for the candidates; and the operation itself, which
takes place in a special hut in the depths of the forest. I had heard
something of all this from Knudsen, who, in the course of his many
years’ residence among the Wayao, has acquired a wonderful
knowledge of their life and customs, and whom I have been pumping
at every spare minute with such persistency that the good fellow has
no doubt often wished one of us elsewhere.
At last, however, our two visitors, becoming more loquacious as
the pombe diminished, reached a part of the subject of which
Knudsen knows very little, but which attracts me most of all. This is
the instruction given to the boys during the months spent in the bush
by their teachers (anamungwi). These instructors, of whom every
boy has one from the time of his initiation into manhood, are
indisputably one of the most sympathetic features in the life of the
people. They watch over their pupils through the painful weeks of the
unyago, teach them what is fitting and unfitting, and remain
responsible for their welfare even after they have left their boyhood
far behind. I was anxious, above all, to ascertain the gist of the moral
teaching given in the bush hut, and, though I only partly succeeded
in doing this, it is a great satisfaction to have taken down verbatim a
fragment of a speech delivered on such an occasion.
Some extra well-filled cups having removed the last scruples of our
two jovial informants, Akundonde, with a little more encouragement
from Knudsen, began in a didactic tone:—
“Mwe mari, sambano mumbēle. Atati na achikuluwēno
mnyōgopĕ́. Nyumba mkasayinjila tinyisimana chimtumbánăgá.
Wakongwe mkasayogopa; mkagononawo, mesi akayasináwo.
Imālagắ akamtikĭté; imālagắ akamila muchisiḗ; masakam.
Munyitikisie: marhaba. Mkuona mwesi sumyógopé, ngakawa
kuulala. Kusimana timchiŭá; Miasi jili kogoya. Chilwele winyi.”[34]
The translation is as follows:—“You, my pupil, now you are
initiated. Your father and your mother, fear (respect) them. See that
you do not enter the house (unannounced), lest you should find them
embracing. Do not be afraid of women, but sleep with them, bathe
with them, when you have finished let her rub (knead) you; when
you have finished she should salute you (saying) ‘Masakam,’ and you
must answer, ‘Marhaba.’ You must be afraid (= take care) when you
see the (new) moon, you might get hurt. Beware of women during
their courses, this is dangerous, (it causes) many diseases.”
My notes were scarcely as complete and connected as the above
when first written down. The native is incapable even when sober of
taking his sentences to pieces, as it were, and dictating them bit by
bit; but taking down the words of these two jovial old sinners was a
difficult task, which, however, we accomplished successfully up to
the point when the inevitable catastrophe set in.
The two had invariably paused for refreshment at the end of every
sentence till they reached the point above indicated, when they
suddenly found the pombe jar empty. They had drunk at least five
gallons at a sitting, but with the strange logic of the intoxicated, they
considered themselves entitled to a further supply, and, when none
proved to be forthcoming, they indignantly broke off their lecture
and left in a huff. This is the reward of being hospitable overmuch.
The address here reproduced, which I have translated with the
help of Knudsen, Daudi, Matola and some others, is said to be the
same, both as to matter and form, at all unyago ceremonies. No
doubt this is correct, for I know nothing which could more exactly
express the feelings of the native than just these precepts. They are a
strange mixture of hygienic rules and moral instruction, and at the
same time contain a good deal of primitive tradition which still forms
part of daily life. I mean by this the fact that the youth, once
recognised as a member of the adult community, is forbidden to
enter his mother’s house unannounced. Here, in East Africa, we are
still in the matriarchal stage, where the husband is nothing, so to
speak, but a connection by marriage. He is his children’s father, but
is not related to them, in fact he belongs to a different clan. This clan,
as so often happens among primitive peoples, is exogamous—that is
to say that there is no impediment to a young man marrying a girl of
any clan but his own. This prohibition goes so far that the young Yao
has, as far as possible, to avoid his nearest female relations who, of
course, are his mother and sisters, and hence the injunction at least
to give warning of his approach when entering his mother’s house.
The stress here as elsewhere laid on the reverence to be shown to
father and mother must strike all right-thinking Europeans as a very
pleasing trait. Respect for parents and for grown-up people in
general is, as I have been told over and over again, the principal and
fundamental feature in native education, and Knudsen testifies that
the young people in general observe it in a marked degree in their
intercourse with their elders. We Europeans might well learn from
the natives in this respect, thinks Nils, who is no doubt, well qualified
to form an opinion.
But, in spite of all pleasant impressions as to native educational
maxims, I have lost the end of the unyago address—a misfortune for
which the good Daudi’s big pombe-jar is to blame. If the mountain
will not come to Muhammad, Muhammad will have to go to the
mountain. In other words, Akundonde having declared that he must
go home to put fresh dawa on his leg and cannot possibly come
again, we shall have to look up the old gentleman at his own
residence.
HERD OF ELEPHANTS. FROM A DRAWING BY BARNABAS,
AN EDUCATED MWERA AT LINDI
 CHAPTER X
FURTHER RESULTS

Chingulungulu, August 31, 1906.

I am still at Chingulungulu, cursing the infernal heat, horrible dust

and dirty natives with more fervour than ever, but unable to get away
from them. The reason for this is the fact that while at first my stay
here seemed utterly barren of scientific results, this state of things
gradually reversed itself, so that the difficulty now lay in dealing with
the mass of new impressions and observations. It is impossible to
relate in full detail the exact way in which I obtained an insight into
native customs and ideas—this would fill several volumes, and my
time is limited. I shall therefore content myself with a few personal
touches and a small selection from the various departments of the
material and mental life of the tribes inhabiting this vast plain.
The most important incident affecting my expedition was the
engagement of Nils Knudsen as a permanent member of its staff,
subject, of course, to the consent of the Agricultural Committee. I
fancy the arrangement is satisfactory to both parties. As I have
already remarked, Knudsen is in the service of the Lindi
Municipality, as master of the Industrial School. At the request of the
District Commissioner, he had been granted leave of absence to
make a tour through the plain west of the Makonde Plateau and
exercise a sort of supervision over the village headmen. For reasons
of which I am not called on to judge, the plan of appointing such
European inspectors has been given up again, and, as the Lindi
municipality naturally saw no occasion to let their industrial teacher
travel about the country for his own amusement, he was recalled. I
must honestly confess that I had long found Knudsen quite
indispensable, and therefore took the opportunity of applying to the
District Commissioner for permission to engage him, when the
latter, a few days ago, visited us on one of his official tours. He has
seemed ever since to enjoy an increased sense of his own importance
and, in fact, the task of initiating a German scholar into the deepest
secrets of alien life is no doubt a far pleasanter one than that of
teaching lazy native boys to plane, saw, forge and solder.
The second incident is a severe attack of fever, with which I have
been laid up during the last few days. I was just about to photograph
the old Sudanese sergeant who had come up with Ewerbeck, and who
was chiefly remarkable for a cough which kept everyone awake at
night. When I saw him going to muster his men for roll-call in the
middle of the afternoon, I went to take down my 9 x 12 cm. camera
which hung from a nail on one of the pillars of the baraza; but let it
fall in lifting it down, and found, on picking it up, that the sliding
front had got bent and the instantaneous shutter injured by the fall.
The first accident was remedied by energetic pressure, for the second
nothing could be done. I do not to this day understand why the loss
of this instrument should have thrown me into such a state of
excitement; but there are moments in life when we do, or omit to do,
things for which we afterwards vainly try to account. I suppose I
never even remembered at the time that I still possessed a 13 × 18
cm. apparatus of excellent quality. That I did not recall the fact later
on, is easier to understand, as by sunset I found that my temperature
was rapidly rising. I tried a remedy previously found effectual for
bringing on perspiration—huge quantities of tea with citric acid in it,
but in vain. After a terrible night with an average temperature of over
104°, the fever had so far abated that I could exert myself to make
the working drawings for additional slides to my 13 × 18 cm. camera,
which I wished to send to the Indian fundi at Lindi. Up to this
moment I had thought my photographic equipment perfect, but the
possibility of such an accident as befell my smaller camera and of
remedying it by the use of simple wooden frames had not occurred
either to me or the firm who supplied me. By exerting all my
energies, I was just able to finish the drawings and send them off by a
runner to Lindi, when my temperature again rose above 100° and I
was forced to go back to bed. The attack then ran its course and came
to an end, as fever always does. To-day I should almost feel inclined
to smoke, if we had any tobacco worthy of the name. However, I have
now had quite enough of Chingulungulu, and as the Rovuma with its
green banks and clear, cool water, its sand-banks and islands is only
a day’s march distant, we intend to go thither shortly for a rest and
change after all the discomforts, great and small, of our stay here.
 Before leaving, I feel that I ought to set down at least a few of the
observations made at this place.
Among many other diseases, such as malarial, black-water and
remittent fever, sleeping-sickness, guinea-worm, beriberi, and
whatever other ills, great or small, mankind may suffer from in these
otherwise favoured regions, leprosy is unfortunately endemic in our
colony on the Indian Ocean. On the coast of the southern district, the
Government is trying to prevent the further spread of this terrible
disease, by establishing an isolation hospital on an island in the
Lukuledi Estuary, where the patients, at present about forty in
number, are treated by the medical staff at Lindi. Here in the
interior, lepers are for the present entirely dependent on the care of
their fellow-tribesmen. Among the Yaos this care is a mixture of
human sympathy and the crudest barbarity. The patient is taken to a
hut built specially for him in a remote part of the bush, where his
friends or relations bring him food, till the end seems to be
approaching. If the wise men of the tribe come to the conclusion that
this diagnosis is correct, a last and very abundant meal is carried out
to the hut, which is then fastened up from the outside, so strongly
that, even had the patient the power and the will to make an effort,
he could not free himself. He is thus, should he still have any vitality
left by the time the last of the food and drink is consumed,
condemned to perish of starvation.
 VILLAGE OF THE NGONI CHIEF MAKACHU

GRAVE OF THE YAO CHIEF MALUCHIRO, AT MWITI

Another picture connected with death presents itself. We have

already seen the mysterious, legend-haunted site of Hatia’s grave on
Unguruwe mountain; those of other mortals are unpretending
enough and quite prosaic in character. In the country round
Chingulungulu I have found graves, both old and recent, at various
places in the bush, none of them outwardly distinguishable from
graves in our own country, except that the mounds over those of
children are round or oval, instead of long like those of adults. So far
I have seen nothing of the custom reported to me by several
informants, of building a hut over the grave, and decorating it with
calico. Only one grave at Masasi had such a hut, but I was told that it
was an Arab grave, and there was no cloth.[35] The grave of Nakaam’s
predecessor, Maluchiro, at Meviti, has unfortunately quite lost the
traditional character. Here the traveller finds a large oval hut, and,
stooping under the wide, overhanging eaves to enter, he sees, in the
solemn twilight within, massive clay pillars at the head and foot of
the grave, and a somewhat lower wall on either side of it. Such
monuments are shown with pride by the natives to the passing
European, but they are a proof how far Islamitic culture has
penetrated the old African life.
European influence also has a share in the disappearance of old
customs, though, in one point, at least, it is less far-reaching than I
had supposed. I imagined that a box of matches would be found in
every native hut, but I have seen nothing of the sort, and, moreover,
have observed no other way of procuring fire. Yet no hut is ever
without it. Here we have the startling solution of a question which
has long occupied the attention of ethnographers. Not so many
decades ago, inquirers of the standing of Tylor and Lubbock
seriously believed in the existence of fireless tribes—even our brown
fellow-subjects in the Marianne Islands being classed with such
unfortunates. The contrary of this hypothesis has now been
irrefutably demonstrated, and it is known that there is no tribe in the
world ignorant of the use of fire, or even of the mode of producing it
artificially. The problem has therefore assumed another aspect. Did
men first use fire, and then learn to produce it? that is to say, did
they begin by making use of its natural sources, such as volcanoes
and lava currents, burning naphtha-beds, trees kindled by lightning,
or heaps of vegetable matter ignited by spontaneous combustion?—
or did they first learn to bring out the divine spark by boring,
friction, or percussion, and then proceed to harness the kindly
element to household tasks? Both sequences of events are a priori
possible, though, of course, the first is much the more probable of
the two. To-day we may say that it is the only one recognised. This
knowledge we owe entirely to ethnography.
 At a time when hundreds of students are continually busy
investigating and describing the remotest and most forlorn of
primitive tribes at present accessible—when the existing
ethnographic museums are filled to overflowing with new collections,
and new museums are opened every year, it is strange to think of the
earlier and less favoured period which had to be content with mere
arm-chair theories. Two branches of a tree rub together in a storm.
As the wind grows stronger, the friction becomes more rapid, till the
surfaces are heated; at last a tiny spark appears, it becomes a larger
spark, and then a devouring flame which consumes the whole tree.
Primitive man, standing under the tree, has been watching the
process with amazement. “Oh!” says he, “is that how it’s done?” and
thereupon takes a couple of sticks and does likewise.
In this description we have a typical
specimen of the old-fashioned theory devoid
of any concrete basis of fact. It is the
hypothesis propounded by Kuhn, the
philologist, who, fifty years ago, was at least as
famous for his “Origin of Fire”[36] as for his
work in comparative linguistics. We of a
generation which knows no reverence have
grown accustomed to laugh at the venerable
scholar; but such is the way of the world.
It is always well to remember, in the case of
a widely-distributed art, like the production of
fire, that it may have originated in more ways
than one. When we see to-day that by far the
greater number of primitive tribes make use
of a boring implement, while a smaller section
KINDLING FIRE BY uses friction, and a third an instrument like a
FRICTION
saw, and the rest have already advanced to the
principles of the flint and steel, the concave
mirror and the pneumatic fire-producer—it follows of itself that such
must be the case. At the same time this variety of method shows us
that the production of fire is everywhere a secondary matter, an
accidental discovery, made while pursuing some entirely different
end. This is even found to be the case with the Malay fire-pump of
South-Eastern Asia. This is a tube, closed below, into which a tightly-