1001 Solved Problems in Engineering Mathematics by Tiong & Rojas

(1)

lOOlsoLvEo PROBLEMs

IN

-

-ENGINEERING MATHEMATICS

, ...

_____

..

_...,.,.

Second Edition

JAIME R.

TIONG

• 1

BSCE,.UV

1985

(Summa Cum Laude)

. , .. ·.. UY

College of Engineering First Summa Cum Laude

,

1st

Placer,

PICE National Students'

Quiz, 1985

. .

Awardee, Outstanding Alumni, UV

. ·' Prestaent, Excel

First

Review

and

Training Center, Inc.

· ·· ·

Past President, Rotary Club of Cebu North

Pr~sident, UV Engineering

&

Arch. Alumi Association

· ·

Del~ate,

Rotary International Convention, Chicago, USA

r(~mt(l~,.

Ramon Aboitiz Foundation Inc. Triennial Awards

' . . ' Former Plant Engineer, University of the Visayas

· ·

Fo~mE!r

Reviewer, Besaviiia Engineering Review Center

f,ormer Reviewer, Salazar Institute of Technology

··

Former Faculty, UV College of .Engineering

Author, Various Engineering Reviewers

·~

ROMEO A. ROJAS Jl-i.

"·'BSEE, CIT

1991 (Cum

Laude), BSECE,

OT

1996

1st

Placer, RME Licensure Examinations, Oc;:tober 1997

8th Placer, REE Licensure Examinations, April 1999

Former Faculty, Cebu Institute of Technology

Former Technical Assistant, CIT Automation Center

Reviewer, Excel Review Center

Author, Various Electrical Engineering Rexiewers

IMPORTANT: Any copy of this book not bearing the signature of any one

of the authors or of the publisher on this page shall be considered as

comingfrom an illegal source.

(2)

TABLE OF

C O N T E N T S

Preface to the First Edition

•ndamenta~~~

2

Preface to the Second Edition Dedication

Algebra

THEORY:

DAY

1

Properties of Integers 23

of Numbers

Properties of Addition Properties of Multiplication 23 23

Conversion

Additive Identity 23

Additive inverse 23

THEORY: Multiplicative Identity 24

Number 1 Multiplicative Inverse 24

Types of Numbers 1 Properties of Equality 24

Numerals 1 Properties of Zero 24

Digits 2 Exponents 24

Real Numbers 2 Radical 25

Imaginary number 2 Surd 25

Complex number 3 Types of Surds 25

System of Numbers 3 Special Products 26

Fractions 3 Proportion 26

Types of Fractions 3 Properties of Proportion 26

Composite numbers 4 Least Common Denominator 26

Prime numbers 4 Least Common Multiple 27

Fundamental Theorem of Greatest Common Factor 27

Arithmetic 4 Remainder Theorem 27

Types of Prime Numbers 5 Factor Theorem 27

Perfect Number 5 Trivia 27

Abundant Number

6

Quote 27

Deficient Number 6

Perfect Number

6

TEST (50 Problems for 3.75 hours) 28

Amicable Number

6

SOLUTIONS 34

Friendly Number

6

Notes 43

Factorial

6

•

Significant Figures & Digits 7

_If

DAY

Forms of Approximation 7

~@tJc

Equation,

3

Conversion 7

Celsius Scale 7

mfal

Theorem &

Fahrenheit Scale 7

_Logarithms

Kelvin Scale 8

Rankine Scale 8 THEORY:

Degrees, Radians, Grads & Mils 9 Quadratic Equation 45

Trivia 9 5 Nature of Roots 45

Quote 9 Properties of Roots _{Binomial Theorem} 46

46

T"EST (50 Problems for 2 hours) 10 Properties of Expansion 46

SOLUTIONS 15 Pascal's Triangle 46

Noles 20 Coefficient of any term 47

(3)

'~'f

II

~

' '

,,

l'i.!'''·'

I. Stjm of Coefficients 47 Sum of Exponents 4 7 Degree of Polynomial/Equation 47 logarithm 47

Common & Natural Logarithms 48

·Euler's Number 48

Binary Logarithm 48

Properties of Logarithms 48

Trivia 48

Quote 48

TEST (40 Problems for 3 hours) 49

SOLUTIONS 54 Notes 60

DAY4

Mixture,

Digit, Motion

Problems

THEORY: Age Problems Work Problems Mixture Problems Digit Problems Motion Problems Coin Problems Trivia Quote

TEST (40 Problems for 4 hours) SOLUTIONS Notes

DAY

Variation,

Problems

&

Progression

THEORY: Clock Problems Variation Problems Diophantine Equations Sequence Series Progression Arithmetic Progression Geometric Progression 63 63 64

64

65

66

73 84

5

87

88

89

Infinite Geometric Progression 89 Harmonic Progression 89 Other related sequences 90

Fibonacci Numbers 90 Lucas Numbers 90 Figurate Numbers 90 Triangular numbers 90 Square numbers 90 Gnomons 90 Oblong numbers 90 Pentagonal numbers 90 Cubic numbers 90 Tetrahedral numbers 90 Cubic numl;>ers 90 Square pyramidal numbers 90 Supertetrahedral numbers 90

Trivia 9t

Quote 91

TEST (65 Problems for4.5 hours) 92

DAY

n Diagram,

trmutation,

Combination

&

Probability

THEORY: Venn Diagram Combinatorics Fundamental Principle of Counting Permutation Inversion Cyclic Permutation Permutation with Identical

Elements Assortment Combination

Relation between Permutation And Combination

Probability

Principles of Probability Mutually Exclusive Events Independent Events Binomial Distribution Odd

6

117

118

118 . 119 119 ' 119

119

119 120 120 120 121

121

Odd For Odd Against Mathematical Expectation Card Games

Probability with Dice Trivia

Quote

DAY

Geometry

THEORY: Definition of Geometry Branches of Geometry Basic Postulates of Euclid Basic Geometry Elements and

Figures Types of Angles Bisector Units of Angles ·Polygons Triangles

Other Types of Triangles Quadrilaterals

Types of Quadrilaterals Bramaguptha's Theorem Ptolemy's Theorem Areas and Perimeters of

Regular Polygons Perimeter

Circles

Useful Theorems involving Circles Ellipses Trivia Quote 122 122 122 122 123 124 124 125 131 141

1

143 143

144

145

146

147

147 148

148

150 150 150 151 151 . 152

153

154

TEST (50 Problems for 3.75 hours) 155

DAY

Geometry

THEORY: Polyhedrons

Five Regular Polyhedrons Platonic Solids Prisms Cylinders Pyramids Cones Frustum of Pyramid Frustum of Cone Prismatoid Prismoidal Formula Sphere Zone Spherical Segment Spherical Sector Spherical Pyramid Spherical Wedge Torus Ellipsoid Spheroid Trivia Quote

TEST (30 Problems for 2.5 hours) SOLUTIONS Notes

DAY

THEORY: Definition of Trigonometry Branches of Trigonometry Classification of Triangles Solution to Right Triangles Pythagorean Theorem Solution to Oblique Triangl(!s Law of Sines Law of Cosines Law of Tangents Trigonometric Identities

8

179 179 179 181 182 182 182 183 183 183 183 184

184

184 184 184

185

185 185 185 186

186

187 191 200

9

203 203 203 204 204 204 204 205 205 205

(4)

Exponential Form of the _{Fundamental Tngonometnc}. =EST (15 Problems for 1.5 hours) _~OLUTION 240 ₂₄₂

.j,·'·:·• .

_r'

_.

DAY

_.

13 .~

...

_:·

.. ·

~

.. ·· .

_.

DAY

15

Function 206 Notes 247 ':

Dlff~re_ntlal

u.ral Calculus

Amb1guous Case 206 .

ulus .(l1m1ts &

0':

Other Parts of Triangle 207

~~,:

·

DAY

11 D ·

f

)

Radius of Inscribed Circle and

'fi:;;

.. ·

enva IVeS

THEORY:

Circumscribing Circle 207 /'''i,X;AJtalytic

Geometry

THEORY: Definition of Integral Calculus 361

Plane Areas (Triangles) 208

~;~~ \~0

'

Points . Lines

&

Definition of Calculus 309 Definite and Indefinite Integrals 361

Plane Areas (Quadrilaterals) 209 '

c·

I

Limits 309 Fundamental Theorem of

Ptolemy's Theorem 209

Ire es

Theorems of Limits 309 Calculus 361

Important Properties of Triangles 209 THEORY One-Sided Limits 310 Basic Integrals 362

Important Points in Triangles 209 Rectangular Coordinates System 249 Continuity 311 Exponential & Logarithmic

Conditions for Congruency 210 Distance Formula 250 Special Limits 311 Functions 362

Conditions for Similarity 210 Distance Between Two Points Derivatives 311 Trigonometric Functions 362

Tnv1a 210 in space 250 Algebraic Functions 311 Inverse Trigonometric Functions 362

Quote 210 Slope of a Line 250 Exponential Functions 312 Hyperbolic Functions 362

Angle Between Two Lines 251 Logarithmic Functions 312 Trigonometric Substitution 363

TEST (50 Problems for 4 hours) 211 Distance Between a Point Trigonometric Functions 312 Integration by Parts 363

SOLUTIONS 217 and a line 251 Inverse Trigonometric Functions 312 Plane Areas 363

Notes 230 Distance Between Two Lines 251 Hyperbolic Functions 31.2 Centroid 364

Division of Line Segment 252 Inverse Hyperbolic Functions 312 Length of Arc 364

1 ·o

Area by coordinates 252 Trivia 313 First Proposition of Pappus 364

DAY

Lines 252 Quote 313 Volume 365

• Conic sections 253 Second Proposition of Pappus 365

Sphencal

General Equation of Conics 253 TEST (40 Problems for 4 hours) 314 Work 366

onometry

Circles 254 SOLUTIONS 319 Hooke's Law 366

Trivia 255 Notes 329 Moment of inertia 366

THEORY: Quote 255 Ml:lltiple Integrals 366

Definition of Spherical

.~;;

14

Trivia 366

Tngonometry 233 TEST (50 Problems for 4 .hours) 256

:.*.

DAY

Quote 366

Great Circle 233 SOLUTIONS 261 1t<~ ..

Small Circle 234 Notes 275

0~,.al

Calculus

TEST (50 Problems for 4 hours) 367

Pole 234

rna/Minima

&

SOLUTIONS 373

Polar Distance 234

Time Rates)

Notes 391

Spherical Wedge 234

~···

12

Spherical Triangle 234

~:'{t:

DAY

THEORY: .

Propositions of Spherical

1\a.lytic Geometry

Max~

mum

a~~

Minimum Values 331 • . • . •.•.·.·· .. ·.· .... ·.

DAY

16

I Tnangle 235 '" '"• . . Max1ma I M1mma 332 { .

· 1

. • Solutions to Right Triangle 235

Parabola, Ellipse

Time :Rates . 332 ₁

.£)ifferential .

~ Nap1er's Rules 235

&

Hyperbola

Relat1on between the vanables

&

>iEouations

'"' · Quadrantal Spherical Triangle 236 maxima I minima values 332

Solution to Oblique Triangles 236 THEORY: Trivia 337

Area of Spherical Triangle 237 Parabola 277 Quote 337 THEORY:

Terrestrial Sphere 237 Ellipse 279 Types of DE 393

Pnme Meridian 237 Hyperbola 281 TEST (35 Problems for 3 hours) 338 Order of DE 394

International Date Line 237 Polar coordinates 284 SOLUTIONS 343 Degree of DE 394

Greenwich Mean Time 238 Trivia 285 Notes 359 Types of Solutions of DE 394

Coordinated Universal Time 238 Quote 285 Applications of DE 395

Latitude i':lnd Longitude 238 Trivia 396

Terrestrial Sphere Constants 239 TEST (55 Problems for 4 hours) 286 Quote 396

Trivia 239 SOLUTIONS 292

(5)

Centripetal Force

443 ; ..

5 '~

.,

-"

DAY

20

TEST

(30

Problems for

2.5

hours)

397

Law of Universal Gravitation

443 ~~~>

DAY

22

SOLUTIONS

₄₀₁

_Work

₄₄₄

,.:

_{., "::<'•···}

~'3',, ', ~~~>l: 'tf;~·.·:·~: :.

~~~t~,£" E~~~~=~ii~~

;~r1: < >«;

Notes

₄₁₀

Energy

₄₄₄

_{,,;~;;:r:>,;:}

_{. ..}

_Engineering

Law of Conservation of Energy

445 _{mfi!fiebhomy (Simple}

_&

DAY

11

Power

₄₄₅

(Dynamics)

Momentum

445 Compound Interest

Law of Conservation of THEORY:

Momentum

445

Types of Rectilinear Translation

487

THEORY:

Impulse

445

Horizontal Translation

487

Definition of Terms

531

Types of Collisions

445

Vertical Translation

488

Consumers & Producers

THEORY: Coefficient of Restitution

445

Free Falling Body

488

Goods and Serv1ces

531

Complex Numbers

413

Gas laws

446

Curvilinear Translation

489

Necessity and Luxury

531

Different Forms of Complex Properties of Fluids

446

Projectile or Trajectory

489

Market Situations

.532

Numbers

413

Archimedes Principle

446

Rotation

490

Demand

533

Mathematical Operation of Trivia

447

D'Aiembert's Principle

490

Supply

534

Complex Numbers

414

Quote

447

Centrifugal force

491

Law of Supply and Demand

534

Matrices

415

Banking of Highway Curve

491

Interest

₅₃₅

Sum 6( two matrices

416

TEST

(40

Problems for

3

hours)

448

Trivia

492

Simple Interest

₅₃₅

Difference of two matrices

416

SOLUTIONS

453

Quote

492

Discount

536

Product of two matrices

416

Notes

462

Compound Interest

_536'

Division of matrices

417

TEST

(45

Problems for

4

hours)

493

Continuous Compounding

537

Transpose matrix

417

SOLUTIONS

499

Nominal

&

effective rates of

Cofactor of an entry of a matrix

417

19

Notes

511

interest

538

Cofactor matrix . •

417

_·~-1-B~%\,·'"t'W

DAY

_Trivia

538

Inverse matrix

417 _·

!~lJtingineering

DAY

21

Quote

538

Determinants

418 _.

_h~rdcs

_(Statics)

Properties of Determinants

418 _{Strength of}

TEST

(40

Problems for

3

hours)

539

Laplace transform

419

SOLUTIONS

₅₄₅

Laplace transforms of elementary THEORY:

Materials

Notes

551

functions

419

Definition of Terms

465

Trivia

420

Branches of Mechanics

465

THEORY:

Conditions for Equilibrium

465

Definition of Terms

513

'iJ'''l[''''''

23 ·,#z~!;<

DAY

TEST

(50

Problems for

4

hours)

421

Friction

466

Simple St~ess

513 .

·~-il;;\

SOLUTIONS Types of Normal Stress

514

' 1Z. ·.' • •

r"'f"''t"' ~

428

_{Parabolic Cable}

466 ::~'~l:t

_Engmeermg

' _Notes

₄₃₈

_{Simple Strain}

₅₁₄

Catenary

467 _{"foilomy (Annuity,}

Moment of inertia

467

Hooke's Law

514

18

Mass moment of inertia

468

Stress-Strain Diagram

515 Depreciation, Bonds,

DAY

Trivia

469

Thermal Stress

515 _{Breakeven analysis,}

'II

-\{•

Quote

469

Thin-Walled Cylinder

516 etc.

'!:"''',

_Physics

~it.'. Torsion

516

TEST

(35

Problems for

3.5

hours)

470

Helical springs

517

THEORY:

SOLUTIONS

475

Trivia

517

Annuity

553

THEORY: Notes

484

Quote

517

Capitalized Cost

555

Vector

&

Scalar Quantities

441

Annual Cost

₅₅₅

Classifications of Vectors

441

TEST

(30

Problems for

2.5

hours)

518

Bonds.

₅₅₅

Speed and Velocity

442

SOLUTIONS

₅₂₂

Depreciation

₅₅₇

Distance and Displacement

442

Notes

528

Break Even Analysis

558

Acceleration

442

Legal Forms of Business

Laws of Motion

442

Organizations

₅₅₈

Force

442

Trivia

₅₅₉

(6)

I

I :RJ·;~i;t~t£

-1ces

A. GLOSSARY

B. UNITS & CONVERSION

c.

PHYSICAL CONSTANTS D. NUMERATION E. MATH NOTATION F. GREEK ALPHABETS G. DIVISIBILITY RULES 560 568 580 583 625 633 634 634 635 636 )",,.._ •;, k. ) ' .. v .)

rr

Pl:R~ON~F\L-

ef(opt:: \<..11'

Cf .

~b~

~

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f'ffl'ff.c:5

(7)

~-'

il

~' ,,./

~

Theory

D

Problems

Cl

Solut1ons

D

~

Mon

D

Tue

D

Wed

,--

_j

Thu

LJ

_Fri

[_]

Notes Sat What is a number?

A number is an item that describes

a

magnitude or a position.

What are the types of numbers? Numbers are classified into two types, namely cardinal numbers and ordinal numbers.

Cardinal numbers are numbers which allow us to count the objects or ideas in a given collection. Example, 1 ,2,3 ... , 1000, 100000 while ordinal numbers state the position of the individual objects in a sequence. Example, First, second, third ..

IIYJ:l.!lt

are numerals?

Numerals CJre symbols, or combination of ·;ymb<)ls wt1ich describe i1 number.

Topics

Cardinal and Ordinal Numbers

Numerals and Digits

System of Numbers

-Natural numbers, Integers,

Rational numbers,

lrrationC!I

numbers

&

imaginary numbers

Complex numbers

Types of Fractions

Composite Numbers

Prime Numbers

Defective and Abundant Numbers

Amicable Numbers

Significant Figures and Digits

Forms of Approximation

Conversion

The most widely used numerals are the Arabic numerals and the Roman numerals.

Arabic numerals were simply the modification of the Hindu-Arabic number signs and are written in Arabic digits. Taken singly, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and in combination 20, 21, 22, ... 1999, ... The Roman numerals are numbers which are written in Latin alphabet. Example MCMXCIV.

The following are Roman numerals and their equivalent Arabic numbers:

I

=

1 C = 100

v

=5 0=500 X = 10 M "' 1000 L :: 50

The Romuns used the following to indicate large nurnbers:

(8)

2 l 00 l s21v~d Problems ·in Engineering Mathematics (2"d Edition) by Tiong & Rojas 1. Bracket - to (11ultiply it by 100

times.

lVI

= 500

2. Vinculum (bar above the humber) - to multiply the number

1000 times.

v

= 5,000

3. Doorframe- to multiply the

number by 1000000 times

I

fVl

= 5, 000,000

What i2J! digit?

A digit is a specific symbol or symbols used alone or in combination to denote a number.

For example, the number 21 has two digits, namely 2 and 1. In Roman numerals, the number 9 is denoted as IX. So the digits I and X were used together to denote one number and that is the number 9.

In· mathematical computations or in some engineering applications, a system of numbers using cardinal numbers was established and widely used.

What ~re real numbers?

The number system is divided into two categories namely, real numbers and imaginary number.

Real numbers are classified as follows:

1. Natural numbers - numbers which

are considered as the "counting numbers".

Examples: 1 ,2, 3 ...

2. Integers- are all the natural number, the negative of the natural numbers and the number zero.

Examples: - 4, -1, 0, 3, 8

3. Rational numbers - are numbers which can be expressed as a quotient (ratio) of two integers. The term "rational" comes from the word "ratio".

2

Examples: 0.5,

3 ,

-3, 0.333 ...

In the above example, 0.5 can be

1

expressed as - and -3 can be

2

-6

expressed as -,.hence the two 2

examples are rational numbers.

The number 0.333 ... can also be express·ed as __!_ and therefore a

3

rational number.

The number 0.333 ... is a repeating and non-terminating decimal. As a rule, a non-terminating but repeating (or periodic) decimal is always a rational number. Also, ajl integers are rational numbers.

4. Irrational numbers - are numbers which cannot be expressed as a quotient of two integers.

Examples:

..J2,

n, e, ...

The numbers in the examples above can never be expressed exactly as a quotient of two integers. They are in fact, a non-terminating number with non-terminating decimal.

What is an imaginary number?

An imaginary number is denoted as "i" which is equal to the square root of

negative one. In some other areas in mathematical computation, especially in electronics and electrical engineering it is denoted as "j".

Day l - Systems of Numbers and Conversion 3

Imaginary number and its equivalent:.

i =~

'

i2 = -1

i3_{=-i =-} _~

i4

=

1

What is a complex number?

A complex number is an expression of both real and imaginary number combined. It takes the form of a + bi, where "a" and "b" are real numbers.

If a

=

0, then pure imaginary number is produced while real number is obtained when b

=

0.

What is a system of numbers?

A system of numbers is a diagram or chart which shows the two sub-classifications of the two basic classifications of numbers, namely real numbers and imaginary numbers.

System of Numbers

/Real Imaginary

Numbers Number

Pi~l<J~::jtn showing the riumbe~ system

What is an absolute value?

The absolute value of a real number is the numerical value of the number neglecting the sign.

For example, the absolute value of- 5 is 5 while of -x is x. The absolute value

I

a

I

is either positive or zero but can never be negative.

What are fractions?

Fractions are numbers which are in the form of

~or

a/b, where a is called!he numerator which may be any integer while b is called the denominator which may be any integer greater than zero. Fraction is also defined as a part of a whole.

What are the types of fractions?

1. Simple fraction - a fraction in which the numerator and denominator are both integers. This is also known as a common fraction. 2 Examples:

3 .

6 7

2. Proper fraction - is one where the numerator is smaller thai! the denominator.

5 2

Examples:

7' 3

3. Improper fraction - is one where the numerator is greater than the denominator.

5 12 Examples:

2' 7

4. Unit fraction - is a fraction with unity for its numerator and positive integer for its denominator.

(9)

"

•r""''"''

II

!_lt:'O 1 Solved Problems in Engineering Mathematics (2"ct Edition) by Tiong & Rojas

1

Examples:

4·

25

5. Simplified fraction - a fraction whose numerator and denominator are 1ntegers and their greatest common factor is 1.

1 8

Examples:

2' -11

6. An lnteJer Represented as fraction

- a fraction in which the denominator is 1.

E xamples: -2 _{1 , 1}--3

7. Reciprocal- a fraction that results from interchanging the numerator and der.1ominator.

Examples: 4 is the reciprocal of

-~

4

8. Complex fraction -a fraction in which the numerator or

denominator, or both are fractions. 3 5 -Examples: __4_

-~

13

7' 1' -- - 2 8 4

9. Similar fractions -two or more simple fraction that have the same denominator.

1 4

!_

Examples:

g-· ·g-·

9

10.

Zero fraction - a fraction in which the numerator is zero. A zero fraction is equal to zero.

0

Examples:

2 11

11.

Undefined fraction- a fraction with a denom1nator of zero. The example below means that 8 is divided by 0, which is an impossibility because nothing can be divided by zero.

8 Examples:

-0

· 12.

Indeterminate fraction - a fraction which has no quantitative meaning.

0

Examples:

0

13.

Mixed number- a number that is a combination of an integer and a proper fraction.

1 8

Examples: 5-. 9 -2 11

What is a composite number?

Composite numbers are positive integers that have more than two positive whole number factors. It can be written as product of two or more integers, each greater than 1. It is observed that most integers are composite numbers. The number 6 is a composite number because its factors are 1, 2, 3 and 6. The number 1 is the only natural nu"'lber ti1at is neither composite nor prime.

What is a prime nu111ber?

A prime number is an integer qreater than

1 that is divisible only by 1 and it~elf.

According to the fundamental theorem of arithmetic, " Every positive integer greater !h.an 1 is a prime or can be expressed as a unique product of primes and powers of primes".

The following 1s a list of the prime numbers less than 1

,000.

Day 1 - Systems of Numbers and Conversion 5

2,3, 5, 7, 11, 13, 17, 19,23,29,31, 37,

41,43,47,53, 59,61,67, 71, 73, 79,83,

89, 97,101, 103,107,109, 113,127,131,

137. 139, 149, 151, 157, 163, 167, 173,

179, 181, 191, 193, 197, 199,211,223,

227,229,233,239,241,251,257,263,

269,271,277,281, 283, 293,307, 311,

3.13, 317,331, 337, 347, 349,353, 359,

367,373,379,383,389,397,401,409,

419,421,431,433,439,443,449,457,

461,463,467,479,487,491,499,503,

509,521,523,541,547,557,563,569,

571,577,587,593,599,601,607,613,

617,619,631,641,643,647,653,659,

661,673,677,683,691,701,709,719,

727,733, 739, 743, 751, 757, 761, 769,

773,787,797,809,811,821,823,827,

829,839,853,857,859,863,877,881,

883,887,907,911,919,929,937,941,

947,953,967,971,977,983,991,997,

The number 2 is the only prime number which is an even number.

What are the types of prime numbers?

Na.tural prime numbers are those that have only two factors; 1 and the number. Twin primes are a set of two consecutive odd primes, which differ by two. The following are twin primes less than

100.

3, 5 5, 7

11' 13

17,

19 29, 31

41,43

59, 61

71, 73

Symmetric primes are a pair of prime numbers that are the same distance from a given number in a number line. Symmetric primes are also called Euler primes. The following are symmetric primes for the number 1 through

25.

Number 1 2 3 4 5 6 7 8 Symmetric prime None None None

3,5

3, 7 5, 7 3, 11 5, 11; "3,

13

9 7,11; 5,13

10

1. 1a a 11

11 5, 17; 3, 19

12 11, 13; 7, 15; 5,19

13 7, 19; 3, 23

14 11,17.5 ;;_· ..

15 13, 17,

''i'

9;

7,

23

16 15, 17;

'!,., 19;

3, 29

17

11,23;6.2~

3,31

18 17, 19;

1.l,

23; 7, 29;

5,

31

19 9,29; 7, 31

20 17, 23; 11, 29; 3, 37

21 19, 23; 13, 29; 11,31;

5,

37

22 13, 31; 7, 37; 3,41

23 17,29; 13, 33; 5,41; 3,43

24 19, 29; 17, 31; 11, 37;

7,41; 5, 43

25 19, 31; 13, 37, 7,43; 3,47

Emirp (prime spelled backwards} is a prime number that remains a prime when its digits are reversed.

The following are emirps less than

500:

11, 13, 17, 31, 71, 73, 79, 97, 101, 107,

113,131, 149,151, 157, 167, 179,181,

191,199,311,313,337,347,353,359,

373,383,389

Relatively prime numbers are numbers whose greatest common factor is 1.

Unique product of power of primes is a number whose factors are prime numbers raised to a certain power.

Example of unique product of pow~r of primes:

360 =

2

3 . 32 .

5

1

What is a perfect number?

A perfect number is an integer that is ~

equal to the sL m of all its possible divisors, except the number itself.

Example:

6, 28, 496 ...

In the case of 6, the factors or diviscrs .-e 1, 2 and 3. When the factors are added the sum is ~qual to the number itself and shown in the following equation.

(10)

6 1001 Solved Problems in Engineering Mathematics (2"d Edition) by Tidng & Rojas

What are an abundant numbers and deficient numbers?

If the sum of the possible divisors is greater than the number, it is referred to as abundant number.

A defective number is an integer'with the sum of all its possible divisor is less than the number itself. It is also called deficient number.

The following is a list of the first 25

numbers with its corresponding type, D for deficient and A for abundant.

Factors Excluding

Number Itself Sum Tvoe

1 0 D 2 1 1 D 3 1 1 D 4 1, 2 3 D 5 1 1 D 6 1, 2, 3 6 Perfect 7 1 1 D 8 1, 2, 4 7 D 9 1, 3 4 D 10 1,2, 5 8 D 11 1 1 D 12 1,2,3,4,6 16 A 13 1 1 D 14 1, 2, 7 10 D 15 1,3,5 9 D 16 1, 2, 4, 8 15 D 17 1 1 D 18 1,2,3,6,9 21 A 19 1 1 D 20 1,2,4,5, 10 22 A 21 1, 3, 7 11 D 22 1' 2, 11 14 D 23 1 1 D 24 1,2,3,4,6,8, 12 36 A 25 1, 5 6 D

What is a perfect number?

Perfect number is a number that is equal to the sum of its factors excluding itself. They are mathematical rarities that have no practical use. The formula to find a perfect number is a follows:

2P-1_(2P

-1)

where:

p and ( 2P

-1)

are prime numbers Formula Perfect number

i

(2

2

_-1)

₆

22 (23 -1) 28_

2

4

(2

5

-1)

496

26 (27 -1) 8128

What are amicable numbers?

Amicable numbers or friendly numbers refer to two integers where each is the sum of all the possible divisors of the other.

The smallest known amicable numbers are

220 and 284.

The number 220 has the following factors/divisors: 1, 2, 4, 5, 10, 11, 20, 22,

44, 55, & 110 which when added sums up to 284, while the number 284 has the following divisors 1, 2, 4, 71, and 142

which adds up to 220.

There a.re more than 1000 pairs of amicable numbers have been found. Th~

following are the ten smallest pairs of amicable numbers. 220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285

and

14595 17296 and 18416 63020 and 76084 66928 and 66992 What is a factorial?

A factorial denoted as n!, represents the product of all positive integers from· t to n, inclusive.

Day I - Systems of Numbers and Conversion_ 7

Example: n! = n(n- 1 ) ... 3, 2, 1 If n = 0, by definition: (n!)(n+1) = (n+1)! ~ (0 !)(0 + 1) = (0 + 1)! 0!(1)=1! 0!

=

1

The factorial symbol ( ! ) was introduced by Christian Kramp in 1808.

What are significant figures or digits?

Significant figures or digits are· digits that define the numerical value of a number.

A digit is considered significant unless it is used to place a decimal point.

The significant digit of a number begins with the first non-zero digit and ends with the final digit, whether zerc, or non-zero.

Examples: 16.72 . 1.672 X 103 0.0016 4 significant figures 4 significant figures 2 significant figures

Example 2 is expressed in scientific notation and figures considered significant are 1, 6, 7 and 2 excluding 103 Example 3 has 2 significant figures only because the 3 zeros are used only to place a deciMal.

The number of significant digit is considered the place of accuracy. Hence, a number with 3 significant digits is said to have a three place accuracy and a number with 4 significant figures is said to have a four place accuracy.

What are the forms of approximations?

There are two forms of approximations, namely rounding and truncation.

Hounding of a number means replacing

IIi" n11rnber with diH >flier JHirnber having

I• ·w• ., •;1qnlfle<HII tl< ,, II I I'll tl1qils, m for

integer number, fewer value-carrying (non" zero) digits.

Examples:

3.14159 shall be rounded up to 3.1416 3.12354 shall be rounded down to 3.1235

Truncation refers to the dropping of the next digits in order to obtain the degree of accuracy beyond the need of practical calculations.This is just the same as rounding down and truncated values will always have values lower than the exact values.

Example:

3.1415 is truncated to 3 decimal as 3. 141

What is

a

conversion?

Conversion is the process of getting the equivalent value in another unit of measure of a certain value with a different given unit of measure.

Most conversions can be done conveniently by using a prepared conversion table while other conversions can be done through mathematical computations using formulas.

The authors suggest to the user of this book to familiarize the values in the conversion table which is found at the last part of this book labeled as "Appendix A".

How

to

convert a temperature in degree Celsius to degree Fahrenheit and vice versa?

The unit "Celsius" was named after the Swedish astronomer, Anders Celsius (1701 -1744). In this unit of temperature, the boiling point and freezing point are 100

degrees and 0 degree, respectively.

The unit "Fahrenheit" was named after the German physicist, Gabriel Daniel Fahrenheit (1686- 1736). In this unit of temperature, the boiling point and freezing point are 212 degrees and 32 degrees, respectively.

(11)

II

8 10.01 Solved Problems in Engineering Mathematics (2nd Edition) by Tiong & Rojas

'iling point 90 80 70 60

100-

0~

1:

oc-212-32 0

30 _~--li

__ _

· 20 'oF- 32

tj·: ''· __ )::"'

-10 Centigrade Scale Fahrenheit Scale

By ratio and proportion:

C·-0 F-32

=

-100-0

c

212-32 F-32 or

=

-100 180 C=-100-(F-32) 180

5

C =~(F-32) 9 9 F=-C+32

5

Problem: Convert 45°C to °F. Solution: 9 F=-C+32

5

9 F=-(45)+32 5 F = 113°F Problema

At what temperature will the Centigrade scale and the Fahrenheit scale will have the same reading?

Solution:

For same reading, F

=

C 9 F=-C+32 5 9 F=-F+32 5 O.BF

=

-32 F = -40°

How to convert temperature in degree Celsius or degree Fahrenheit to absolute temperature?

Absolute temperature may be expressed in Kelvin or in Rankine.

Kelvin was named after British physicist, William Thompson (1824-1902) the First Baron, Kelvin.

Rankine was named after Scottish engineer and physicist, William John Macquom Rankine (1820-1872). The formulas for conversion to absolute temperature are as follows:

°K=

°C+273 0R=°F+490

Problem:

Express the temperature of 60°C to absolute temperature. Solution: °K = °C+273 °K=60+273 °K

=

333° Problem:

Express the temperature of 150°F to absolute temperature. Solution: '0R=°F+490 0 R = 150+490 0 R = 640°

Day .1 - Systems of Numbers and Conversion · 9

How to convert one unit of an angle to another unit?

There are four units commonly used to measure an angle. They are degrees, radians, grads and mils.

The degree is the mostpommonly used measure of an angle. The radian is an angle subtended by an arc equal to the length of the radius of the circle.

The following is a tabulation of the unit of angle measurement and the

corresponding value in one revolution:

Unit 1 Revolution Degree 360 Radian 21t Grad 400 Mil 6400 Problem&

How many mils are there in 90 degrees? Solution:

x

90°

6400 mils 360° x =1600 mils Problema

How many radians is equivalent to 320 grads? · Solution: _x __ = 320 grads 2Tt radians 400 grads · x = 5.03 radians Problema

145 degrees is equivalent to how many grads? Solution: X 145° 400 grads = 360" x = 161.11 grads Problema

There are how many grads in 1200 mils? Solution:

x

1200 mils 400 grads 6400 mils

x=75 grads.

Study Appendix B - Prefixes which is found in the last part of this book.

Proceed to the next page for your first test. Detach and use the answer sheet provided at the last part of this book. Use pencil number 2 in shading your answer.

GOOD LUCK!

'Ol:ribia:

Did you know that... the symbol 1t (pi), which is the ratio of the circumference of a circle to its diameter was introduced by William Jones in 1706 after the initial letter of the Greek word meaning "periphery".

~uote:

"I could hardly ever known

a

mathematician who was capable of reasoning."

(12)

il

~

Mon

D

_Tue

D

Theory

~

Problems

D

_Wed

D

_Thu

D

Solutions Fri

D D

_Notes

_Sat

•• ME Board October 1996

"ifow many significant digits do 10.097

have?

A. 2

B. 3

c.

4

D. 5

,£icE Board April1991

Round off 0.003086 to three significant figures.

A. 0.003 B. 0.00309

C. 0.0031

D.

0.00308

Y,ECE Board April1991

Round off 34.2814 to four significant figures.

A. 34.2814

B.

34.281 C. 34.28

Topics

Cardinal and Ordinal Numbers

Numerals and Digits

System of Numbers

-Natural numbers, Integers,

Rational numbers, Irrational

numbers

&

imaginary numbers

Complex numbers

Types of Fractions

Composite Numbers

Prime Numbers

Defective and Abundant Numbers

Amicable Numbers

Significant Figures and Digits

Forms of Approximation

Conversion

D. 34.0

4: ME Board April 199&

·'which number has three significant figures?

A. 0.0014

B. 1.4141 C. 0.01414

D. 0.0141

~1!:CE Board April1991

'Round off 149.691 to the nP.arest integer

A. 149.69 B. 149.7 C. 150 D. 149

__.x.

&~CECE Board April1991

Round off 2.371 x 10"8 to two significant figures.

A. 2.4 X 10·8

Dayl -Systems of Numbers and Conversion 11

B. 2.37 X 10"8 C. 0.2371 X 10"9 D. o.oo2371

x

1 o·11 7. EE Board October 1994 7 + Oi is A. irrational number B. real number C. imaginary number D. a variable

8. ECE Board Marc:h 1996

The number 0.123123123123 ... is

A. irrational

B. surd C. rational D. transcendental

/«"ECE Board April1991

Round off 6785768.342 to the nearest one-tenth. A. 6785768 B. 6785768.4 C. 6785768.3 D. None of these 10. EE Board April1993

Express decimally: Fourteen Ten thousandths

A. 0.0014

B. 0.00014

C. 0.014

D. 0.14

u . ECE Board Marc:h 1996

MCM~CIV is equivalent to what number?

A: 1964

B. 1994

C. 1984 D. 1974

12. EE Board April 1993

Express decimally: Fourty-Sevenmillionth

A. 0.00000047

B. 0.0000047

c

0 000047

D 0. 00000004 7

13. EE Board April1993

Express decimally: Seven hundred twenty-five hundred thousandths

A. 0.000725

B. 0.00725

c.

0.0725

D.

0.725

14. EE Board April1993

Express decimally: Four and two tenth

A. 0.042

B. 4.02

C. 4.2

D. 0.42

.A$~/

ECE Board November 1995

Express 45" in mils

A. 80 mils

B. 800 mils

C. 8000mils D. 80000 mils

~.fi: ME Board April1997

What is the value in degrees of 1 radian?

A. 90 B. 57.3

C. 100

D. 45

Jl'/•

CE Board May 1993

3200 mils is equal to how many degrees?

A. 45"

B. 90"

C. 180"

D. 270"

18. ECE ~rd November 1995

An angular unit equivalent to 1/400 of the circumference of a circle is called

A. mil B. degree

C. radian D. grad

(13)

II

I)

12 lOOi Solved Problems in Engineering Mathematics (2nd Edition) by Tiong & Rojas

.19. EC:E Board April1999

4800 mils is equivalent to _ _ degrees.

A.

135

B. 270

C. 235

D.

142

,.a(

ME Board April 199ft

How many degrees Celsius is 100 deg~::ees

Fahrenheit?

A.

2.667° C B. 1.334°C

c.

13.34°

c

D. 37.8° C

u.

EE Board October 1990

What is the absolute temperature of the freezing point of water in degree Rankine?

A.

492

B. 0

c.

460

D.

273

-~

"8

Board October 1994

·What is the Fahrenheit equivalent of 100 degrees Celsius?

A ..

200

B. 180

c.

212

D. 100

Z31EE Board Aprii199:J

The temperature 45° C is equal to

A.

45°

F

B.

113°

F

c. 5rF

D.

81°

F

Z4;ME Board October 1994

· How many degrees Celsius is 80 degrees Fahrenheit? A. 13.34 B. 1.334

c.

26.67 D. 2.667

zs.

ME Board October 1996 1 0 to the 121

h power is the value of the

prefix A. micro

B.

femto

C. tera

D. atto

:&Cn EE Board October 1994

The micro or ~ means

A. 10"2

B. 10-e

C. 10"3

D. 10"12

:&7: RME Board October 1994

The prefix pico means

A.· 10"12 of a unit

B. 10-e of a unit

C. 10"15 of a unit

D. 10·9 of a unit

:&8. ME Bo8rd April1999

The prefix nano is opposite to A. mega

B. giga

C. tera

D. hexa

:&9•1 foot is to 12 inches as 1 yard is to _· _spans. A.

4

B. 6

c.

9

D.

24

/ :J{).

EE Board .Juae 1990

A one-inch diameter conduit is equivalent to

A. 254mm

B. 25.4 mm

C. 100mm

D. 2.54mm

Day 1 - Systems of Numbers and Conversion

13 ,:Ji:·

If a foot has 12 inches, then how many hands are there in one foot? A. 3

B. 4

c.

6

D. 8

:JZ~

How many feet difference is 1 nautical ' mile and 1 statute mile?

A. 100 feet

B. 200 feet C. 400 feet

D. 800 feet

:J~ In a hydrographic survey, a certain point below the surface of the water measures 12 fathoms. It is equivalent to a deep of how many feet?

A. 72 B. 60

C. 48

D.

36

w;

The legendary ship, Titanic that sunk in 1912 was estimated to be at the sea bottom at a deep of 18 cables. How deep it is in feet? A. 12,000

B.

12,343

c.

12,633 D. 12,960

.

:~s: ME

Board

October 1994

How many square feet is 1 00 square meters?

A 328.1

B.

929

C. 32.81

D.

1076

:56.

A certain luxury ship cruises Cebu to Manila at 21 knots. If it will take 21 hours to reach Manila from Cebu, the distance traveled by the ship is nearly

A. 847.5km

B 507 15 statute mile

C. 441 statute mile D. 414 nautical mile

57· EE Board October 1994

' Carry out the following multiplication and express your answer in cubic meter:

8 em

x

5 mm

x

2in.

A. 8x 10"2

B. 8x 102

C. 8 X 10"3

D. 8x 10_.

:JS(vVhich of the following is equivalent to

·'1

hectare?

A. 100 ares

a.

2 acres

C. 1000 square meters

D. 50000 square feet

:J9.

Ten square statute miles is equivalent

· to sections.

A. 100

B. 5

C. 10

D.

20

40.

The land area of the province of Cebu 'is 5088.39 sq. km. This is equivalent to

A.

5088.39 hectares

B.

1964.64 sq. miles

C. 2257907.2 acres

D.

5.08839 acres

.u~"'u

Board October 1994

,/How many cubic feet is 100 gallons of liquid? A. 74.80

B.

1.337 C. 13.37

D.

133.7 /

,.u;·io

~rd

Octo.,...1994 ' ME Board April1998

How many cubic meters is 100 gallons of liquid?

A. 1.638

(14)

!l'>

14 .1001 Solved Problems in Engineering- Mathematics (2nd Edition) by Tiong & Rojas

C. 0.164 D. 0.378

4~.-ME Board October 1994

.·How many cubic meters is 100 cubic feet of liquid?

A. 3.785 B. 28.31

C. 37.85 D. 2.831

4.4•-'f~n

(10) cubic meters is equivalent to r'llow many stere?

A. 5

B. 10

C. 20 D. 100

45. ME Board Aprii199S

The standard acceleration due to· gravity is

A. 32.2 ft/s2

B.

980 fUs2

C. 58.3 fUs2

D.

35.3 fUs2

46. ME Board October 1996

A ?kg mass is suspended in

a

rope. What is the tension in the rope in Sl?

A. 68:67 N B. 70 N C. 71 N D. 72 N

47. A 1 0-liter pail is full of water. Neglecting the weight of the pail, how heavy is its water content?

A. 5kg

B. 6.67 kg

c.

10 kg D. 12.5 kg

48;;'The unit of work in the mks system is .<known as joule (J) and the unit of work in the cgs system is erg. How many ergs are there in one joule?

A.

106

B.

107

C. 105

D. 104

49_.ME Board April ~:99s

c5'ne horsepower is equivalent to

A. 746 watts

B.

7460 watts

C. 74.6 watts

D.

7.46 watts

$0(ME

Board October 1994 "'1iow many horsepower is 746 kilowatts?

A. 500 B. 74.6

c.

100 D. 1000 ·:;..:. 1. D 2. B 3.

c

4. D 5.

c

6.A 7. B 8.

c

9.

c

. 10. A 11. 8 12.

c

13. 8 ,.,._ l<

0

Theory

0

Problems Solutions

0

_Notes ANSWER KEY 14.

c

27. A 40. B 15. B 28. B 41.

c

16. B 29. A 42. D 17.

c

30. B 43. D 18. D 31. A 44. B 19. 8 32. D 45. A 20. D 33. A 46.A 21. A 34. D 47.

c

22.

c

35. D 48. B 23. 8 36. 8 49.A 24.

c

37. D 50. D 25.

c

38. A 26. 8 39.

c

~ '-~

~

Mon

0 _Tue

0 _VVed

0 _Thu

0

_Fri

D

_Sat

'

Topics

Cardinal and Ordinal Numbers

Numerals and Digits

System of Numbers

- Natural numbers, Integers,

Rational numbers, Irrational

numbers

&

imaginary numbers

Complex numbers

Types of Fractions

Composite Numbers

Prime Numbers

Defective and Abundant Numbers

Amicable Numbers

Significant Figures and Digits

I

Forms of Approximation

Conversion

RATING

c:J

43-50 Topnotcher

c:J

30-42 Passer

c:J

25-29 Conditional

0

0-24 Failed

(15)

:t·

16 .1061 Solved Problems in Engineering Mathematics (2nd Edition) by Tiong & Rojas

a

_{The number 10.097 has 5 significant} figures.

II

The number 0.003086 when rounded off to three significant digitsbecomes 0.00309.

II

The number 34.2814when rounded off to four significant digitsbecomes 34.28.

II

0.0014 has two significant figures 1.4141 has five significant figures 0.01414 has four significant figures 0.0141 has three significant figures m-Ans

II

The number 149.691 when rounded off to the nearest integer becomes 150.

II

The number 2.371

x

10'8 when rounded off to two significant digitsbecomes 2.4 x 10'8.

II

7 + Oi = 7 thus, the answer is, "real number".

II

Repeating decimal number is a "rational number".

II

The number 6785768.342 when rounded off to the nearest one-tenth becomes 6785768.3.

II

₁₄

Fourteen ten thousandths= 10000 Fourteen ten thousandths= 0.0014

Ill

MCMXCIV

=

M CM . XC IV

=

1000 + 900 + 90 + 4 = 1994

El

Fourty-seven millionth

=

-~

1000000 Fourty-seven millionth = 0.000047

lEI

Seven hundred twenty-five hundred 725

thousandths =

100000 = 0.00725

Ill

Four and two tenth

=

4.2

•

By ratio and proportion: X 45° 6400 mils= 360°

x =800 mils

lrl

x

1 rad 360° .= 2Jt rad

X= 57.3°

Ill

m

Grad

Ill

x 3200 mils 360° = 6400 mils X= 180°

x

4800 mils 360° = 6400 mils

X =270°

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ D_a ... y_l_-_S_y._s_t..,..e_ms __ o_f_N_u_m __ ,bers and Conversion 17

II

Using the formula,

·c=~(F-32)

9

Ell

·c=%(100-32)

°C=37.8°C

The freezing point of water is equal to 32•F oro•c.

El

0 R= "F+460 0 R=32°+460 0 R

=

492°R • Using the formula,

•F

=~(•c)+32

°F=~(100)+32

. 5

°F = 212°F

m

Using the formula,

m

°F=*(•c)+32

°F=~(45)+32

°F=113°F

Using the formula,

·c=~(F-32)

9

m

oc

=~(a0-32)

9 oc = 26.67•c

I hu prefix tera is equivalent to 1012 _{of a}

unit

m

10·6 means micro

El

The prefix pico is equivalent to 10'12 _{of a}

unit

El

The prefix nano is equivalent

to

1

o·

9 _{of a}

unit while the prefix giga is equivalent to

109 of

a

unit . ·

m

1 span is equivalent to 9 inches 1 yard

=

3 ft

=

36 inches, thus

. 1 span 36 mches

x - - - - ::::

4 spans 9inches

El

. 10mm 1 mch

=

2.54

em x - - -

=·25.4 mm 1 em

II

1 hand is equivalent to 4 inches, thus

. hand 1 foot= 12 mches

x - .

-h-4

1nc es 1 foot

=

3 hands

m

1 statute miie

=

5280 ft. 1 nautical mile

=

6080 ft

Let: x

=

the difference between a nautical

mile and a statute mile X :: 6080 - 5280

x = 800 feet

El

1 fathom is equivalent to 6 feet, thus 12 fathoms= 6(12) = 72 feet

Ell

1 cable is squivalent to 120 fathoms, thus:

18 cables= 120(18)

=

2160 fathoms 6feet 18 cables= 2160 fathoms

x

(16)

18 100 1 'Solved Problems in Engineering Mathematics (2"d Edition) by Tiong & Rojas

ft3

18 cables= 12 960 feet 100 gallons x _ = 13.37 ft3

' 7.48 gallons

-

1 meter is equivalent to 3.281 ft, thus 1 m2 = (3.281 )2 ft2

1 m2 = 10.76 ft2

100m2_{= 100(10.76) ft}2

100m2= 1076 ft2

El

Solving for distance, D = Vt

V = 21 knots = 21 nautical miles hour D=21(21) D = ₄₄₁nm x 1.15 statute miles nautical mile D

=

507.15 statute mile

Iii

1m 8 em x-~-= 0.8 m 100 em 5 mm x 1 m = 0.005 m 1000 mm 0.08(0.005)(2) = 8 x 1

o·

4 m3

m

1 hectare = 100 ares

&Core: 1 are= 100 sq. meters

1 hectare = 1 00 ares x -1 0.:..0.::_::_sq.:.o.·-.-m.:..

1 are 1 hectare= 10,000 sq. meters

m

1 square statute mile = 1 section 10 square statute mile = 10 sections

m

1 square km = 0.386102 sq. miles A= ₅₀₈₈_₃₉km2 x 0.3861 02" miles2 km A = 1964.64 sq. miles

-

1 cubic ft. = 7.48 gallons

m

1 gallon = 3.79 liters 1000 liters = 1 cubic meters

3.79 li m3 100 gallons x x

-gallon 1000 li 100 gallons= 0.379 m3

m

Given volume is 100 cu. ft.

V

=

1 00 ft3

x (-m-)

3 3.281 ft

V

=

2.831 m3

m

1 cubic meter = .1 stere, thus,

10m3= 10 steres

m

The following are the standard gravitational acceleration:

32.2 ft/s2 981 cm/s2 9.81. m/s2

m

The unit of force (tension) in the Sl system is newtons (N). Tension= 7 kg

x(

9

-~~

m) = 68.67 N

El

Density of water ( p ) = 1000

_k~

mo

,,~ kg Density of water ( p ) = 1 -. -liter W= p·V W = 1

~

x

10 liters = 10 kg liter

m

1 joule= 107 ergs

Day 1 -Systems of Numbers and Conversion 19

II

1 horsepower

=

7 46 watts

II

1 hp = 746 watts 1 hp = 0.746 kilowatts 746

kW

x.

hp

=1000

hp

(17)

22 100 1 Solved Problems in Engineering Mathematics ~2~d Editionl by Tiong &: Roiaa

I"

\.

0

Mon Tue

0

Theory

0

Problems

Wed

0 Thu

0

Solutions Fri

0 LJ

Notes Sat

Topics

Properties of Addition of 1 nteger

Properties of Multiplication of

Integers

Properties uf Equality

Properties of Zero

Properties of Exponents

Properties of Radicals

Surds

Special Products

Properties of Proportion

Least Common Denominator

Least Common Multiple

Greatest Common Factor

Remainder Theorem

Factor Theorem

What are properties of integers? 4. Identity property Integers have special properties.

Computations of integers will become easier by understandir;g these special

properti~s. The commutative property, for

instance, allows you to change the order of adding or multiplying while the associative property allows you to change grouping. The properties of adpitiori of integers:

Closure property a + b = integer 2. Commutative property a+b=b+a :\ Associative proper1y (a 1 ll) 1 < " 1

II'

1 ' ) a+O=a

The num 2r 0 is called the additive identiy

5. Inverse property a +(-a)= 0

The number -a is called the additive inverse

6. Distributive property

a(b+c) = ab:t-ac

The properties of multiplication of integers:

Closure properly

(18)

~r

24 ~ 00 I· Solved Problems in Engineering Mathematics (2nd Edition) by Tiong & Rojas

2. Commutative· property ab=ba

3.

Associative property (ab)c=a(bc) 4. Identity property a+1=a

The number 1 is called the multiplicative identiy

5.

Inverse property

a(;)=1

The number

~-

_a

is called the

_.

multiplicative inverse

6.

Distributive property

a(b+c)=ab+ac

7. Multiplication property of zero a(O) =0

The

properties ofequality of integers: Consider a, b and c as integers or real numbers or variables of an algebraic expression. ·1. Reflexive property a=a 2. Symmetric property If a

=

b, then b

=

a 3. Transitive property

If

a = b and b

=

c, then a = c 4. Substitution property

If a

=

b, then a can be replaced by b in any expression inl'olving .

a

5. Addition I Subtraction property

If a = b, then

a

+ c

=

b + c If a

=

b, then a - c = b - c 6. Multiplication I Division property

If

a

= b, then ac = be If a = b, then

!

=

~

with c

~

0

c c

7. Cancellation property lfa+c=b+c,thena=b If ac = be and c

*

0 , then a

=

b The properties of zero:

Consider a, b and c as integers or real numbers or variables of an algebraic expression.

1. ·. a+O=a and a-O=a 2. a(O)=O

3.

~

== 0 , with

a

;t 0

a

4.

~

is undefined

5. If ab = 0, then

a

= 0 or b

=

0. This is known as Zero-Factor property

What Is an exponent?

Exponent is a number that gives the power to which a base is raised. For example, in 32, the base is 3 and the exponent is 2.

Exponent should not be misunderstood as "power" Power is a word that is almost never used in its correct, original sense any more. Strictly speaking, if

we

write 32 = 9, then 3 is the base, 2 is the exponent and 9 is the power. But almost everyone, including most mathematicians,

Day 2 -Fundamentals in Algebra 25 would say that 3 is the power and that Property Example

"power" and "exponent" mean the same

thing. The misuse has probably come from ₀r,;; m

a misunderstanding of statements such 1.

v

am = (

cya)

~

=(Wf =22

~=4

"nine is the second power of three".

The exponential notation states that if a is 2.

cya ·

cyb

=

'fab

_¥5

_·~675

a real number, variable or algebraic

expression and n is a positive number, then:

a" = a · a · a · a · · · ·

"---y---J

n factors

The properties of exponents with corresponding examples: Property Example

1.

am +a"= am+n x2 + xa = x2+3 = xs 2. am m-n X 8 a-3 s - = a -=X =X a" x3

3. (am)" =amn (y6)2 = y12 4. (abt =ambm (2x)4 =24x4 =16x4 5.

(~r

= : :

(~r=24=~

_x _x4 x4 m 5

((44

6.

a"=~

(4x)3 = 3(4x) 7. a -m = -1 am -5 1 I( = -x5 8. a0 = 1 (a;;, 0) _(x2

+2t

_{= 1} What is a radical?

Radical refers to the symbol that indicates a root,

F .

It was first used in 1525 by Christoff Rudolff in his Die Coss. In the expression,

cya ,

n

is called the Index,

a

(the expression inside the symbol) is called the radicand while the symbol

J

is called radical

cya~

3. - = n - b;t:O

%

b' 4.

'ifiFa

= mzya 5.

(cya)"

=a

?t

!fa"= lal !fa"= lal What is a surd? =

~(5)(675)

= ~3375 = 15

we=

3rso

=¥5

V10 \/10

~~=1tfl5

(~f

=2x

~(-12)

4

₌

l-121 :::12

(For n =even no.)

~(-12)

3

= -12 (For n =odd no.)

Surd is a radical expressing an irrational number. The surd is described after the index of the radical. For example,

.J3 is a

quadratic surd, ~ is a cubic surd, ~ is a quartic surd and so on.

Different types of surds:

Pure surd, sometimes called an entire surd contairls no rational number and all its terms are surds.

Example:

.J3 +

J2 .

Mixed surd is a surd that contains at least one rational number.

s.J3 is a mixed surd

because 5 is a rational number while

.J3

is a surd.

(19)

26. 1001 Solved Problems in Engineering Mathematics (2nd Edition) by Tiong & Rojas

Binomial surd is an expression of two

ter111s with at least one term a surd.

Example: 5 +

F2

Trinomial surd is an expression of three terms with at least two or. them are surds and cannot be expressed as a single surd, otherwise it will become a binomial surd. Example: 5 +

F2

+

J3

What is a special product?

Special products are the expressions where the values can be obtained without execution of long multiplication.

With x, y and z as real numbers or variables or algebraic expression, the following are the special products.

1. Sum and difference of same terms or Difference of two squares

(X

+

y

){X -

y)

= x

2 -

y

2

2. Square of a binomial ( x + y )2 = x2 + 2xy + y2 ( x -

y )

2 = x2 - 2xy +

l

3. Cube of a binomial (x + y)3 = x3 + 3x2y + 3xy2 + y3

(x-

yf = x3- 3x2y + 3xy2- y3

4. Difference of two cubes

x3 -y3 =(x-y)(x2

+xy+l)

5. Sum of two cubes

x3 + y3

=

(x- y)(x2- xy +

l)

6, Square of a trinomial

( x + y + z )2 = x2 + y2 + z2 + 2xy

:t

2xz + 2yz

What is a proportion?

Proportion is a statement that two ratios are equal. Properties of proportion a x 1. If - = - . then a : x = y : d y d

a c

a

b 2. If - then - =-b d.

c

d

a c

b d 3. If - , then - =-b d

a c

4.

If~=~

then a- b = c- d b d. . b d 5.

If~

=

~

then a + b = c + d b d. b d 6 If

~

=

~

then a + b = c + d · b d' a-b c-'d

In number ( 1 ), quantities a and d are called extremes while x and y are called means. If x = y, then its value is known as mean proportional. In the ratio xly, the first term x is called the antecedent while the second term y is called the

consequent. extremes

~

l

a:x=y:d

u

means antecedent

r:

·o

consequent

What is a least common denominator (LCD)?

Least common denominator (LCD) refers .to the product of several prime numbers occurring in the denominators, each taken with its greatest multiplicity.

Problem:

What is the least common denominator of

8,9,12and15? • ' Solution: 8

=

23 9

=

32 12=3·22 15=3·5 LCD=

2

3

(3

2

)(5)

LCD= 360

What is a least common multiple (LCMI?

A common multiple is a number that two other numbers will divide into evenly. The least common multiple (LCM) is the lowest multiple of two numbers.

Problema

What is the least common multiple of 15 and 18? Solution: 15=3·5 18

=

32_.2 LCM = 32 (5}(2) LCM

=

90

What is a greatest common factor (GCFI?

A factor is a number that divides into a larger number evenly. The greatest common factor (GCF) is the largest number that divides into two or more numbers evenly.

Problema

What Is the greatest common factor of 70 and 112?

Solution:

70

=

2. 5. 7 112=24-7

Common factors aro 2 and 7.

Day 2 - Fundamentals in Algebra 27

GCF = 2(7) GCF = 14

What is a Remainder Theorem?

Remainder Theorem states that if a polynomial in an unknown quantity x is divided by a first degree expression in the same variable, (x-k), where k may be any real number or complex number, the remainder to be expected will be equal to the sum obtained when the numerical . value. of k is substituted for x in the polynomial. Thus,

remainder = f(X)

x->k

What is a Factor Theorem?

Factor theorem states that if a polynomial is divided by (x - k) will result to a remainder of zero, then the value (x-k) is a factor of the polynomial.

Both remainder theorem and factor theorem were suggested by a French mathematician, Etienne Bezout (1730-1783).

Proceed to the next page for your second test. Detach and use the answer sheet provided at the last part of this book. Use pencil number 2 in shading your answer.

GOOD LUCK I

'Orribia:

Did you know that. .. the two long parallel lin;)S (=)as a symbol for equality was introduced by Robert Recorde in 1557!

~note:

"Among the great things which are found among us, the existence of Nothing is the greatest,"