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Page i
ENGINEERING
COMPUTATIONS
An Introduction Using
MATLAB® and Excel®
Joseph C. Musto
Milwaukee School of Engineering
William E. Howard
East Carolina University
Richard R. Williams
Auburn University

Page ii
ENGINEERING COMPUTATION
Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY
10121. Copyright ©2021 by McGraw-Hill Education. All rights
reserved. Printed in the United States of America. No part of this
publication may be reproduced or distributed in any form or by any
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to, in any network or other electronic storage or transmission, or
broadcast for distance learning.
Some ancillaries, including electronic and print components, may not
be available to customers outside the United States.
This book is printed on acid-free paper.
1 2 3 4 5 6 7 8 9 QVS 24 23 22 21 20
ISBN 978-1-260-57071-7
MHID 1-260-57071-1
Cover Image: ©Ingram Publishing

All credits appearing on page or at the end of the book are
considered to be an extension of the copyright page.
The Internet addresses listed in the text were accurate at the time of
publication. The inclusion of a website does not indicate an
endorsement by the authors or McGraw-Hill Education, and McGraw-
Hill Education does not guarantee the accuracy of the information
presented at these sites.
mheducation.com/highered

Page iii
C ON T EN T S
Preface v
P A R T 1
Computational Tools 1
C H A P T E R 1
Computing Tools 3
Introduction 3
1.1 Analytic and Algorithmic Solutions 4
1.2 Approaches to Engineering Computation 11
1.3 Data Representation 13
C H A P T E R 2
Excel Fundamentals 21
Introduction 21
2.1 The Excel Interface 21

2.2 Tutorial: Entering and Formatting Data With Excel 24
2.3 Tutorial: Entering and Formatting Formulas With Excel 29
2.4 Tutorial: Using Built-in Functions 37
2.5 Tutorial: Performing Logical Tests Using the IF Statement
42
2.6 Tutorial: Using Lookup Tables 49
2.7 Tutorial: Interpolating With Excel 53
C H A P T E R 3
MATLAB Fundamentals 63
Introduction 63
3.1 The MATLAB Interface 63
3.2 Tutorial: Using the Command Window for Interactive
Computation 65
3.3 Tutorial: Using MATLAB Script Files 74
3.4 Tutorial: Using MATLAB Function Files 81
3.5 Tutorial: Computing With One-Dimensional Arrays 85
3.6 Tutorial: Computing With Two-Dimensional Arrays 92
3.7 Tutorial: Saving a MATLAB Session 96
C H A P T E R 4
MATLAB Programming 103

Page iv
Introduction 103
4.1 Flowcharts 103
4.2 Tutorial: Loop Commands 106
4.3 Tutorial: Logical Branching Statements 115
4.4 Tutorial: Combining Loops and Logic 125
4.5 Tutorial: Formatting MATLAB Output 134
C H A P T E R 5
Plotting Data 143
Introduction 143
5.1 Types of Graphs 143
5.2 XY Graphs 147
5.3 Guidelines for Producing Good Graphs 178
5.4 Tutorial: Creating Other Types of Graphs With Excel 180
P A R T 2
Engineering Applications 193
C H A P T E R 6
Finding the Roots of Equations 195
Introduction 195

6.1 Motivation 196
6.2 Roots of Equations: Theory 197
6.3 Tutorial: Solution of General Nonlinear Equations Using
MATLAB 207
6.4 Tutorial: Solution of Polynomial Equations Using MATLAB
210
6.5 Tutorial: Solution of General Nonlinear Equations Using
Excel 213
C H A P T E R 7
Matrix Mathematics 219
Introduction 219
7.1 Properties of Matrices 219
7.2 Tutorial: Matrix Operations Using Excel 223
7.3 Tutorial: Matrix Operations Using MATLAB 228
C H A P T E R 8
Solving Simultaneous Equations 237
Introduction 237
8.1 Systems of Linear Equations 237
8.2 Tutorial: Solutions of Linear Equations Using Excel 238
8.3 Tutorial: Solutions to Simultaneous Linear Equations Using
MATLAB 244

8.4 Tutorial: Solving Nonlinear Simultaneous Equations Using
Excel 248
8.5 Tutorial: Solving Nonlinear Simultaneous Equations Using
MATLAB 250
C H A P T E R 9
Numerical Integration 263
Introduction 263
9.1 Concepts From Calculus 263
9.2 Tutorial: Numerical Integration of Functions 267
9.3 Tutorial: Numerical Integration of Measured Data 279
C H A P T E R 1 0
Optimization 289
Introduction 289
10.1 Engineering Optimization 290
10.2 Formulating an Optimization Problem 292
10.3 Solution of an Optimization Problem 294
10.4 Solution of an Optimization Problem Using MATLAB 302
10.5 Solution of an Optimization Problem Using Excel 309
10.6 Tutorial: Engineering Application of Linear Constrained
Optimization 317

Page v
P R EF A C E
This text has grown out of the authors’ experiences teaching
introductory computation courses to engineering students from a
variety of disciplines at three different institutions. The integration of
computational tools in engineering programs is a constant challenge
for educators. The broad goals associated with an introductory
course in computer applications often include:
▪ Teaching the concept of “procedural thinking” and algorithm
development.
▪ Teaching the mechanics of the computational tools required in
both the subsequent academic program and professional
practice.
▪ Teaching the techniques for developing a computational solution
to a physical problem.
▪ Providing the context for the selection of a computational tool
appropriate for the task at hand.
▪ Teaching the accepted techniques for documenting and verifying
computer-based solutions to engineering problems.
▪ Stimulating interest in upper-division coursework by introducing
the students to realistic, interesting, and exciting problems and
applications.

This text will emphasize these concepts, using MATLAB®
and Excel®
as the software packages of choice. These packages were chosen
because:
▪ MATLAB is widely accepted as a first computation tool in
numerous engineering programs.
▪ MATLAB has the unique ability to be both used as an
introductory programming tool and a high-level computational
tool; the programming constructs (loops and logic) allow it to be
used as a first programming language for engineering students,
while the numerous mathematical and analysis “toolboxes” allow
it to be readily applied to high-level engineering applications.
▪ Excel is a ubiquitous spreadsheet application, which nearly every
engineering student will have access to during their academic
and professional careers. Excel has powerful built-in functions
that allow it to be applied to high-level engineering problems.
▪ Since spreadsheet solutions are so fundamentally different than
the procedural solutions developed using programming tools like
MATLAB, the contrasting approach allows for demonstration and
discussion about implication of the choice of software tool on the
type and complexity of the solution technique.
Philosophy of the Text
The underlying philosophy behind the approach taken in this text is:
▪ Computer tools will change during the professional careers of a
freshman engineering student. While it is important to teach the
mechanics of using the relevant tools, the focus of this text
should be on the fundamentals of engineering computing:

Page vi
algorithm development, selection of appropriate tools,
documentation of solutions, and verification and interpretation of
results.
▪ Programming is a fundamental concept for engineers; while
“shortcut” solutions (such as implied loops in MATLAB) and
“canned” software are certainly appropriate for upper-division
students and practicing engineers, introductory students should
be focused on the basics of structured programming: loops,
logic, and array structures. These basic concepts, which are
language-independent, are the critical building blocks for
programming, and should be introduced early.
With this in mind, the text was developed in two parts. The
first part generally covers the mechanics of programming
and spreadsheet usage; including:
▪ An introduction to computational theory,
▪ An introduction to number representation (scalars, arrays, and
matrices),
▪ An introduction to programming constructs, including algorithm
development and flowcharting,
▪ The mechanics of MATLAB and Excel usage, and
▪ Best practices in computer tool usage, including tool selection,
documentation of solutions, and checking of results.
These chapters include detailed “keystroke-level” instructions, which
will guide the reader through the use of the MATLAB and Excel tools.
The second part focuses on typical applications of engineering
computation; these applications are motivated with engineering

problems, and include:
▪ Root finding,
▪ Matrix methods,
▪ Simultaneous equations,
▪ Numerical integration, and
▪ Optimization.
These applications are intended to motivate not only engineering
computation, but the use of concepts from upper-division
engineering courses as well. Both theoretical concepts and
“keystroke-level” tutorials are presented in these applications
chapters.
New In This Edition
This edition has been updated for the latest versions of MATLAB and
Excel. In addition, many new end-of-chapter problems have been
created, and a tutorial involving symbolic computation in MATLAB
has been developed.
Instructor Resources
Additional resources for instructors are available on the web at
www.mhhe.com/musto2e. These resources include solutions to the
end-of-chapter problems and book figures in PowerPoint format.
Instructors can contact a McGraw-Hill representative for a password.
Acknowledgments

We are grateful to our friends at McGraw-Hill for their support and
encouragement during this project. Theresa Collins, our production
developer, has provided invaluable support and guidance during this
project. We wish to thank Fleck’s Communications for page layout,
and Kim Haas for copyediting. Additionally, the cooperation and
support of the MathWorks Book Program was invaluable.
Feedback provided during the review process was greatly
appreciated, and helped to shape the final form of this text. We wish
to thank the following reviewers for their careful reviews of the initial
manuscript:
Ali Elkamel, University of Waterloo
Bill Elmore, Mississippi State University
Howard Fulmer, Villanova University
Brian Grady, Oklahoma University
Mark Kerstetter, Western Michigan University
Leo Pérez y Pérez, California State University at Long Beach
Michael Robinson, Rose-Hulman Institute of Technology
David Rockstraw, New Mexico State University
Scott Short, Northern Illinois University
Elisa H. Barney Smith, Boise State University
J. Steven Swinnea, University of Texas at Austin
Michael Weinstein, University of Rochester
Students in the Computing Applications in Engineering course at East
Carolina University class tested an early version of this text; their
feedback was appreciated. Also at East Carolina University, Scott
Martin provided a thorough reading and review of this text, and we
thank him for his insight and input.
Joe Musto
Ed Howard
Rick Williams

MATLAB is a registered trademark of TheMathWorks, Inc.
Excel is a registered trademark of The Microsoft Corporation.

Page 2
PART 1
COMPUTATIONAL TOOLS
Chapter 1: Computing Tools
Chapter 2: Excel Fundamentals
Chapter 3: MATLAB Fundamentals
Chapter 4: MATLAB Programming
Chapter 5: Plotting Data

C H A P T E R 1
Computing Tools
Introduction
The engineering profession is a discipline devoted to problem
solving, applying principles of mathematics and science to develop
solutions to practical problems involving structures, machines,
electrical circuits, and various other physical systems and devices.
With their ability to perform numerical analysis and data handling,
computers are important tools for practicing engineers. Engineering
graduates from all disciplines are expected to have proficiency in a
wide range of computational tools and software. New engineering
graduates should expect to have proficiency in:
▪ communication tools (for e-mail and messaging),
▪ Internet search tools (for research),
▪ word processing tools (for report preparation and memo writing),
▪ presentation tools (for audiovisual presentation),
▪ data acquisition tools (for running and reading data from
experiments), and
▪ computational tools (for programming, data analysis, equation
solving, and plotting).

In addition to these basic tools that cross all engineering disciplines,
there are specific computer tools that are considered part of the core
skill set for each engineering discipline. Examples of these tools
include:
▪ solid modeling and computer-aided design/drafting software (for
mechanical and civil engineers),
▪ electric circuit simulation software (for electrical and computer
engineers),
▪ finite element analysis software (for mechanical, civil, and
electrical engineers),
▪ advanced programming languages (for computer and software
engineers), and
▪ statistical analysis software (for industrial engineers).
This text is devoted to an introduction to the application of
computational tools to the solution of engineering problems.
These are the types of tools used for mathematical analysis and data
processing across a wide variety of engineering disciplines. While no
individual software product can truly be considered a “standard” for
all engineers, two widely available and widely used software
packages are introduced in this text: MATLAB®
and Excel®
. Besides
being two of the most popular computational packages for
engineers, they also provide the opportunity to demonstrate two
very different approaches to engineering computation: programming
tools and spreadsheet tools. While these two platforms offer very
different ways to approach the solution of engineering problems,
there are some similarities between them, particularly in the way
data is represented, stored, and handled. These similarities are

important to understand, in that they offer a common language for
these, and other, computational tools.
In this chapter, you will:
▪ learn the difference between an analytic and an algorithmic
solution,
▪ learn the essentials of algorithm development and pseudocode,
▪ learn the basic difference between programming tools and
spreadsheet tools,
▪ learn the basic terminology of data storage and handling, and
▪ learn the difference between accuracy and precision, and the
implications of both in engineering computation.
1.1 Analytic and Algorithmic Solutions
The computational tools introduced in this text allow us to automate
the mathematical analysis required to solve an engineering problem.
In order to understand both the advantages and limitations of the
application of computational tools to engineering problems, we must
first understand the essential difference between the analytic
solution and the algorithmic solution to an engineering problem.
Consider the classic projectile motion problem addressed in
introductory physics classes. A ball will be fired from a toy cannon,
with an initial speed of 10.0 meters per second at an angle of 35.0°,
as shown in Figure 1.1. An engineer has been asked to predict the
peak height that the cannonball will reach, the location at which the
cannonball will hit the ground, and the total flight time of the
projectile.

1.1.1 The Mathematical Model
The first step in the solution of this problem is the development of
the mathematical model that the engineer will use to predict the
behavior of the system. In this case, it requires the application of the
principles of the physics of mechanics. The engineer creates a sketch
of the system, as shown in Figure 1.2.
In developing the mathematical equations that will be used to
predict the behavior of the system, the engineer must make some
decisions as to what to include in the model. In doing so, the
engineer balances the accuracy of the model (the ability of the
equations to properly predict the behavior of the system) and the
simplicity of the model. In this case, the engineer makes the
following decisions:
Figure 1.1
MATLAB® is a trademark of The MathWorks, Inc. Excel® is a trademark of the
Microsoft group of companies.
Figure 1.2

(1.1)
(1.2)
▪ The ground will be considered flat and level.
▪ The launch point will be considered to be at ground level.
▪ Wind resistance will not be considered in the equations.
These decisions, called simplifying assumptions, require considerable
engineering judgment. The engineer must decide that the
complexity introduced by including these effects into the equations
will not lead to any significant increase in the accuracy of the
solution. In this case, with these assumptions made, principles of
physics can be used to write the following equations for height and
horizontal distance as functions of time:
where h is the height of the cannon ball, x is the horizontal distance
travelled, ν is the initial speed of the cannonball, θ is the launch
angle, g is gravitational acceleration, and t is the time after launch
(in seconds). With this model in place, the engineer must now select
a solution technique to solve the equations. We will now contrast the
analytic solution to the algorithmic solution of this problem.
1.1.2 The Analytic Solution
An analytic solution is an exact solution, based on the application of
the mathematical principles of algebra, calculus, etc. In the model
we have developed, an analytic solution is possible. In order to find
the peak height the ball will reach, the engineer uses principles of
calculus to take the first derivative of Equation 1.1:

(1.3)
Page 6
(1.4)
(1.5)
(1.6)
(1.7)
When this derivative is equal to zero, the height is at an
extreme (maximum or minimum) value. Setting the
derivative equal to zero and solving for t yields:
Substituting in the known values for launch speed, angle, and
gravitational acceleration and carrying out the arithmetic yields:
or t = 0.585 seconds. This indicates that the ball will reach its peak
height after 0.585 seconds of flight time. Substituting this value for
time into Equation 1.1 yields:
or the peak height is determined to be hmax = 1.68 meters.
To determine the total flight time and horizontal distance travelled,
the engineer uses Equation 1.1 to determine the time at which the
height of the ball is zero:
The engineer uses algebra to factor out t, leading to two solutions:

(1.8)
Page 7
(1.9)
The engineer recognizes that t = 0 corresponds to the launch time,
and t = 1.17 seconds corresponds to the time at which the ball hits
the ground again. This value can be substituted into Equation 1.2 to
determine the horizontal distance travelled during 1.17 seconds of
flight:
yielding a horizontal distance travelled of 9.58 meters.
The engineer reports the results in Table 1.1, with appropriate units:
1.1.3 The Algorithmic Solution
An algorithmic solution is an approximate solution, based on the
application of a computational procedure. In an algorithmic solution,
the engineer will define a series of steps or rules to be followed that
will lead to the discovery of the solution of the problem. The
algorithm will generally rely on principles of arithmetic only to solve
the problem; therefore, while the solution is approximate, it also
eliminates the need to apply more complicated mathematics to the
problem. An algorithmic approach will be demonstrated in our
sample problem.
Table 1.1 Results of the Analytic Solution

The engineer has an equation to compute the height of the ball at
any time t. The engineer recognizes that the cannonball starts and
ends at a height of zero, and will reach its peak somewhere in the
middle. For the first part of the flight, height is increasing; during the
second part of the flight, height is decreasing. If the engineer can
identify the point where the height stops increasing and starts
decreasing, the point of peak height will be identified. The engineer
constructs the following algorithm for identifying the peak height;
the solution steps, provided in verbal descriptions called
pseudocode, are as follows:
▪ Step 1: Start at a time value of t = 0 and h = 0.
▪ Step 2: Increase time by adding some small value ∆t to t (e.g.,
tnew = t + ∆t).
▪ Step 3: Plug the new value of tnew into Equation 1.1 to get a new
value of h, which we will call hnew.
▪ Step 4: Compare h and hnew:
▪ If h < hnew, then the height is still increasing, and the peak
has not been reached. Set t = tnew, h = hnew, and return to
Step 2.
▪ If h > hnew, then the height has started decreasing. This tells
us that the ball reached its peak somewhere in the
neighborhood of h (either in the interval between h and hnew,
or in the previous interval).
▪ Step 5: Assume that the maximum height occurs at the height at
the start of the interval, or hmax = h.

The algorithmic solution is a “road map” to the solution of
the problem; it is not an answer itself, but is a series of
specified steps that will lead to an answer. The main computational
part of the algorithm, embodied in Steps 2 through 4, may need to
be repeated multiple times before a solution is reached; there is no
particular way to tell ahead of time how many times the algorithm
will “loop” back to Step 2 before a solution is found. Note that this
algorithmic solution uses arithmetic operations only; unlike the
analytic approach, no principles of calculus or algebra are required.
However, note that there is a critical approximation used in the
algorithm; the height is only computed for specific values of t, but it
is likely that the actual peak value occurs at some intermediate
value.
The engineer carries out the algorithmic solution, using a time step
value of ∆t = 0.1 seconds. The values of each variable at each
“loop” through Steps 2 through 4 are shown in Table 1.2:
The algorithm leads to a solution value of hmax = 1.68 meters.
The engineer proceeds to construct a solution algorithm for finding
the point at which the cannonball strikes the ground. The engineer
recognizes that a height value of zero indicates the point at which
the cannonball hits the ground, and the time at which this happens
is the total flight time. The algorithm is as follows:
Table 1.2 Step-by-Step Solution Algorithm
for Finding hmax

▪ Step 1: Start at a time value of t = 0 and h = 0.
▪ Step 2: Increase time by adding some small value ∆t to t (e.g.,
tnew = t + ∆t).
▪ Step 3: Plug the new value of tnew into Equation 1.1 to get a new
value of h, which we will call hnew.
▪ Step 4: Check the value of hnew:
▪ If hnew > 0, then the cannonball is still in flight. Set t = tnew,
h = hnew, and return to Step 2.
▪ If hnew < 0, then the cannonball hits the ground somewhere
between h and hnew.
▪ Step 5: Approximate the total flight time by setting
▪ Step 6: Find the horizontal distance travelled during the flight by
substituting tflight into Equation 1.2. End the algorithm.
The engineer again executes the algorithm, with ∆t = .1 seconds.
The values at each step of the algorithm are shown in Table 1.3:
Table 1.3 Step-by-Step Solution Algorithm
for Finding Flight Time and Distance Travelled

The results from these two algorithms are reported by the engineer,
as shown in Table 1.4:
1.1.4 Comparison of the Analytic and the
Algorithmic Solutions
Examining both the process and results of the analytic and
algorithmic solutions can tell us much about the nature of each
solution. The most important differentiation between the two
solutions is that the analytic solution is exact. As long as the
mathematical techniques of algebra and calculus were properly
applied, and the arithmetic computation was performed correctly,
the result is valid to the appropriate number of significant digits
allowed by the given data. The algorithmic solution is
approximate; the equations are not solved exactly, but are
Table 1.4 Results of the Algorithmic
Solution

merely evaluated at specified values of the independent time
variable. These specified values of time, known as discrete values,
limit the accuracy of the final solution. By the nature of the
algorithm, the only possible solutions occur at time values at, or
midway between, our discrete points. However, while the algorithm
will always produce an approximate solution, the spacing between
the discrete points is under the engineer’s control; the solution can
be made more accurate by decreasing the value of ∆t used in the
algorithm. For example, if the algorithm was repeated, but with a
value of ∆t = 0.001 seconds, the results would be the same as those
reported for the analytic solution to three significant digits. However,
to achieve this increased accuracy, the algorithm would need to
“loop” through Steps 2 through 4 many more times (1680 times to
find hmax, instead of seven).
While algorithmic solutions are approximate by their nature, we can
increase their accuracy by decreasing the interval between discrete
points and simply running through the algorithm more times.
Approaching an algorithmic solution “by hand” would prove
impractical if increased accuracy was required. Within practical limits,
however, it is no more difficult to run through the algorithm tens,
hundreds, or even thousands of times. This is because algorithmic
solutions lend themselves readily to implementation with computer
tools. The algorithms developed in this chapter were presented as
verbal descriptions of the step-by-step problem-solving strategy;
these verbal descriptions are known as pseudocode. A computer
programming language, such as the MATLAB software introduced in
this text, can be readily used to translate our software-independent
pseudocode into software-specific computer code, providing clear
instructions that the computer can follow. While computers can be
applied to automate the arithmetic part of an analytic solution, it is
in the implementation of algorithmic solutions where computational
tools are best suited.
Since analytic solutions are exact, and algorithmic solutions are
approximate, why should algorithmic solutions be used at all? In our
example case, there is really no need for an algorithmic solution,

since the problem is readily solved using analytic techniques. An
engineer with a background in differential calculus and algebra could
readily arrive at an analytic solution. When this is the case, an
analytic solution is the preferred approach. However, in the
professional practice of engineering, this is not always the case.
Even in your undergraduate studies, you will soon encounter
problems that you lack the mathematical background to solve, and
even some where no analytic solution exists. It is at these times that
algorithmic solutions become an important option. You have likely
already used algorithmic solutions to solve otherwise difficult or
unsolvable problems; if you have used a root finding function on
your programmable graphic calculator to find the roots of a higher-
order algebraic equation, you have used an algorithmic solution
technique derived and implemented by the calculator’s manufacturer.
(Did you realize that the answers you obtained with your calculator
were approximate?)
Another hallmark of an algorithmic solution is that there is
not one unique algorithm that will solve a specific problem.
Development of solution algorithms requires a combination of
mathematical reasoning and creativity. Computer scientists focus on
the development of efficient computational algorithms that minimize
the computer time and memory used when running a program
based on their algorithm. In this text, we will focus on implementing
and using many standard solution algorithms; however, there is still
much room in the field of engineering computation for the
development of new and creative algorithms for solving engineering
problems.
1.2 Approaches to Engineering Computation
In this text, we will contrast two distinct approaches to the
implementation of engineering computation: programming tools and
spreadsheet tools.
Programming tools allow the translation of pseudocoded algorithms
into instruction sets that can be followed by the computer. These
instruction sets are called computer programs or computer code.

There are many programming languages used by practicing
engineers; we will use the MATLAB platform to introduce the concept
of programming tools in this text. As an example, the MATLAB
implementation of the pseudocoded algorithm for finding maximum
height is shown below. The details of developing your own MATLAB
code will be covered in Chapters 3 and 4 of this text.
Programming tools allow us to specify the logic and
decision-making structure that the computer will follow in
the implementation of an algorithm. From the earliest days of
computing, users communicated with the machines through
instruction sets provided by the programming tools available. While
the look of the programming languages and the interface used to
communicate with the machine have evolved significantly, the idea
of translating an algorithm into software-specific instructions is a
classic paradigm for computation.

A spreadsheet tool is a completely different paradigm for
computation; a spreadsheet resembles a large data table. The user
of a spreadsheet tool fills some of the cells of the table with data,
and fills other cells with mathematical equations and logical
expressions that use the data cells as their input. This tabular
structure provides an intuitive graphical approach to data
manipulation and computation, but it differs significantly from the
step-by-step instruction sets used in programming approaches.
While the graphical approach to computation is appealing, the direct
link with pseudocoded algorithms is sometimes lost with spreadsheet
implementations. Although other spreadsheet products do exist, we
will use the Excel product from Microsoft Corporation as the
spreadsheet platform in this text. As an example, a spreadsheet
used to implement the maximum height algorithm and generate the
data shown in Table 1.1 is shown in Figures 1.3 (with numerical
values shown) and 1.4 (with formulas relating the cells shown).
Figure 1.3

As we introduce the various problem-solving approaches in this text,
we will highlight the advantages and disadvantages of these two
approaches to engineering computation. Some problems are best
suited to solution by programming, while others are best suited to
spreadsheet solutions. That said, the MATLAB and Excel tools that
we introduce in this text are very advanced and capable
computational tools, and have to some extent adapted the best
features of each approach into their functionality. Excel has
implemented a programming interface, where more traditional
computer code can be developed to operate on and fill cells in the
spreadsheet. MATLAB has developed an array editing interface that
looks and acts similar to a spreadsheet. Both have developed
preprogrammed utilities for plotting, root finding,
optimization, and other common operations that allow the
user access to advanced algorithms for engineering problem solving.
The goals of this text involve both proficiency with the computational
tools and the insight into selection of an appropriate tool for a given
engineering application.
1.3 Data Representation
Despite the differences between various computational platforms,
there are some similarities between packages in the way data is
represented, stored, and manipulated. In this section, the
Figure 1.4

(1.10)
(1.11)
(1.12)
terminology of data representation will be introduced and related
back to the MATLAB and Excel tools used in this text.
1.3.1 Variables and Functions
A variable is the symbolic representation of a quantity that can take
on more than a single value. Consider the equation:
In this equation, x and y are variables, since they can take on many
values. We often refer to variables as independent or dependent. A
variable is dependent when its value depends on the value of other
variables. Usually we write equations so that the dependent variable
is on the left side of the equal sign. For example, in Equation 1.10,
we would assume that y is the dependent variable, since its value
depends on the value of the independent variable x. Of course, we
could rearrange the equation so that x is on the left side:
Does this mean that y is the independent variable? We must go
beyond the equation and examine the problem that it represents in
order to determine which one is the independent variable. We do
know that both x and y cannot be independent, because when we
assign a value to one of them, the value of the other can be
determined from the equation.
Let’s go back to the example of the cannonball’s trajectory. We wrote
an equation for the height h of the cannonball as:
where ν = initial velocity
t = time
θ = launch angle

g = gravitational acceleration
There are five quantities related by symbols in this equation;
however, they are not all variables in the problem. We are
considering the initial velocity, launch angle, and gravitational
acceleration to be constants rather than variables. This is
not evident in the equation, but rather in the problem
statement. Suppose the problem statement was changed so that we
were asked to calculate the height at time t = 2 seconds for various
values of the launch angle. In this case, time would be a constant
and the launch angle a variable, but the equation would be
unchanged. In another version of the problem, we might be asked to
find the maximum height that can be attained for any launch angle.
In this case, height, launch angle, and time would all be variables.
Going back to the original problem, with the initial velocity, launch
angle, and gravitational acceleration all considered to be constants,
we say that the height is a function of time, and will often write the
left side of the equation as h(t), as in Equation 1.1. Therefore, time t
is the independent variable, while height h is dependent on time. To
be more precise, a function is defined as follows: A function is a
mathematical operation that returns a single value for a given input
value or set of values. The input values are called the arguments of
the function.
While not identical, computational functions are similar to
mathematical functions in that they compute an output value from a
number of input arguments. Both Excel and MATLAB have many
built-in functions. Many of these functions require a single argument.
For example, the cos function in MATLAB returns the cosine of an
input value of an angle in radians. Other functions require a specific
number of multiple arguments. For example, Excel has a function
called ROUND which requires two arguments: the number to be
rounded off and the number of decimal places to which the number
is to be rounded. Still other functions have a variable number of
arguments. An example is the AVERAGE function in Excel, which
finds the average of a group of values entered as arguments. In
MATLAB, there are a number of functions that use arrays or matrices

as arguments, which we shall discuss in subsequent chapters. (Note
that the convention in this text is to refer to MATLAB functions in
italics, and Excel functions in capital letters).
1.3.2 Scalars and Arrays
In the algorithmic solution presented in Section 1.1, we chose the
values of the independent variable t (time) to evaluate, and
calculated the value of the height at each increment. We continued
to make calculations until the value of the height began to decrease,
indicating that the peak height was attained. We chose a time
increment of 0.1 seconds. The results of this analysis are repeated in
Table 1.5:
In a spreadsheet solution, the calculations would be
performed in the cells of the spreadsheet, with the results
shown in a form similar to that of Table 1.5. In a spreadsheet, the
numerical value in a cell is a scalar—a single value that can be
represented on a numerical scale. However, in a computing language
like MATLAB, results of calculations are stored differently. It is
possible that we could have variables named t, tnew, h, and hnew. In
each calculation loop, we could write over the previous value of the
variable. In doing so, we would also be treating each variable as a
scalar, with a single value. But what if we wanted to keep the results
for each loop? We may want to plot height versus time. To do so, we
Table 1.5 Algorithmic Solution to
Cannonball Problem

need to have those values stored in memory. It would be difficult to
give each value a unique name (for example, t1, t2, t3, etc. for time
values), and doing so would require making every calculation
sequentially, rather than in a repeating loop. Instead, we use arrays
to store data. An array is a single variable that has multiple values
associated with it. In our example, time t would be an array with
seven values. The values within an array are referenced by an index.
An index is an integer that refers to the position of the value within
the array. You can think of an index as being an address. Our
variable t has seven addresses, labeled 1–7. Into each address we
place the value of time (as in Table 1.6):
We refer to an individual value of the array by including the index
number in parentheses following the array name, or as a subscript
to the variable name. For example, t (5) = 0.4 seconds, or t3 = 0.2
seconds.
It is very important to remember that index numbers must be
integers, beginning with one and progressing by one for subsequent
values. Some of the common errors that are made by beginning
programmers include:
▪ Trying to use zero as an index. In our example, the first value of
time is zero. Therefore, it is tempting to start an array with t (0)
= 0. This will result in the following error in MATLAB:
Table 1.6 Structure of Array t

▪ Trying to use non-integer indices. For example, the statement “t
(.1) = 0.1” would result in the same error as above.
▪ Progressing index values by increments other than one. For
example, suppose that you are performing an experiment in
which you take temperature readings every 10 seconds. Your
first reading is 100°C, so you enter this as T(10) = 100.
In this case, there will not be an error, but rather an
array is created with T(1) through T(9) all having values of zero:
In all of these cases, the fundamental error is confusion between
independent variables and indices. Remember that indices are simply
counting values representing addresses within an array, and are not
variables themselves.
The arrays shown above are one-dimensional arrays. That is, a
single index number is used to establish the identity of a value in the
array. Arrays can also be multidimensional. In the example illustrated
in Table 1.5, note that there are two values of time for each loop: t
and tnew. Instead of storing these values in two one-dimensional
arrays, we can store them in a single two-dimensional array. If we
assign the first index a value of 1 or 2, representing t and tnew,
respectively, and assign the second index the value of the loop
number, then all 14 values will be stored in an array. As an example
of this scheme, t (1,5) = 0.4 and t (2,5) = 0.5.
1.3.3 Matrices and Vectors
One- and two-dimensional arrays are often referred to as matrices.
In addition to being an efficient method for storing data, many
mathematical operations can be performed directly with matrices. In

fact, the name MATLAB stands for Matrix Laboratory, and the
program was originally created to perform matrix operations. In
Chapters 7 and 8, we will learn some simple matrix mathematics and
use matrix methods to solve a series of simultaneous equations.
The size of a matrix is defined by its number of rows and columns.
For example, the matrix below is a (3 × 2) matrix (pronounced
“three by two matrix”), with three rows and two columns:
One-dimensional arrays are often called vectors in engineering
computation. If the values are arranged in a single row, then the
array is called a row vector; if the values are arranged in a single
column, the array is called a column vector. One-dimensional arrays
can also be called column matrices and row matrices.
It is important to note here that there is another definition of the
term vector that you will encounter in physics and engineering
mechanics. In that context, a vector quantity is one that is defined
by a magnitude and a direction. For example, velocity is a vector
quantity. In addition to its magnitude (speed), the direction
of motion is necessary to completely define a velocity. One
method to define a vector quantity is to define its components in the
x, y, and z directions. Of course, these three components can be
placed in a one-dimensional array, fitting the computational
definition of a vector. Because of the confusion that can be created
by the two definitions, in this text we will avoid using the term
vector when referring to a one-dimensional array. Instead, we will
use the more general term array when referring to the storage of
multivalue variables, and matrix when referring to one- and two-
dimensional arrays for which we will perform matrix mathematics
operations.
Excel also has the ability to represent and manipulate arrays. Data
entered into a region of adjacent cells in a spreadsheet can be

interpreted as a matrix, and operated upon using matrix
mathematics. In Excel, these operations are performed using
prewritten functions (like those described in Section 1.3.1)
specifically developed for matrix computation. Unlike MATLAB, which
was developed specifically for matrix operations and where matrix
and scalar computations are performed with the same mathematical
operators, matrices require special handling when using Excel. These
methods will be described in Chapter 7 of this text.
1.3.4 Accuracy and Precision
The terms accuracy and precision are often used interchangeably,
but have different meanings in computing applications. Accuracy
refers to the closeness of the calculated solution to the actual value,
and is a function of the model itself. For example, when finding the
height reached by the cannonball, we noted that if we decreased the
size of the time step, our solution would approach the “exact”
analytical solution. We also noted that there were several
assumptions present in our model. For example, the effects of wind
resistance were neglected. This assumption also affects the accuracy
of the solution.
The precision of the solution depends on how well the input
variables are known, and on how numerical values are stored from
one calculation to another. In the cannonball problem, the launch
angle was given as 35°. But how precise is this value? Depending on
how the cannon’s launch angle is set and measured, the value might
be precise to the nearest degree, the nearest one-tenth of a degree,
or the nearest 5°.
In the sciences, the precision of measured input variables are usually
known, and calculation results are reported based on the number of
significant digits of the input. For numbers containing decimal points,
the number of significant digits of a number is defined as the
number of digits between the first non-zero digit and the last digit.
Consider these examples:

When making calculations, the answer can only be as
precise as the least precise of the input values. For addition
and subtraction, this means that the number of digits to the right of
the decimal point in the answer must be equal to the least number
of digits to the right of the decimal point in any of the inputs.
Examples include:
For multiplication and division, the number of significant digits in the
answer must equal the least number of significant digits of the input
values. Examples include:
Some quantities are exact. For example, there are exactly 12 inches
in a foot. So if we want to convert 11.556 inches to feet, the answer
is:
In this case, the exact value of 12 inches per foot is considered to
have an infinite number of significant digits.
The precision of quantities without a decimal point is not always
known. For example, as we discussed previously, we may not know
the precision of the 35° elevation angle in the cannonball problem.
This is a typical situation in most engineering problems, with at least
some of the input quantities of unknown precision. Therefore, the
rules for calculations described previously cannot be applied. Rather,
a reasonable number of significant digits should be reported. Many
engineering texts suggest three significant digits for final answers

(some recommend four significant digits if the first significant digit is
a one). When performing calculations by hand, intermediate results
should be carried to more significant digits than will be reported for
the final answer. For example, you cannot round the value of the
sine of 35° to 0.57 and then report the final answer to more than
two significant digits.
With computing solutions, intermediate calculations are not rounded
off, so the precision of the final answer is usually dependent only on
the precision of the input values. The qualifier “usually” in the
previous sentence must be added because there are some instances,
when working with combinations of very large and very small values,
where errors will accumulate in computing solutions as well. For
example, when analyzing mechanical structures using a
computational technique known as finite element analysis, tens or
hundreds of thousands of simultaneous equations are solved. If the
numerical values in these equations differ by orders of magnitude,
then the solution algorithm of the program must be designed in a
way that minimizes computational errors. For the problems
encountered by most engineering students and practicing
engineers, this is not a concern. How precise are the values stored in
Excel and MATLAB? Excel carries values to 15 significant digits. By
default, MATLAB stores values as double-precision values, which also
have approximately 15 significant digits. The term double-precision
refers to the fact that these values require two 8-bit units of
computer memory to store, while single-precision values are stored
in a single unit of memory. In the early days of computing, storage
space was severely limited, so double-precision values were used
only when necessary to ensure sufficient precision of calculations.
Processing times were also increased when double-precision values
were used. With today’s inexpensive computer hardware and fast
processors, there is rarely a need to use single-precision values,
although MATLAB does support single-precision values for working
with extremely large data sets.
One final thought about accuracy and precision is warranted. When
formulating a computer solution to a problem, many students will

report the final answer to whatever precision is displayed on the
computer screen, even though they routinely round the answers of
hand calculations to a reasonable precision. In doing so, they are
treating the computer solution as a “black box,” with no
consideration of what is happening between the inputs and outputs.
When a computer solution is reported to a reasonable number of
significant figures, a student conveys the impression that he or she
is aware of the assumptions and approximations associated with the
problem. Engineering students and practicing engineers should take
care to interpret the results of their computations, and report the
results to a reasonable level of precision, regardless of the
computer’s output.
Problems
1.1 Describe the differences between analytic and algorithmic
solutions.
1.2 Develop the pseudocode for an algorithmic solution for
finding the two points where the function f(x) = 3x2
–
12.4x + 3 crosses the x-axis.
1.3 Consider the cannon model developed in Section 1.1.1.
a. Using the equations developed and a launch speed of
10.0 m/s, develop the pseudocode for an algorithmic
solution to determine the launch angle required to reach
a peak height of at least 2.5 meters.
b. Using discrete values spaced 5° apart, carry out the
algorithmic solution by hand. Report each step of the
algorithm in a table.
c. Perform an analytic solution for this problem, and
compare the result with your algorithmic solution.
1.4 Consider the cannonball problem described in Section
1.1.1. You have been asked to determine a combination of
launch speed and angle required to clear a 5-meter wall
erected 8 meters from the launch point. The maximum

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203
“Avast there, Monahan!” he growled. “Have a care how
you blacken my good name! Now stand against yon
rock, all of you! And reach for the stars!”

204
Chapter 24
CAPTAIN CARTER’S SCHEME
Slowly, Mr. Monahan and the Scouts obeyed the
captain’s terse command to raise their hands. He lined
them up against the rock, but, observing Mr. Livingston’s
weakened condition, did not force him to arise.
“I should do you all in now and put an end to this cat-
mouse game,” he said in a bored tone. “It would be so
easy.”
“I rather doubt that, captain,” Mr. Monahan answered,
matching his cool, detached manner. “True, you might
shoot and toss us to the fish, but in doing so, you
certainly would bring the wrath of the natives down
upon your head. Don’t forget that as Ino, the Medicine
Man, I still swing a little weight. Do away with me, and
you’ll bring the pack down on your back!”
“You over-estimate your quack medical powers,
Monahan,” Captain Carter sneered. “But that’s beside
the point. Why work against each other when we can
make a deal?”
“A deal?”
“This lake holds enough treasure for both of us, with a
few trinkets left over for the Scouts to take home to
their mamas. Why not team together to get it out?”

205
“Team with you!” Mr. Monahan exclaimed. “You’ve
already betrayed and cheated me! Instead of revealing
to my brother that I was safe, you gave him quite the
opposite impression. You defrauded him.”
“He’ll get his cash back,” the captain retorted. “I was
stony broke when we parted company, and didn’t have
enough money to pay off my crew. I had to raise cash
fast to get back here with the equipment we needed to
pull off the job.”
“Apparently, it never occurred to you to tell my brother
the truth. Or to go to government authorities. That was
because you expected to do me in and grab everything
for yourself!”
“Oh, I considered it,” Captain Carter admitted with a
shrug, “but the scheme offers risks. First, the Scouts
loused up my deal by bringing the authorities down on
my head. As a result, I got here with a minimum of the
explosives I’ll need.”
“You intend to dynamite the lake?”
“That might be the general idea. Know of a better way
to get rid of those man chewin’ fish?”
“It might work,” Mr. Monahan conceded grudgingly. “But
the point is, what will the natives do when you set off
an explosion?”
“I always was one to go for the big chance—take all,
lose all, that’s me. First, I aim to set myself up as the
big Chief, deposing old mud-in-the-mouth Panomuna.
Once that’s done, I’ll say hocus-pocus and toss some
grenades into this lake. That should do the trick.”

206
“You make it sound very easy,” Mr. Monahan replied.
“Just how do you propose to depose Panomuna?”
“It’s simple,” Captain Carter boasted. He flashed a
cigarette lighter. “I’ll do a snappy job of starting a fire
with this little gadget.”
“You think of everything, captain!” Mr. Monahan
remarked sarcastically.
“That’s me. Well, what do you say? Are you playing
along?”
“Just what is your proposition?”
“We’ll split the treasure two ways—half yours, half mine.
You let me get out of the country before you tip
government officials. That’s all I ask.”
“No! All of the treasure must be turned over to the
proper authorities.”
“You’re a stubborn fool!” Captain Carter asserted angrily.
“Okay, if you don’t want to play along, I’ll take all the
treasure and you can’t stop me. You and your boys can
take your chances on getting out of here alive. Your
decision is final?”
“It is.”
“Okay then,” Carter said, lowering his automatic. “If we
can’t be friends, then it’s each man for himself, and the
Devil catch the hindmost. I’m warning you though—
don’t try any tricks either tonight or tomorrow. I’m
setting myself up as a ruler, and if you try to interfere,
I’ll turn the natives loose on you.”

207
The captain started to leave. In passing Mr. Livingston,
he scrutinized him briefly.
“Fever, eh?” he remarked. “You’ll all be down with it
before long.”
“Could you spare me a cigarette?” the Scout leader
asked.
“Sure, anything for a pal,” the captain replied
sarcastically.
Mr. Livingston fumbled with the cigarette which the
seaman gave him, and then asked for a match. Captain
Carter offered him the cigarette lighter. As he lit the fag,
Jack suddenly moved forward as if to attack the captain.
“Oh, no you don’t!” the officer snarled, whipping out his
automatic again. “No tricks, I warned you!”
“Jumpy, aren’t you?” Jack taunted. “I wasn’t even
starting your way.”
“No? Well, remember what I told you, or it will be the
worse for you all.” His gaze upon the grinning Scout,
Captain Carter reached out to snatch the cigarette
lighter from Mr. Livingston’s fingers.
Then, his automatic still trained upon the group, he
backed slowly toward the tunnel.
“You’re all invited to the ceremony at dawn,” he called in
parting. “I advise you though, to watch from a distance.
If I catch a glimpse of you, I’ll sick my natives onto you.
Furthermore, once I’ve finished off old Mud-in-the-
Mouth, I may find it expedient to purge the Forbidden
City of strangers.”

208
After the captain had gone, the Scouts, Mr. Livingston
and Mr. Monahan, put in uncomfortable hours by the
lake. Though they discussed any number of plans, none
of them seemed feasible.
Captain Carter, they knew, was quite capable of carrying
out his threat. Aware of their hostility, he would be more
than ever on the alert.
“If we show ourselves in the city, he’ll finish us off,” Mr.
Monahan asserted. “My advice is to wait here until
dawn. Even then, I don’t know what we can do. If we
try to overpower Carter, the natives will turn on us.”
“Don’t give up hope,” Mr. Livingston encouraged the
little band. “The captain may outsmart himself. I
thought of a scheme, but we can’t know until tomorrow
whether or not it will work.”
Near exhaustion, the Scout leader closed his eyes and
slept. Toward morning he was aroused by his
companions, who whispered that the hour of dawn was
upon the mountain.
“Willie will stay here with you,” Jack told him. “The rest
of us are going to sneak down to the plaza to see what
happens.”
Mr. Livingston aroused himself. “I’m stronger,” he
insisted, stretching his cramped legs. “My fever is down
again. We’ll all go together.”
The others could not dissuade him. Aided by Jack and
Ken, the Scout leader made it through the tunnel. Still
shielded by semi-darkness, the group found a hiding
place not far from the scene of activity.

209
“This is going to be like watching a spectacle movie!”
War remarked, thrilled by the sight.
In the plaza, hundreds of chanting natives knelt before
the temple, their heads bowed. As a prelude to the
ceremonial test between Captain Carter and the Inca
ruler, replicas of the Sun and Moon were paraded on the
temple steps. An impressive silence fell upon the
throng.
“This is it,” Jack whispered to his crouching companions.
“Here comes Panomuna!”
A procession of priests wound its way to the broken
stone steps. Moving with great dignity, the Inca ruler
took his place in front of the great crowd. He wore a
flame colored robe and held aloft a magnificent golden
bowl.
As the first rays of the sun came over the mountain
peak. Panomuna turned to face the horizon. Raising his
hand, he chanted:
“Capak inti-illariymin.”
The Indians bowed before him, replying in chorus to the
chant.
“Now, Panomuna will kindle the sacred fire on the altar,”
Mr. Monahan informed the group. “He will concentrate
the rays of the sun upon tinder in the golden bowl.
Then Captain Carter will do the trick faster.”
The native ruler held his great bowl aloft, catching the
rays of the sun as he pronounced his weird chant.

210
Soon he had created his fire, which he deposited with
ceremony on the altar. The multitude cheered.
Gradually, the cries subsided and deep silence came
upon the throng. Every eye fastened upon Captain
Carter. Confident and sure of himself, he strode down
the temple steps.
“I hope he uses that cigarette lighter!” Mr. Livingston
murmured. “It would be just our luck for him to use a
match.”
“The natives already are familiar with matches,” Mr.
Monahan commented. “That wouldn’t impress them and
Carter knows it.”
By this time Jack had caught the gleam of bright metal
in the captain’s hand.
“He’s using the cigarette lighter!” he exclaimed
jubilantly.
Carter raised his hands and in an imitation of
Panomuna, entoned a meaningless chant to the Sun
God.
“Now, I produce fire!” he shouted.
But the flames were not forthcoming.
Three times the captain tried with the cigarette lighter
and failed completely to produce a spark. The natives,
at first attentive, began to rumble with displeasure.
“His silly old lighter won’t work!” War chortled, scarcely
able to control his laughter. “Serves him right for trying
to set himself up as king. Say—” Warwick’s gaze sought

211
first Mr. Livingston and then Jack. Both were grinning
from ear to ear. “I get it!” he cried. “Mr. Livingston, you
emptied the fluid out of that lighter, didn’t you?”
“While Jack created a diversion,” the Scout leader
confessed. “Captain Carter doesn’t have a very good
memory, or he would have recalled that I never smoke
cigarettes. He was easy to fool. I was afraid though,
that he’d check the lighter before the ceremony.”
“Hey, watch!” Willie interrupted the conversation.
“There’s going to be fireworks now! Not created by his
royal highness, Captain Carter, either!”
The captain appeared stunned by his failure to produce
fire, and then dismayed. Well he might be fearful.
Triumphant that his rival had failed, Panomuna now
danced down the temple steps, inciting the natives to
take their revenge upon the intruder.
“Keep back, you!” the captain snarled. “Keep back I
say!”
He drew his automatic and as a native came up the
temple steps to seize him, deliberately fired. The man
fell, moaning.
Captain Carter fired twice into the crowd. Then, leaping
down from the temple steps, he fled up the trail toward
the entrance to the treasure lake.
“The man is mad!” Mr. Monahan exclaimed. “Now that
he has discredited himself, he should try to escape
before the natives turn upon him completely.”
“He’s heading straight for the treasure lake!” Jack cried
in alarm. “I’ll bet he has explosives hidden up there

212
somewhere!”
Minutes passed. From their hiding place, the Scouts
watched the angry natives pursue the fleeing seaman.
Their own position, they realized, was highly precarious.
But escape, even through the lower passageway, was
cut off. They could only wait and hope that if the
situation became critical, Ino might influence the natives
in their favor.
Suddenly the Scouts heard a series of muffled
explosions which shook the earth.
“What was that?” Ken demanded, startled. “Sounded
like dynamite all right!”
“Hand grenades being exploded under water,” Mr.
Monahan informed the group. “Carter brought in a
supply of them. He’s determined to get the gold, even if
it costs him his life. And I think it will. Nothing can save
him now.”
In the plaza, a native was pounding an alarm on the
temple gong. Bong! Bong! Bong! Weirdly the sound
echoed through the streets of the village.
“Even if Captain Carter succeeds in killing the cannibal
fish, how can he hope to hold the natives at bay while
he brings up the treasure?” Jack speculated.
“It’s madness!” Mr. Monahan asserted.
“Maybe he thinks we’ll help him,” Willie began. “Maybe
—”
His speculation ceased at that point, for the ground
beneath his feet began to shake and tremble.

213
For an instant the Scouts thought that Captain Carter
had touched off another mighty explosion, more
powerful than anything that had preceded it.
But their reasoning told them better. No man-made
dynamite could cause an entire area to be so convulsed.
Walls of stone houses lining the streets were weaving
and crackling. A massive pillar came tumbling down.
Great chasms had developed in the earth, so deep that
they seemed without bottom. Monoliths of immense size
were hurled down.
“An earthquake!” cried Willie, seizing a rock for support.
“One of the worst this area has had since I’ve been
here!” gasped Mr. Monahan.
A great dust rose from the ruined city. Everywhere there
was screaming, shouting and terror as natives sought
refuge.
“The wrath of the Gods is being visited upon the city,”
murmured Mr. Monahan.
“Surely, you don’t really believe that,” returned Mr.
Livingston.
“Of course not,” the other admitted. “But that is what
the natives will think, if any survive this awful upheaval.”
Another hard tremor shook the area, leveling the statue
in the plaza. Crouching together for protection against
the falling stone, the Scouts tensely waited.

214
215
No further upheavals followed. After awhile, Mr.
Monahan decided to creep from the shelter to see what
could be done to help the injured.
“Stay here until I test the temper of the natives,” he
warned the others. “In their present mood, there’s no
telling what they may do. Those explosions and the
quake have thrown them into a panic.”
Cautiously, Mr. Monahan moved out into the devastated
street. But before he could start toward the shattered
temple, he was brought up short by the wild cries of a
mob which approached the plaza from the inner lake
trail.
Into view came the Indian warriors, their dust-streaked
faces contorted with both fear and fierce triumph. On
their shoulders they bore the lifeless, battered body of
Captain Carter.
“They’ve done for him!” exclaimed the Scout leader.
“They have,” grimly agreed Mr. Monahan. “He brought it
on himself by setting off those explosions!”
“Now what?” Jack asked, watching as the strange
procession proceeded to the temple steps. “Are they
offering prayers to the Gods?”
Mr. Monahan nodded. “And may they be appeased!” he
murmured. “If they show displeasure by further earth
tremors, all our lives may be forfeit!”

216
Chapter 25
INCA GOLD
In the garden of Father Francisco’s mission, the Scouts,
their leader and Mr. Monahan sat sipping limeade from
tall, frosted glasses.
Three weeks had elapsed since the fateful morning
when Captain Carter had set off a series of explosions in
the lake within the mountain. Since that day, many
events had transpired, some of which were not pleasant
to recall.
The terrifying earthquake had completed the wreckage
of many of the impressive structures in the hidden Inca
city. The great temple had been half destroyed. Five
natives had died in the disaster, and many more had
suffered injury.
That the earth tremor had been caused by the wrath of
the gods over Carter’s desecration of the treasure lake,
the natives had become firmly convinced. Angered, they
had set upon him, taking his life.
“So you see,” Mr. Livingston soberly related to the
missionary, “everything considered, we are fortunate to
have escaped. The natives accepted us only because Mr.
Monahan was able to convince them that we were not
there to loot the pool. After the quake we cared for the
injured, and that too, helped win friendship.”

217
“What of the treasure?” the missionary inquired.
“A major portion already has been removed and
transported to Lima under guard,” Mr. Monahan
answered. “Government officials are at the scene to
complete the job. Our responsibility is ended.”
From a jacket pocket, Ken removed the ancient
parchment he first had seen in Father Francisco’s library.
“We return this to you, Father,” he said, offering the
manuscript. “It was found in Captain Carter’s dunnage
after his death.”
“That old parrot woman must have stolen it and turned
it over to him,” contributed Jack. “Captain Carter knew
you had the parchment, Father. He probably wanted it
to prevent adventurers, and particularly our party, from
seeking the lost city.”
“Your theory must be correct,” the missionary said
meditatively. “However, Captain Carter did not arrive in
Cuertos until after your party came.”
“We figure he probably tipped Lolita off about the
parchment before he left here for America,” Willie
offered his opinion. “She must have watched her chance
to snatch it, and probably was paid well, either in
jewelry or cash.”
“I’m afraid my directions for reaching the lost city were
not very helpful,” Father Francisco apologized. “I gave
you the best information available, but unfortunately, I
was deceived.”
“Deceived?” Warwick asked quickly. “In what way?”

218
“I have always believed that according to the story, the
mountain of the lost city could be seen from the
doorway of this mission.”
“Actually, it can’t be,” remarked Ken.
“The wording of the manuscript was not incorrect—only
our interpretation,” declared the missionary. “Come, I
will show you.”
Walking with difficulty, Father Francisco led the party
through the garden, into the mission. Surprisingly he did
not conduct them to the door with which they were
familiar. Instead, he took them once more to the half-
underground library.
There, the Scouts were astonished to see that the walls
had been severely cracked. Plaster still lay untouched on
the carpets.
“The quake which was so severe where you were, also
struck here,” the missionary disclosed. “The mission as
you have noted, suffered some damage. In taking down
a wall here in the library, another door, which had been
plastered over, was revealed.”
“And from this original door, one would gaze directly
toward the treasure mountain!” exclaimed Jack. “No
wonder so many explorers were thrown off the track!”
Mr. Livingston told Father Francisco that he and the
Scouts planned to return to the United States as soon
as flight tickets could be obtained. Burton Monahan
would remain a few weeks longer to assist government
officials in cataloging the treasures taken from the Inca
city.

219
Mr. Monahan turned gratefully to the Scouts. “I can’t
thank you fellows enough for undertaking a dangerous
mission in my behalf,” he told them. “If it hadn’t been
for you, I’m afraid Captain Carter would have
accomplished his evil purpose. Alone, I’d never have
been a match for him.”
“It was Mr. Livingston’s trick with the cigarette lighter
that proved his undoing,” Ken chuckled at the
recollection. “’Course, the earthquake helped. Even now,
the natives can’t be convinced that Carter didn’t set off
the earth tremors with those grenade explosions.”
“All in all, it’s been a real trip of exploration,” Jack
contributed. “One we’ll never forget. After Peru though,
it will be hard to tame ourselves down enough to
schedule a canoe trip to Minnesota.”
“Oh, I don’t know,” drawled War. “Right now, I can’t
imagine anything that would be more fun than to hit
white water.”
“Or a quiet fishing trip,” added Ken.
“Depends on the kind of fish you go after,” declared
Willie with a grin. “Perch or cannibals?”
“I’ll settle for muskies,” Ken laughed. “Even a nice peppy
bass!”
“The Minnesota trip may have to wait awhile,” Mr.
Livingston told the Explorers.
“Oh, that’s all right,” Ken assured him. “After a long,
hard trip such as this, we won’t need another vacation
for awhile. Belton is good enough for us.”

220
“How long you fellows stay there will be strictly up to
you,” the Scout leader hinted. “The truth is—I hate to
tell you this—”
“Go ahead,” Willie urged. “After what we’ve been
through, we can take anything.”
“You can, and that’s a fact,” Mr. Livingston responded
warmly. “I’ve told you before, and I repeat, you fellows
more than lived up to my hopes and expectations on
this trip.”
“Tell us the news,” Jack interrupted impatiently. “What’s
in the wind, Hap?”
“Word of our successful mission here has spread. I’ve
already had an offer of another expedition—one which
would bring us back to South America.”
“To Peru again?” questioned Ken.
“No,” Mr. Livingston replied, “but possibly to an even
more interesting country. How does that strike you?”
“It hits me from the ground floor up,” asserted War.
“When do we take on this new job?”
“Not for awhile,” Mr. Livingston said, smiling at his
eagerness. “We all need a little rest, and I want to rid
myself completely of fever before I lead you off on
another jaunt. For that matter, other offers may
develop.”
“Then, for the immediate future, it’s Belton?” Ken asked.
“Right. We should have our flight tickets by tomorrow.”

“Just think of the yarns we can spin when we tie up
with the fellows again,” chuckled Ken, relishing the
prospect. “Lucky we still have a few Inca trophies, or
I’m afraid no one would believe our story.”
“So it’s back to the USA and good old Post 21,”
announced Jack with a flourish.
“To paved roads and plenty of hot running water,” added
War.
“To hamburgers and double-dip ice cream sundaes,”
completed Willie, his eyes twinkling. “Peru’s great, but
right now, I’d trade every souvenir in the world for a
nice restful day at home!”
Boy Scout Explorers at Treasure Mountain
The lure of Inca gold led Burton Monahan on a
dangerous trip to the mountains of Peru. When word is
brought back that he has apparently disappeared, his
brother asks “Hap” Livingston and his Boy Scout
Explorers to try to find out what happened.
An ancient parchment provides clues to the location of
the Treasure Mountain but it cannot forewarn the
Explorers of the many hazards—both natural and man-
made—that must be surmounted before their goal can
be reached.
An exciting, live-action story, filled with thrilling
incidents of courage and bravery, sure to hold the

interest of every adventure enthusiast.
BOY SCOUT EXPLORERS
By Don Palmer
The BOY SCOUT EXPLORERS is a part of the BOY
SCOUTS OF AMERICA, a world wide organization that
helps to mold the boys of today into the men of
tomorrow.
The Boys of Explorer Post No. 21 have a very good
leader named George (Happy) Livingston. He directs the
regular meetings of the Post, and also takes them on
various outings, camping trips, etc.
Follow the adventures of this group of boys, as they
search for a lost treasure, etc. and run into many
thrilling experiences.
THE TITLES ARE:
1. BOY SCOUT EXPLORERS at EMERALD VALLEY.
2. BOY SCOUT EXPLORERS at TREASURE MOUNTAIN.
3. BOY SCOUT EXPLORERS at HEADLESS HOLLOW.
DAN CARTER BOOKS
By MILDRED A. WIRT

An exciting series of action stories about a group of
youngsters and the fun they enjoy as CUB SCOUTS, the
junior organization of the BOY SCOUTS. Every boy will
get a kick out of the adventures of DAN CARTER and
how he and his Pack help to solve some thrilling
mysteries. For boys eight to eleven.
1 Dan Carter Cub Scout
2 Dan Carter and the River Camp
3 Dan Carter and the Money Box
4 Dan Carter and the Haunted Castle
5 Dan Carter and the Great Carved Face
6 Dan Carter and the Cub Honor

THE BOY SCOUT EXPLORERS
By DON PALMER
THE BOY SCOUT EXPLORERS, senior branch of THE
BOY SCOUTS of AMERICA, is an organization dedicated
towards molding the good character of our boys and
promoting their good citizenship. It seeks to instill them
with a spirit of civic duty and readiness to help others
by stimulating their interest in wholesome and creative
activities.
Every boy will enjoy reading about the interesting
adventures of The Boys of Explorer Post No. 21 and

their capable Scout Leader, George (Happy) Livingston.
Ideal stories for boys from 10 to 14.
1 Boy Scout Explorers at Emerald Valley
2 Boy Scout Explorers at Treasure Mountain
3 Boy Scout Explorers at Headless Hollow
For sale at all book and department stores.
CUPPLES & LEON COMPANY
200 Fifth Avenue New York 10, N. Y.

Transcriber’s Notes
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is public domain in the country of publication.
Silently corrected palpable typos; left non-standard
spellings and dialect unchanged.
In the text versions, delimited italics text in
_underscores_ (the HTML version reproduces the font
form of the printed book.)

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