Example

Size: px
Start display at page:

Download "Example"

Transcription

1 6. SOLVING RIGHT TRINGLES In the right tringle B shwn in Figure 6.1, the ngles re dented y α t vertex, β t vertex B, nd t vertex. The lengths f the sides ppsite the ngles α, β, nd re dented y,, nd. Nte tht ngles α nd β re mplementry ute ngles, tht ngle is right ngle, nd tht is the hyptenuse f right tringle B. Therefre, sinα = s β = sα = sin β = sα = se β = seα = s β = β B tnα = t β = tα = tn β = α Figure 6.1 If the lengths f tw sides f right tringle re given, r if ne side nd n ute ngle re given, then these frmuls sn e used t slve fr the remining ngles nd sides f the tringle. The predure is lled slving the right tringle. Exmple Slve the right tringle B leled s in Figure 6. if β = 30 nd = 4. Figure 6. α 30 B 4 We must find, nd α. Beuse α nd β re mplementry, α = 90 β = = 60. T find, we use the ft tht tn β =, s = tn β. Therefre, = tn β = 4 tn 30 = = 8 3. T find, we use the ft tht se β =, s = se β. Therefre, = se β = 4 se 30 = 4 Of urse, we uld hve insted used the Pythgren therem t find = 3 3 = Unless the speil ngles 30, 45, r 60 re invlved, it is neessry t use lultr r tle f trignmetri funtins t slve right tringle. Yu shuld lwys keep in 41

2 mind tht the slutins tined in these wys re pprximtins. In this setin, whenever suh pprximtins re invlved, we will rund ff ll ngles t the nerest hundredth f degree, nd ll side lengths t fur signifint digits. Exmple In eh se, ssume tht right tringle B is leled s in Figure 6.1. Then slve the tringle nd sketh it. ( i ) = 31, α = We must find,, nd β. Beuse α nd β re mplementry, β = 90 α = = T find, we ntie tht tn α =, s = tnα. Using lultr, we find tht tnα = tn = Therefre, runded ff t fur signifint digits, = tnα = 31 tn = 9.4. T find, we ntie tht s α =, s =. sα Therefre, runded ff t fur signifint digits, s = = = = 4.6. sα s =4.6 α = Figure 6.3 =31 β = B =9.4 One gin, we uld hve insted used the Pythgren therem t slve fr : = + = (.4) = The disrepny in the lst deiml ple ws used y using the runded ff vlue fr. The result = 4.6 is tully mre urte. The tringle is shwn in figure 6.3. ( ii ) = 4, = 3. By the Pythgren therem: = + = = 5. Nw we hve sin α = = 5 4 = 0.8. It fllws tht α = rsin 0.8 = Figure 6.4 =5 β = B =4 β = 90 α = = The tringle is shwn in Figure 5.4. α = =3 4

3 Exmple nsider hurh with steeple, s shwn in the Figure 6.5. The prlem is t lulte the height f the steeple while stnding n the grund. T find the height, mrk pint B n the grund diretly elw the steeple nd nther pint tht is 00 feet wy n the grund. t, mesure the ngles f elevtin α nd β t the tp nd ttm f the steeple. This is ll the infrmtin yu need. Figure 6.5 Let x e the height f the steeple nd y e the distne frm the grund t the ttm f the steeple. Suppse tht α = 35 nd β = 6. x + y y Then tn 35 = nd tn 6 = Slving fr y in the send equtin nd sustituting int the first yields: x + 00 tn 6 tn35 =. 00 α = 35 x y S x = 00 tn tn 6 =00 ( ) = 4.5 feet. β = 6 00 ft B Setin 6 Prlems In prlems 1-3, ssume tht the right tringle is leled s in the figure shwn. In eh se, slve the tringle nd sketh it. When pprximtins re invlved, rund ff ngles t the nerest hundredth f degree nd side lengths t fur signifint digits. 1. = 10, =10. =100, β = = 91, β = 30 α β B 43

4 Hin 4. Find the re f the shded regin f the right tringle B shwn in the figure. 5. The huse in the figure is 4 ft. wide. The ridge is 9 ft higher thn the side wlls, nd the rfters prjet 1.5 ft eynd the sides f the huse. Hw lng re the rfters? 6. Find the distne B etween the enters f the tw rllers in the figure. 7. highwy uts rner frm prel f lnd. Find the numer f res in the tringulr lt B. (Nte: 1 re = 43,560 ft ). (Hint: First use the Lw f sines, pge 54, t find the the mesure f ne ngle in B.) 44

5 8. em B is supprted y tw rssed ems. Find the distne x. 9. Frm the tp f lighthuse 10 feet ve se level, the ngle f depressin (the dwnwrd ngle frm the hrizntl) t t drift in the se is 9.4. Hw fr frm the ft f the lighthuse is the t? 10. Frm windw in n ffie uilding, I m lking t televisin twer tht is 600 meters wy (hrizntlly). The ngle f elevtin f the tp f the twer is 19.6 nd the ngle f depressin f the se f the twer is 1.3. Hw tll is the twer? 11. Find x in the figure. 1. The vertil distne frm the first t the send flr f ertin deprtment stre is 8 feet. The esltr, whih hs hrizntl reh f 86 feet, tkes 5 sends t rry persn etween flrs. Hw fst des the esltr trvel? 13. nsider tw irles th f rdius r nd with the enter f ne lying n the rim f the ther. Find the re f the mmn prt f the tw irles. 45

6 14. The Gret Pyrmid is ut 480 feet high nd its squre se mesures 760 feet n side. Find the ngle f elevtin, β, f ne f its edges. 15. Find the ngle etween prinipl dignl nd fe f ue. 16. regulr hexgn (6 equl sides) is insried i irle f rdius 4. Find the perimeter, P, nd the re,, f this hexgn. 17. Find the re f the regulr 6-pinted Str f Dvid tht is insried in irle f rdius Find the re f the regulr 5-pinted str (the pentgrm) tht is insried in irle f rdius 1. 46

The remaining two sides of the right triangle are called the legs of the right triangle.

The remaining two sides of the right triangle are called the legs of the right triangle. 10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

More information

SECTION 7-2 Law of Cosines

SECTION 7-2 Law of Cosines 516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished

More information

State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127

State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127 ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not

More information

Ratio and Proportion

Ratio and Proportion Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty

More information

End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.

End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below. End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.

More information

Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Mean and the Pythagorean Theorem Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

More information

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed

More information

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example 2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel

More information

Chapter. Contents: A Constructing decimal numbers

Chapter. Contents: A Constructing decimal numbers Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting

More information

Lesson 12.1 Trigonometric Ratios

Lesson 12.1 Trigonometric Ratios Lesson 12.1 rigonometric Rtios Nme eriod Dte In Eercises 1 6, give ech nswer s frction in terms of p, q, nd r. 1. sin 2. cos 3. tn 4. sin Q 5. cos Q 6. tn Q p In Eercises 7 12, give ech nswer s deciml

More information

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5. . Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry

More information

1. Area under a curve region bounded by the given function, vertical lines and the x axis.

1. Area under a curve region bounded by the given function, vertical lines and the x axis. Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given

More information

NCERT INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS. Trigonometric Ratios of the angle A in a triangle ABC right angled at B are defined as:

NCERT INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS. Trigonometric Ratios of the angle A in a triangle ABC right angled at B are defined as: INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: side opposite to A BC sine of A = sin A = hypotenuse

More information

Section 5-5 Solving Right Triangles*

Section 5-5 Solving Right Triangles* 5-5 Solving Right Tringles 379 79. Geometry. The re of retngulr n-sided polygon irumsried out irle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml

More information

Words Symbols Diagram. abcde. a + b + c + d + e

Words Symbols Diagram. abcde. a + b + c + d + e Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

More information

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter

More information

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929  Math Learning Center Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

Angles and Triangles

Angles and Triangles nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir

More information

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

Heron s Formula for Triangular Area

Heron s Formula for Triangular Area Heron s Formul for Tringulr Are y Christy Willims, Crystl Holom, nd Kyl Gifford Heron of Alexndri Physiist, mthemtiin, nd engineer Tught t the museum in Alexndri Interests were more prtil (mehnis, engineering,

More information

2 2 The kinetic energy of a rotating shaft is MUCH higher!!!

2 2 The kinetic energy of a rotating shaft is MUCH higher!!! Hmewrk #6 Slutin Aignment: 4., 4.4, 4.5, 4.8, 4.9, n 4. Bergen & Vittl 4. Prt A: H W S Slutin: kinetic Wkinetic H * S 3 b 5*M 5 3 B Prt B: T fin the kinetic energy f -tn truck (ee imge belw which i ctully

More information

c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00

c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00 Chter 19, exmle rolems: (19.06) A gs undergoes two roesses. First: onstnt volume @ 0.200 m 3, isohori. Pressure inreses from 2.00 10 5 P to 5.00 10 5 P. Seond: Constnt ressure @ 5.00 10 5 P, isori. olume

More information

Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn

Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?

More information

What is the phase sequence of a balanced three-phase circuit for which V an = 160 30 V and V cn = 160 90 V? Find V bn.

What is the phase sequence of a balanced three-phase circuit for which V an = 160 30 V and V cn = 160 90 V? Find V bn. Chter 1, Prblem 1. f b 400 in blnced Y-cnnected three-hse genertr, find the hse vltges, ssuming the hse sequence is: () bc (b) cb Chter 1, Slutin 1. () f b 400, then 400 n bn cn - 0 1-0 1-150 1-70 (b)

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,

More information

Right-angled triangles

Right-angled triangles 13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Right-ngled tringles Syllus referene Mesurement 4 Right-ngled tringles

More information

Chapter. Fractions. Contents: A Representing fractions

Chapter. Fractions. Contents: A Representing fractions Chpter Frtions Contents: A Representing rtions B Frtions o regulr shpes C Equl rtions D Simpliying rtions E Frtions o quntities F Compring rtion sizes G Improper rtions nd mixed numers 08 FRACTIONS (Chpter

More information

The Pythagorean Theorem

The Pythagorean Theorem The Pythgoren Theorem Pythgors ws Greek mthemtiin nd philosopher, orn on the islnd of Smos (. 58 BC). He founded numer of shools, one in prtiulr in town in southern Itly lled Crotone, whose memers eventully

More information

GED MATH STUDY GUIDE. Last revision July 15, 2011

GED MATH STUDY GUIDE. Last revision July 15, 2011 GED MATH STUDY GUIDE Last revisin July 15, 2011 General Instructins If a student demnstrates that he r she is knwledgeable n a certain lessn r subject, yu can have them d every ther prblem instead f every

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

Calculating Principal Strains using a Rectangular Strain Gage Rosette

Calculating Principal Strains using a Rectangular Strain Gage Rosette Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used

More information

MATH PLACEMENT REVIEW GUIDE

MATH PLACEMENT REVIEW GUIDE MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Network Theorems - Alternating Current examples - J. R. Lucas

Network Theorems - Alternating Current examples - J. R. Lucas Netwrk Therems - lternating urrent examples - J. R. Lucas n the previus chapter, we have been dealing mainly with direct current resistive circuits in rder t the principles f the varius therems clear.

More information

Interior and exterior angles add up to 180. Level 5 exterior angle

Interior and exterior angles add up to 180. Level 5 exterior angle 22 ngles n proof Ientify interior n exterior ngles in tringles n qurilterls lulte interior n exterior ngles of tringles n qurilterls Unerstn the ie of proof Reognise the ifferene etween onventions, efinitions

More information

Review Problems for the Final of Math 121, Fall 2014

Review Problems for the Final of Math 121, Fall 2014 Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since

More information

In order to master the techniques explained here it is vital that you undertake the practice exercises provided.

In order to master the techniques explained here it is vital that you undertake the practice exercises provided. Tringle formule m-ty-tringleformule-009-1 ommonmthemtilprolemistofindthenglesorlengthsofthesidesoftringlewhen some,utnotllofthesequntitiesreknown.itislsousefultoeletolultethere of tringle from some of

More information

Surface Area and Volume

Surface Area and Volume Surfce Are nd Volume Student Book - Series J- Mthletics Instnt Workooks Copyright Surfce re nd volume Student Book - Series J Contents Topics Topic - Surfce re of right prism Topic 2 - Surfce re of right

More information

Getting Your Fingers In On the Action

Getting Your Fingers In On the Action Rry Garfrth Getting Yur Fingers In On the Actin Once yu are able t strum with yur fingers, yu can begin fingerpicking! The first task is t learn yur hand psitin and t learn which fingers are used n which

More information

esupport Quick Start Guide

esupport Quick Start Guide esupprt Quick Start Guide Last Updated: 5/11/10 Adirndack Slutins, Inc. Helping Yu Reach Yur Peak 908.725.8869 www.adirndackslutins.cm 1 Table f Cntents PURPOSE & INTRODUCTION... 3 HOW TO LOGIN... 3 SUBMITTING

More information

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

More information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Real Analysis HW 10 Solutions

Real Analysis HW 10 Solutions Rel Anlysis HW 10 Solutions Problem 47: Show tht funtion f is bsolutely ontinuous on [, b if nd only if for eh ɛ > 0, there is δ > 0 suh tht for every finite disjoint olletion {( k, b k )} n of open intervls

More information

Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb.

Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb. - Wht You ll ern o use sine nd osine to determine side lengths in tringles... nd Wh o use the sine rtio to estimte stronomil distnes indiretl, s in Emple Sine nd osine tios hek Skills You ll Need for Help

More information

Lesson 22. 3-Dimensional Solids. Objectives. Classify 3-Dimensional solids Determine the Volume of 3-Dimensional solids. Student Name: Date:

Lesson 22. 3-Dimensional Solids. Objectives. Classify 3-Dimensional solids Determine the Volume of 3-Dimensional solids. Student Name: Date: Student Name: Date: Cntact t Persn Name: Phne Number: Lessn -Dimensinal Slids Objectives Classify -Dimensinal slids Determine the Vlume f -Dimensinal slids Authrs: Jasn March,.A. Tim Wilsn,.A. Editr: Graphics:

More information

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

More information

Mathematics Higher Level

Mathematics Higher Level Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

More information

10.6 Applications of Quadratic Equations

10.6 Applications of Quadratic Equations 10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given,

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL 1 TRIGONOMETRIC RATIOS, TRIGONOMETRIC TECHNIQUES AND GRAPHICAL METHODS

MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL 1 TRIGONOMETRIC RATIOS, TRIGONOMETRIC TECHNIQUES AND GRAPHICAL METHODS MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL 1 TRIGONOMETRIC RATIOS, TRIGONOMETRIC TECHNIQUES AND GRAPHICAL METHODS This is the ne f a series f basic tutrials in mathematics aimed at beginners r anyne

More information

SOLVING EQUATIONS BY FACTORING

SOLVING EQUATIONS BY FACTORING 316 (5-60) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting

More information

Module 5. Three-phase AC Circuits. Version 2 EE IIT, Kharagpur

Module 5. Three-phase AC Circuits. Version 2 EE IIT, Kharagpur Module 5 Three-hse A iruits Version EE IIT, Khrgur esson 8 Three-hse Blned Suly Version EE IIT, Khrgur In the module, ontining six lessons (-7), the study of iruits, onsisting of the liner elements resistne,

More information

SOLVING QUADRATIC EQUATIONS BY FACTORING

SOLVING QUADRATIC EQUATIONS BY FACTORING 6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving

More information

Business Digital Voice Site Services - Phone & User Assignments

Business Digital Voice Site Services - Phone & User Assignments Feature Overview The Phnes and Users must be assigned befre setting up ther cmpnents f Business Digital Vice. The system is designed t allw custmers t quickly setup and mdify phne assignments in real time

More information

Radius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002

Radius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002 dius of the Erth - dii Used in Geodesy Jmes. Clynh vl Postgrdute Shool, 00 I. Three dii of Erth nd Their Use There re three rdii tht ome into use in geodesy. These re funtion of ltitude in the ellipsoidl

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Further applications of area and volume

Further applications of area and volume 2 Further pplitions of re n volume 2A Are of prts of the irle 2B Are of omposite shpes 2C Simpson s rule 2D Surfe re of yliners n spheres 2E Volume of omposite solis 2F Error in mesurement Syllus referene

More information

Trigonometry Chapter 3 Lecture Notes

Trigonometry Chapter 3 Lecture Notes Ch Ntes F07 O Brien Trignmetry Chapter Lecture Ntes Sectin. Radian Measure I. Radian Measure A. Terminlgy When a central angle (θ) intercepts the circumference f a circle, the length f the piece subtended

More information

Maximum area of polygon

Maximum area of polygon Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is

More information

Quick Guide to Lisp Implementation

Quick Guide to Lisp Implementation isp Implementtion Hndout Pge 1 o 10 Quik Guide to isp Implementtion Representtion o si dt strutures isp dt strutures re lled S-epressions. The representtion o n S-epression n e roken into two piees, the

More information

Volumes of solids of revolution

Volumes of solids of revolution Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the x-xis. There is strightforwrd technique which enbles this to be done, using

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

11.2 The Law of Sines

11.2 The Law of Sines 894 Applitions of Trigonometry 11. The Lw of Sines Trigonometry literlly mens mesuring tringles nd with Chpter 10 under our belts, we re more thn prepred to do just tht. The min gol of this setion nd the

More information

Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m

Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m . Slr prout (ot prout): = osθ Vetors Summry Lws of ot prout: (i) = (ii) ( ) = = (iii) = (ngle etween two ientil vetors is egrees) (iv) = n re perpeniulr Applitions: (i) Projetion vetor: B Length of projetion

More information

Gain vs. Proportional Band

Gain vs. Proportional Band Gain vs. Prprtinal Band The end result f the analysis f a typial lp tuning predure, whether pen lp r lsed lp, is a set f parameters with whih t adjust the ntrller. One f these parameters is ntrller gain.

More information

UNCORRECTED SAMPLE PAGES

UNCORRECTED SAMPLE PAGES 6 Chpter Length, re, surfe re n volume Wht you will lern 6A Length n perimeter 6B Cirumferene of irles n perimeter of setors 6C Are of qurilterls n tringles 6D Are of irles 6E Perimeter n re of omposite

More information

Broadband Task Force Team C: Permitting Process

Broadband Task Force Team C: Permitting Process Bradband Task Frce Team C: Permitting Prcess 1/24/2014 Cunty Office Building, Chris Greene Cnference Rm Attendees: Sarah Baldwin, Michael Culp, Baylr Fks, Philip Freeman, Bill Fritz, Pat Grt, Valerie Lng,

More information

The NBEO National Center of Clinical Testing in Optometry (NCCTO) How to Access the NCCTO in the BB&T Building

The NBEO National Center of Clinical Testing in Optometry (NCCTO) How to Access the NCCTO in the BB&T Building The NBEO Natinal Center f Clinical Testing in Optmetry (NCCTO) Sectins Hw t Access the NCCTO in the BB&T Building 200 Suth Cllege Street, Suite 2020 Charltte, Nrth Carlina 28202 1.800.969.3926, Extensin

More information

Using PayPal Website Payments Pro UK with ProductCart

Using PayPal Website Payments Pro UK with ProductCart Using PayPal Website Payments Pr UK with PrductCart Overview... 2 Abut PayPal Website Payments Pr & Express Checkut... 2 What is Website Payments Pr?... 2 Website Payments Pr and Website Payments Standard...

More information

OUTLOOK All About Archives

OUTLOOK All About Archives OUTLOOK All Abut Archives Why d we have Zantaz Enterprise Archiving Slutin (EAS)? Archiving prvides verall lng term strage EAS has been implemented t help manage the 300MB mailbx size limit in Outlk/Exchange

More information

PEARL LINGUISTICS YOUR NEW LANGUAGE SERVICE PROVIDER FREQUENTLY ASKED QUESTIONS

PEARL LINGUISTICS YOUR NEW LANGUAGE SERVICE PROVIDER FREQUENTLY ASKED QUESTIONS PEARL LINGUISTICS YOUR NEW LANGUAGE SERVICE PROVIDER FREQUENTLY ASKED QUESTIONS 1) Hw d I determine which service I need? 2) Hw d I bk face t face interpreters? 3) Hw d I bk telephne interpreters? 4) Hw

More information

Times Table Activities: Multiplication

Times Table Activities: Multiplication Tny Attwd, 2012 Times Table Activities: Multiplicatin Times tables can be taught t many children simply as a cncept that is there with n explanatin as t hw r why it is there. And mst children will find

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

If two triangles are perspective from a point, then they are also perspective from a line.

If two triangles are perspective from a point, then they are also perspective from a line. Mth 487 hter 4 Prtie Prolem Solutions 1. Give the definition of eh of the following terms: () omlete qudrngle omlete qudrngle is set of four oints, no three of whih re olliner, nd the six lines inident

More information

!"#$%&'()%"*#%*+,-./-*+01.2(.*3+456789*!"#$%&"'()'*+,-."/01&2#."'3424,'

!#$%&'()%*#%*+,-./-*+01.2(.*3+456789*!#$%&'()'*+,-./01&2#.'3424,' "#$()"*#*+,-./-*+01.2(.*3+456789* "#$"()*+,-."/012#."3424, Dr. Peter T. Gallagher Astrphysics Research Grup Trinity Cllege Dublin :2";,

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Tallahassee Community College. Simplifying Radicals

Tallahassee Community College. Simplifying Radicals Tllhssee Communit College Simplifing Rdils The squre root of n positive numer is the numer tht n e squred to get the numer whose squre root we re seeking. For emple, 1 euse if we squre we get 1, whih is

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

GRADE 4. Fractions WORKSHEETS

GRADE 4. Fractions WORKSHEETS GRADE Frtions WORKSHEETS Types of frtions equivlent frtions This frtion wll shows frtions tht re equivlent. Equivlent frtions re frtions tht re the sme mount. How mny equivlent frtions n you fin? Lel eh

More information

WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS?

WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS? WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? There s n ol syng, you n t pples n ornges. Mthemtns hte n t; they love to throw pples n ornges nto foo proessor n see wht hppens. Sometmes they

More information

Volumes by Cylindrical Shells: the Shell Method

Volumes by Cylindrical Shells: the Shell Method olumes Clinril Shells: the Shell Metho Another metho of fin the volumes of solis of revolution is the shell metho. It n usull fin volumes tht re otherwise iffiult to evlute using the Dis / Wsher metho.

More information

Formal Languages and Automata Exam

Formal Languages and Automata Exam Forml Lnguges nd Automt Exm Fculty of Computers & Informtion Deprtment: Computer Science Grde: Third Course code: CSC 34 Totl Mrk: 8 Dte: 23//2 Time: 3 hours Answer the following questions: ) Consider

More information

Typical Interview Questions and Answers

Typical Interview Questions and Answers Typical Interview Questins and Answers Why d yu want t wrk fr this cmpany? Why are yu interested in this jb? The interviewer is trying t determine what yu knw and like abut the cmpany, whether yu will

More information

LISTENING COMPREHENSION

LISTENING COMPREHENSION PORG, přijímí zkoušky 2015 Angličtin B Reg. číslo: Inluded prts: Points (per prt) Points (totl) 1) Listening omprehension 2) Reding 3) Use of English 4) Writing 1 5) Writing 2 There re no extr nswersheets

More information

CHAPTER 31 CAPACITOR

CHAPTER 31 CAPACITOR . Given tht Numer of eletron HPTER PITOR Net hrge Q.6 9.6 7 The net potentil ifferene L..6 pitne v 7.6 8 F.. r 5 m. m 8.854 5.4 6.95 5 F... Let the rius of the is R re R D mm m 8.85 r r 8.85 4. 5 m.5 m

More information

EXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE Real-World Connection EXAMPLE

EXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE Real-World Connection EXAMPLE - Wht You ll Lern To use the Pthgoren Theorem To use the onverse of the Pthgoren Theorem... nd Wh To find the distne etween two doks on lke, s in Emple The Pthgoren Theorem nd Its onverse hek Skills You

More information

CSE 231 Fall 2015 Computer Project #4

CSE 231 Fall 2015 Computer Project #4 CSE 231 Fall 2015 Cmputer Prject #4 Assignment Overview This assignment fcuses n the design, implementatin and testing f a Pythn prgram that uses character strings fr data decmpressin. It is wrth 45 pints

More information

Basic Guide line for The Sportident system

Basic Guide line for The Sportident system Basic Guide line fr The Sprtident system SI-statins BSF7-8, Master statins BSM7, SI-Cnfig versin 1.4.1, 2006-10-24-1 - Index This guide line... 3 General... 3 Knwledge... 3 Mre infrmatin... 3 System requirements...

More information

DocuSign Quick Start Guide

DocuSign Quick Start Guide DcuSign Quick Start Guide DcuSign Lbby Management Applicatin v1.0 The DcuSign Lbby Management Applicatin is designed fr cmpanies wh need a lightweight way t replace utdated paper sign in sheets in the

More information

Merchant Management System. New User Guide CARDSAVE

Merchant Management System. New User Guide CARDSAVE Merchant Management System New User Guide CARDSAVE Table f Cntents Lgging-In... 2 Saving the MMS website link... 2 Lgging-in and changing yur passwrd... 3 Prcessing Transactins... 4 Security Settings...

More information

Live Analytics for Kaltura Live Streaming Information Guide. Version: Jupiter

Live Analytics for Kaltura Live Streaming Information Guide. Version: Jupiter Live Analytics fr Kaltura Live Streaming Infrmatin Guide Versin: Jupiter Kaltura Business Headquarters 250 Park Avenue Suth, 10th Flr, New Yrk, NY 10003 Tel.: +1 800 871 5224 Cpyright 2015 Kaltura Inc.

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Software Distribution

Software Distribution Sftware Distributin Quantrax has autmated many f the prcesses invlved in distributing new cde t clients. This will greatly reduce the time taken t get fixes laded nt clients systems. The new prcedures

More information