1001 Solved Engineering Fundamentals Problems

1001_Solved_Engineering_Fundamentals_Problems-2C_3rd_ed._by_Michael_R._Lindeburg

Engineering Fufdamentals

Third Edition

MichaEf R Hndebdtu, PE

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How to Locate and Report Errata for This Book

At PPI, we do our best to bring you error-free books But when errors do occur, we

so the errors cause as little confusion as possible

A current list of known errata and other updates for this book is available on the PPI website at www.ppi2pass.com/errata We update the errata page as often as

necessary, so check in regularly You will also find instructions for submitting suspected errata We are grateful to every reader who takes the time to help us improve the quality

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3 1 New edition Copyright update

All units converted to SI,

3 2 Minor corrections

Copyright (©) 2005 by Professional Publications, Inc (PPI) All rights reserved No part

of this publication may be réproduced, stored in a retrieval system, or transmitted, in

any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the publisher

Printed in the United States of America

ACKNOWLEDGMENTS vũ

HOW TO USE THISBOOK ix MATHEMATICS 1-1 ECONOMICS .0.0 0004 2b ae 2-1 ĐVSTEM§OFUNITS 3-1 FLUID STATICS AND DYNAMICS 4-1 THERMODYNAMICS 5-1 POWERCYCLERS 6-1 CHEMISTRY 7-1 ĐIATIOS Q Q Q Q cv 8-1 MATERIALSSCIENCE 9-1 MECHANICS OF MATERIALS ., 10-1 DYNAMICS .0 0 00020004 11-1 DCBELECTRICITY 19-1

AC ELECTRICITY .004 13-1

PHYSOS 14-1

SVYSTEMS MODELING 15-1 COMPUTER SCIENCE 16-1

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PREFACE TO THE THIRD EDITION

Tn the good old days, when examinations for engineering licensing were in their infancy, most review books were mainly compilations of problems with little supporting theory—much like this book Such books placed the burden on the examinees to accumulate and become familiar with numerous textbooks and

literally brought shopping carts of books to the exam), it also changed the na-

ture of how most examinees conduct their review The scope of the exam has

narrowed significantly, and the scope of the examinees’ review has shrunk ac-

cordingly, placing more emphasis on the ability to quickly work certain standard types of problems

Since you cannot use your own reference books in the Fundamentals of Engi-

neering (FE) exam, working countless practice problems has become the review method of choice for many examinees This book contains the prist for the

problem review mill

Parallel to the actual exam, this edition features problems that all use the SI

system of units Approximately half of the problems have been revised or re- placed in order to use these units New problems have been added to cover high-probability subjects that had no representation in previous editions The

entire book has been edited to incorporate PPI’s rigorous style and quality stan-

ACKNOWLEDGMENTS

The FE examination is a secure examination, which means problems are not

made public after the exam is given Therefore, all of the problems in this

book were developed from scratch, based on our knowledge of the examination

Working from detailed outlines that I prepared, a team of more than 20 engineers and engineering students fleshed out the problems in the first edition

This third edition shares a similar heritage with the first edition in that many

people contributed to it Timothy W Zeigler, PE, revised, wrote, and rewrote approximately 750 problems for the Mathematics, Economics, Systems of Units, Fluid Statics and Dynamics, Thermodynamics, Chemistry, Statics Materials Science, Mechanics of Materials, Dynamics, and Physics chapters Similarly, Yanqing Du, PhD, PE, revised, wrote, and rewrote almost 300 problems for the

Power Cycles, DC Electricity, AC Electricity, Systems Modeling, Computer Sci- ence, and Atomic Theory chapters Without their authorship, this new edition would still be sitting in the in-basket on my desk

Christopher Lew, a master’s program candidate at Stanford University and for-

mer engineering intern in the PPI editorial department, performed the first round

of calculation and unit checks You can thank him for the accuracy of the cal-

culations

PPI editorial and production ‘staff who worked diligently on this book include:

Sarah Hubbard, editorial director; Cathy Schrott, production director; Amy and Sean Sullivan, project editor These people gave the book a consistency and

personality of its own The styles and conventions you see in this book are the

result of their work

Many of the thousands of readers of the previous two editions sent in their com-

ments and suggestions Though they are nameless, this book is better because

of their help

Michael R Lindeburg, PE

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HOW TO USE THIS BOOK

If you never read the material at the front of your books anyway, and if you are

in a hurry to begin and you only want to read one paragraph, here it is All chapters in this book are independent Start with any one Solve all the problems for which you have time Don’t peek at the answers Work

problems in your weak areas as well as in your strong areas Keep studying

until the exam Good luck ,

However, if you want a thorough review, you will probably want to know a little more about reviewing for the exam The rest of this introduction is for you

This book doesn’t contain any supporting theory and is not really meant to be

used as a stand-alone exam review It was meant to be used in conjunction

with two other references: a FE exam review book (e.g., either the FE Review Manual or Engineer-In-Training Reference Manual) and the NCEES Handbook All three of the books listed are available from PPI

Depending on your preference, you might decide to first review a subject and then work practice problems Or, you might decide to jump right in and try to

work the problems, reviewing only those subjects that are rusty or unfamiliar

This book can be used either way

While working through the problems, you should try to use the NCEES Hand-

book as your sole reference source Refer to the FE Review Manual or Engineer- In-Training Reference Manual when you need to refresh your memory about subjects that have become dim However, use the NCEES Handbook when you with the only reference that is permitted in the examination room

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The intersection of {2,4,5, 7,8} and {1,2,6,7} is the set of all ele-

ments appearing in both

When f(t) = f(—t), the function is “mirrored” on either side of the

vertical axis This is known as even symmetry

1-2 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS | MATHEMATICS-3

What is the value of each interior angle of a regular pentagon?

MATHEMATICS—4

A cubical container that measures 2 m on a side is tightly packed with eight

balls and is filled with water All eight balls are in contact with the walls of the

container and the adjacent balls All of the balls are the same size What is the

volume of water in the container?

(A) 0.38 mề (B) 2.5 m3 (C) 3.8 m3 (D) 4.2 m3

Since the balls are tightly packed, ryan = 0.5 m

Vwater = Vhax — 8MWban

MATHEMATICS 1-3 The number of significant figures, or digits, for each choice is (A) 2,

greater than or equal to zero, since all terms containing z are squared There is no solution to this equation for real values of z

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1-4 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS

MATHEMATICS-9

What is the radius of the circle defined by x? + y? — 4x + 8y = 7?

Since the general equation for a circle is (2 — a)? + (y — 6)? = r?,

rearrange the equation given to fit the general equation

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MATHEMATICS 1-5 MATHEMATICS-10

What is the natural logarithm of e747?

What is the value of (0.001)?/3?

(A) antilog (# log 0.001)

the top (fifth) level to the second level, which results in a reduction of $122.00

per month What is the employee’s present salary per month?

(A) $440/mo (B) $570/mo (C) $680/mo (D) $900/mo

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1-6 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS The salary levels represent a geometric sequence Let 5; be the salary

at level i

S3 = 1.055, S4= 1.0553

MATHEMATICS 1-7 The determinant, D, is calculated as follows

To find the determinant, expand by minors across the top row

1-8 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS

D is the determinant of X For matrix A,

D = cos’ @ — (sin @)(—sin 9)

= cos” @ + sin? @

=]

-1 cos@ sind

Av = (35) ne) The answer is (C)

MATHEMATICS 1-9 The rank of a matrix is the number of independent vectors (rows) The rank can be found by row-reducing (diagonalizing) the matrix and counting the number of pivots in the row-reduced form of the

matrix

Row 2 = (—2)(Row 2) + (3)(Row 4)

Row 4 = Row 4 — Row 2

Row 3 = (2)(Row 3) + Row 2

The row-reduced form of A is

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1-10 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS

Use Cramer’s rule to solve the system of equations

If sina = x, what is seca?

Since sina is the side facing angle « divided by the hypotenuse, the

MATHEMATICS MATHEMATICS—20

Experimental data show that a body’s temperature declines exponentially in time

according to the expression T(t) = 50e—°-4* (where 50 is a constant expressed

1-12 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS

a,

A als “ncaa” oT >10 ———

A) The tangent is positive

(B) The cotangent is positive

(C) The cosine is negative

(D) The sine is negative

In the second quadrant, the natural functions and their signs are as

follows _

sin positive

cos negative tan negative cot negative sec negative csc positive

The answer is (C)

MATHEMATICS~—23

What is the inverse natural function of the cosecant?

(A) secant (B) sine (C) cosine (D) tangent

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MATHEMATICS 1-13

opposite side The sine is the opposite side divided by the hypotenuse

In a right triangle, the cosecant is the hypotenuse divided by the

1 sin Q =

The double angle formula for the sine function is

sin 27 = 2sinz cosxr

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1-14 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS

MATHEMATICS-—27

A transit set up 40 m from the base of a vertical chimney reads 32°30! with the

crosshairs set on the top of the chimney With the telescope level, the vertical

rod at the base of the chimney is 2.1m Approximately how tall is the chimney?

At approximately what time between the hours of 12:00 noon and 1:00 p.m

would the angle between the hour hand and the minute hand of a continuously

driven clock be exactly 180°?

1-16 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS

In the preceding equations, f is in minutes past 12:00 noon The angle between the two hands is a1 — ao

Qy—- a9 = 180°

(6£)° — (0.5¢)° = 180°

(5.5t)° = 180°

t = 32.7 min The time is approximately 12:33 p.m

The slope-intercept form of the equation of a straight line is y =

mz +b, where m is the slope and b is the y-intercept

— -18 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS

MATHEMATICS-33

Find the equation of the line that passes through the points (0,0) and (2, —2)

Since the line passes through the origin, the y-intercept is 0 Thus, the equation simplifies to y = max Substituting for the known points,

The answer is (D)

MATHEMATICS-34

What is the name for a vector that represents the sum of two vectors?

(A) scalar (B) resultant (C) tensor (D) moment

By definition, the sum of two vectors is known as the resultant

What type of curve is generated by a point that moves in uniform circular motion

about an axis, while travelling with a constant speed, v, parallel to the axis?

(A) a cycloid (B) anepicycloid (C) ahypocycloid (D) a helix

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1-20 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS

A curve generated by the method described is called a helix and is illustrated in the following figure

What is the term that describes a possible outcome of an experiment?

(A) a sample space (B) a random point

(C) an event (D) a finite set

By definition, an event is a possible outcome of a trial or experiment

(D) defined only over a finite horizon

The values of a random variable can be continuous or discrete over

a finite or infinite domain The values in the sample space can be shared by other sample spaces However, the values of a random variable must be numerical

The answer is (B)

MATHEMATICS—41

If two random variables are independently distributed, what is their relationship?

(A) They are not identically distributed

(B) They are uncorrelated

(C) They are mutually exclusive

(D) Either option (A) or option (B) is true

By definition, two independently distributed random variables are un- correlated Any two random variables may or may not be identically

1-22 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS MATHEMATICS—42

Which of the following properties of probability is NOT valid?

(A) The probability of an event is always positive and less than or equal to one

(B) If & is an event which cannot occur in the sample space, the probability

of Eg is zero

(C) If events EH, and E» are mutually exclusive, then the probability of both

events occurring is zero

(D) If events E, and EF, are events from the same sample space, then P(E, +

Ey) = P(E)) + P(e) — P(E Es)

The probability law given in option (D) is valid for events from two sample spaces, not events from a single sample space The correct

rule for events from a single sample space is

P(E, + Ey) = P(E;) + P(E)

Which one of the following functions cannot be a probability density function

for the variable 2?

MATHEMATICS 1-23 (C)

If n is the number of trials, and m is the number of successes, what is the

frequency based interpretation of the probability of event E?

The probability of an event can be interpreted as the fraction of suc-

cessful outcomes when the experiment is performed an infinite number

1-24 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS

MATHEMATICS—45

For a continuous random variable X with probability density function f(x), what

is the Ser value of X?

(A) E(X) = Jy? xf (x) dx

(B) Z sự ye = [3 ef (a) de

(C) E(X) = fo° f(w) dx

(D) E(X) = fo° ada

The expected value or average of X can be defined mathematically

If P(B) #1, and A and B are not independent events, what is P(.A|B)?

(A) (P(A))(P(B)) (B) (P(BIA)) (Fa)

The probability of event A occurring, given that the dependent event

B has occurred, is predicted by the conditional probability law, com-

monly known as Bayes theorem

MATHEMATICS 1-25 MATHEMATICS-47

If the discrete random variable X has a geometric distribution parameter P and smallest mass point 0, what is the expected value of X?

(A) P (B) P-} (C) PP (D) ———

The geometric distribution is a special case of the negative binomial

distribution The mean is (1— P)/P, and the variance is (1—P)/P?

Note: Some authors define the geometric distribution with the small-

est mags point being 1 (instead of 0) In that case, the mean is 1/P

and the variance is the same as before

(A) +? (B) A(1— A) (C) AT! (D) À

For the Poisson distribution, both the mean and variance are equal

to A

The answer is (D)

1-26 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS

MATHEMATICS-50

For a discrete random variable X with probability mass function P(X), what is

the expected value of X?

(A) E(X) = $0 2:P(ai)

(A) $2.5 (B) $2.9 (C) $3.0 (D) $3.1

The expected value is the sum of the products of the individual values

and their respective probabilities

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One fair die is used in a dice game The player wins $10 if he rolls either a 1 or

a 6 He loses $5 if he turns up any other face What is the expected winning for

one roll of the die?

(A) $0.00 (B) $3.33 (C) $5.00 (D) $6.67

For a fair die, the probability of any face turning up is 1/s Therefore,

the expected value is

1-28 1001 SOLVED ENGINEERING FUNDAMENTALS PROBLEMS MATHEMATICS-B4

The probability that both stages of a two-stage rocket will function correctly

is 0.95 The reliability of the first stage is 0.98 What is the reliability of the

What is the exponential form of the complex number 3 + 4i?

(A) @83-1° (B) 5ef53-1° (C) 5et126.9° (D) 7et531

Any complex number a + bi can be converted to its equivalent expo-

nential form as follows

a+ bi= Ja? + be! arctan b/a

MATHEMATICS 1-29 MATHEMATICS-—56

What is the product of the complex numbers 3 + 4i and 7 — 2i?

= 29 + 221 The answer is (D)