# Calculus 1 Optimization Problems

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle and the semicicle Find the dimensions of the ectangle (and hence the semicicle) that will maimize the aea of the window ) You ae building a clindical bael in which to put D Bent so ou can float him ove Niagaa Falls I can fit in a bael with volume equal cubic mete The mateial fo the lateal suface costs \$8 pe squae mete The mateial fo the cicula ends costs \$9 pe squae mete What ae the eact adius and height of the bael so that cost is minimized? ) A ectangula sheet of pape with peimete 6 cm is to be olled into a clinde What ae the dimensions of the sheet that give the geatest volume? 4) A ight tiangle whose hpotenuse is m long is evolved about one of its legs to geneate a ight cicula cone Find the adius, height, and volume of the cone of geatest volume Note: V = π h h 5) Detemine the clinde with the lagest volume that can be inscibed in a cone of height 8 cm and base adius 4 cm

2 9 Calculus Optimization Poblems 6) A staight piece of wie 8 feet long is bent into the shape of an L What is the shotest possible distance between the ends? 7) Find the dimensions of the ectangle of lagest aea that has its base on the -ais and its othe two vetices above the -ais and ling on the paabola = 9 8) A closed clindical containe is to have a volume of 00 π in The mateial fo the top and bottom of the containe will cost \$ pe in, and the mateial fo the sides will cost \$6 pe in Find the dimensions of the containe of least cost a)daw a pictue, label vaiables and wite down a constained optimization poblem that models this poblem (5 Pts) b) Using calculus, solve the poblem in pat (a) to find the dimensions 9) A closed ectangula containe with a squae base is to have a volume of 00 in The mateial fo the top and bottom of the containe will cost \$ pe in, and the mateial fo the sides will cost \$6 pe in Find the dimensions of the containe of least cost 0) You deam of becoming a hamste beede has finall come tue You ae constucting a set of ectangula pens in which to beed ou fu fiends The oveall aea ou ae woking with is 60 squae feet, and ou want to divide the aea up into si pens of equal size as shown below The cost of the outside fencing is \$0 a foot The inside fencing costs \$5 a foot You wish to minimize the cost of the fencing a) Labeling vaiables, wite down a constained optimization poblem that descibes this poblem b) Using an method leaned in this couse, find the eact dimensions of each pen that will minimize the cost of the beeding gound What is the total cost?

3 9 Calculus Optimization Poblems Solutions: ) We will assume both and ae positive, else we do not have the equied window Let P be the wood tim, then the total amount is the peimete of the ectangle the cicumfeence of a cicle of adius, o π Hence the constaint is The objective function is the aea P = π = 8 + π A = + π 4 + plus half Solving the constaint fo gives 8 + π (4 +π ) = and so A = ( 8 + π (4 + π ) ) + π o A = + π + π + π 8 + π And so we wish to maimize A ove the inteval 0, 4 + π d A = ( 8 + π ) (4 + π ) + π d ( 8 ) (4 ) d A Which is 0 when = Since = ( 8 π ) < 0, we have indeed have a maimum d Since = implies =, the dimensions of the ectangle ae b feet Student ma choose the altenative wa to solve the poblem Assume is a function of and using implicit diffeentiation: π = 8 + π and A = + π

4 9 Calculus Optimization Poblems o so d d d A d ( π ) = 0 and = ( + π ) d d d π = 0 d and d A d = + +π d d d d 4 + π = d and + + π = 0 d 4 + π + π = 0 Using this in the oiginal constaint eq gives o = π = (8 + π ) = 8 + π So =, and =, and the dimension of the ectangle is b feet +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+= ) The volume is given b V = π h The cicula ends have total aea π The lateal suface aea is π h Cost is C = (\$8)(π h) + (\$9)(π ) h So the poblem at hand is minimize C = 6π h + 8π subject to the constaint π h = π h = means h = Using this hee gives π and so 6 C = + 8π C = 6 6π +, which gives citical points as = 0, and = π

5 9 Calculus Optimization Poblems The phsicall easonable solution is = m which gives π h = m π Note: If ou wish to solve the poblem using implicit diffeentiation The steps follow h The volume is given b V = π h The cicula ends have total aea π The lateal suface aea is π h Cost is C = (\$8)(π h) + (\$9)(π ) So the poblem at hand is minimize C = 6π h + 8π d d subject to the constaint π h = Assuming h = f (): d d d ( π h) = () and C = (6π h + 8π ) d d d dh dc d h π h + π = 0 and = 0 = 6π h + 6π + 6π d d d which gives d h h = d h 6 π h + 6π + 6π = 0 o = h Using = h in π h = gives = h = m π +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=

6 9 Calculus Optimization Poblems ) Let and be the dimensions of the sheet of pape Since + = 6, + = 8 is the constaint The adius is given b π =, so =, and the volume is V = π = Using π 4π (8 ) dv 6 =8, V = is the function to be optimized =, so citical numbes 4π dt 4π ae = 0, Maimum volume occus when =, (Wh?) so dimensions ae 6 cm b cm 6 and the volume is cm π +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+= 4) The volume of the cone is: V = π h h The constaint equation is: + h = ( ) o + h = ) Solving fo gives = h, so V ( h ) h = π o V = π (h h ) dv Taking the h-deivative of V gives = π ( h ) = π ( h ) dh Stationa points ae h = ±, and the phsicall easonable one is h = If h =, then = h = =, so =, and the volume is V = π ) Solving fo h is FAR moe difficult + h = means h =, and so V dv So = π d Rewiting this as dv = π d + π + π π 6 = ( ) = = π So stationa points ae = 0, ±, ± The onl phsicall easonable one is =, so h = =, and V = π

7 9 Calculus Optimization Poblems +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+= 5) Detemine the clinde with the lagest volume that can be inscibed in a cone of height 8 cm and base adius 4 cm (5 points) (0, 8) (, ) (4, 0) The volume of the clinde is V = π The equiement that the clinde is inscibed in the cone leads to the pictue on the ight Since the edge of the cone is a staight line, we can use the 8 0 two points (0, 8) and (4, 0) to detemine the elation between and Slope is = The 0 4 line is = + 8 So the volume becomes V = π (8 ) = π (8 ) and [0,4] V = π (6 6 ) = π (8 ) 8 8 And so citical numbes ae = 0, The maimum occus when the adius is = which 8 5π means the height is =, and the volume is V = cubic cm 7 6)A staight piece of wie 8 feet long is bent into the shape of an L What is the shotest possible distance between the ends? (0 Points) d

8 9 Calculus Optimization Poblems Let d be the distance between the ends of the L then and so = 8, and then d d + = The constaint is + = 8 = f ( ) = + (8 ) which, when simplified, gives, f ( ) = Since d and d have the same citical points we wok with f () To detemine citical numbes we compute f ( ) = 4 6, and solve 4 6 = 0, and so the citical numbe is = 4, and so = 8 = 4 also The distance is then d = + = = 4 We know this is a minimum since f ( ) = 4 > 0 7) Find the dimensions of the ectangle of lagest aea that has its base on the -ais and its othe two vetices above the -ais and ling on the paabola = 9 = 9 - ^ The ectangle aea is A = The equiement that the ectangle lies on the gaph of = 9 means A = (9 ) = 8 The vaiable is esticted between [0,] at which both points ield a minimum value of no ectangle o 0 aea Since When A ( ) = 8 6 = 6( ), A ( ) = 6( ) = 0 we get

9 9 Calculus Optimization Poblems citical points at = ± Fo ou geomet we choose the positive oot =, and so, = 6, so the dimensions ae units b 6 units, and the aea is = squae units 8) h Volume: V = π h Cost: \$(π ) + \$6(π h) So Poblem is minimize costc = 4π + π h subject to the constaint V = π h = 00π and so h = b) Solving this last equation fo h gives: h =, which when substituted into the cost equation 600π ields C = 4π + The geomet gives ( 0, ) To minimize the cost we detemine 600π citical numbes fom C = 8π = 0 hence = 450 so the citical numbe is / / 00 (450) 700 = (450) This gives h = = in Since C = 8π + π = 0π > 0, the / (450) / dimensions ield the minimum cost The clinde should have a adius = (450) in, and a 00 height of h = in ode to minimize the cost / (450) 9) A closed ectangula containe with a squae base is to have a volume of 00 in The mateial fo the top and bottom of the containe will cost \$ pe in, and the mateial fo the sides will cost \$6 pe in Find the dimensions of the containe of least cost (0 Points) Volume: V = h Cost: \$( ) + \$6(4 h) So Poblem is minimize C = h subject to the constaint h = 00 Solving this last equation fo h gives: h 00 h =, which when substituted into the cost equation 700 ields C = 4 + The geomet gives ( 0, ) 700 Since C = 8 = 0 gives = 900 whose solution is = 900 / 9 65 This gives / / / h = = in Note h = 900 = = 00 in so the volume is / C = 4 +

10 9 Calculus Optimization Poblems 4,400 coect And since C = 8 + = > 0, the dimensions ield the minimum cost The bo / 900 / should have a squae base of side length 900 in, and a height of in FYI 600 C () ) The cost of the outside fencing is \$0 a foot The inside fencing costs \$5 a foot You wish to minimize the cost of the fencing a) Let be the width of each individual pen, and be the length as shown above Since the total aea is 60 sq ft, each individual pen will have an aea of 0 sq ft The constaint is = 0 The objective function is the cost Eamining the fencing above, thee is 5 feet of inteio fencing, and + feet of eteio fencing So the total cost is C = \$ 5 (5) + \$0 ( + ), o C = The constained optimization poblem is: Minimize =0 C = subject to the constaint b) Solving = 0 of gives = Substituting this into C gives C = + 0, as the function to minimize ove ( 0, ) 450 C = + 0, and so citical points ae = 0, and = = The cost is C = = 0 5 \$ = 5, so

### Quantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w

1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

### Model Question Paper Mathematics Class XII

Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat

### arxiv: v2 [math.ho] 13 Jul 2016

axiv:1603.0854v [mat.ho] 13 Jul 016 IT IS NOT A COINCIDENCE! ON CURIOUS PATTERNS IN CALCULUS OPTIMIZATION PROBLEMS MARIA NOGIN Abstact. In te fist semeste calculus couse we lean ow to solve optimization

Road tunnel This activity is about using a gaphical o algebaic method to solve poblems in eal contets that can be modelled using quadatic epessions. The fist poblem is about a oad tunnel. The infomation

### 9.5 Volume of Pyramids

Page of 7 9.5 Volume of Pyamids and Cones Goal Find the volumes of pyamids and cones. Key Wods pyamid p. 49 cone p. 49 volume p. 500 In the puzzle below, you can see that the squae pism can be made using

### Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

### 2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90

. Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal

### Problems on Force Exerted by a Magnetic Fields from Ch 26 T&M

Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuent-caying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

### LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

### Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

### Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such

### Algebra and Trig. I. A point is a location or position that has no size or dimension.

Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite

### 4.4 VOLUME AND SURFACE AREA

160 CHAPTER 4 Geomety 4.4 VOLUME AND SURFACE AREA Textbook Refeence Section 8.4 CLAST OBJECTIVES Calculate volume and uface aea Infe fomula fo meauing geometic figue Select applicable fomula fo computing

### In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

Radians At school we usually lean to measue an angle in degees. Howeve, thee ae othe ways of measuing an angle. One that we ae going to have a look at hee is measuing angles in units called adians. In

### Experiment 6: Centripetal Force

Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

### Thank you for participating in Teach It First!

Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational Mathematics teache lesson followed by the coesponding student lesson. We ae confident

### CHAT Pre-Calculus Section 10.7. Polar Coordinates

CHAT Pe-Calculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to

### In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea

Double Integals in Pola Coodinates In the lectue on double integals ove non-ectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example

### 11.4 Surface Area and Volume

I. Suface Aea and Volume of any Polyhedon A. Suface Aea um of the aea of it face. (Enough pape to wap it) B. Volume the meaue of pace taken up by a olid in thee-dimenional pace. (box of chocolate laye)

### New proofs for the perimeter and area of a circle

New poofs fo the peimete and aea of a cicle K. Raghul Kuma Reseach Schola, Depatment of Physics, Nallamuthu Gounde Mahalingam College, Pollachi, Tamil Nadu 64001, India 1 aghul_physics@yahoo.com aghulkumak5@gmail.com

### 4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

### Section 5-3 Angles and Their Measure

5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

### Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

### Mechanics 1: Motion in a Central Force Field

Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

### Exam I. Spring 2004 Serway & Jewett, Chapters 1-5. Fill in the bubble for the correct answer on the answer sheet. next to the number.

Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 1-5 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE

### PY1052 Problem Set 3 Autumn 2004 Solutions

PY1052 Poblem Set 3 Autumn 2004 Solutions C F = 8 N F = 25 N 1 2 A A (1) A foce F 1 = 8 N is exeted hoizontally on block A, which has a mass of 4.5 kg. The coefficient of static fiction between A and the

### Trigonometric Functions of Any Angle

Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

### XIIth PHYSICS (C2, G2, C, G) Solution

XIIth PHYSICS (C, G, C, G) -6- Solution. A 5 W, 0 V bulb and a 00 W, 0 V bulb ae connected in paallel acoss a 0 V line nly 00 watt bulb will fuse nly 5 watt bulb will fuse Both bulbs will fuse None of

### mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !

Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!

### 4.1 - Trigonometric Functions of Acute Angles

4.1 - Tigonometic Functions of cute ngles a is a half-line that begins at a point and etends indefinitel in some diection. Two as that shae a common endpoint (o vete) fom an angle. If we designate one

### SHORT REVISION SOLUTIONS OF TRIANGLE

FREE Download Study Package fom website: wwwtekoclassescom SHORT REVISION SOLUTIONS OF TRINGLE I SINE FORMUL : In any tiangle BC, II COSINE FORMUL : (i) b + c a bc a b c sin sinb sin C o a² b² + c² bc

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

### Multiple choice questions [60 points]

1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions

### Gauss Law. Physics 231 Lecture 2-1

Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

### Multiple choice questions [70 points]

Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions

### PHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013

PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80-kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,

### Review of Coordinate Systems

Review o Coodinate Sstems good undestanding o coodinate sstems can be ve helpul in solving poblems elated to Mawell s Equations. The thee most common coodinate sstems ae ectangula (,, ), clindical (,,

### 9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics

Available Online Tutoing fo students of classes 4 to 1 in Physics, 9. Mathematics Class 1 Pactice Pape 1 3 1. Wite the pincipal value of cos.. Wite the ange of the pincipal banch of sec 1 defined on the

### Physics 107 HOMEWORK ASSIGNMENT #14

Physics 107 HOMEWORK ASSIGNMENT #14 Cutnell & Johnson, 7 th edition Chapte 17: Poblem 44, 60 Chapte 18: Poblems 14, 18, 8 **44 A tube, open at only one end, is cut into two shote (nonequal) lengths. The

### Semipartial (Part) and Partial Correlation

Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

### Problem Set # 9 Solutions

Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

### Problem Set 6: Solutions

UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 16-4 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente

### PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

### Lab M4: The Torsional Pendulum and Moment of Inertia

M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disk-like mass suspended fom a thin od o wie. When the mass is twisted about the

### A discus thrower spins around in a circle one and a half times, then releases the discus. The discus forms a path tangent to the circle.

Page 1 of 6 11.2 Popeties of Tangents Goal Use popeties of a tangent to a cicle. Key Wods point of tangency p. 589 pependicula p. 108 tangent segment discus thowe spins aound in a cicle one and a half

### Carter-Penrose diagrams and black holes

Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

### Unit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.

Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads

### Review of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction.

Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD:

### So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)

Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing

### Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

### The Critical Angle and Percent Efficiency of Parabolic Solar Cookers

The Citical Angle and Pecent Eiciency o Paabolic Sola Cookes Aiel Chen Abstact: The paabola is commonly used as the cuve o sola cookes because o its ability to elect incoming light with an incoming angle

### Fluids Lecture 15 Notes

Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit

### Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

### Do Vibrations Make Sound?

Do Vibations Make Sound? Gade 1: Sound Pobe Aligned with National Standads oveview Students will lean about sound and vibations. This activity will allow students to see and hea how vibations do in fact

### Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

### Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth

Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate

### Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

### FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

### SL Calculus Practice Problems

Alei - Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram

### CLOSE RANGE PHOTOGRAMMETRY WITH CCD CAMERAS AND MATCHING METHODS - APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT

CLOSE RANGE PHOTOGRAMMETR WITH CCD CAMERAS AND MATCHING METHODS - APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT Tim Suthau, John Moé, Albet Wieemann an Jens Fanzen Technical Univesit of Belin, Depatment

### Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43

Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.

### NURBS Drawing Week 5, Lecture 10

CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu

### Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

### Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

### Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

### 1.4 Phase Line and Bifurcation Diag

Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes

### 12. Rolling, Torque, and Angular Momentum

12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

### On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.

C.Candan EE3/53-METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof

### Week 3-4: Permutations and Combinations

Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication

### rotation -- Conservation of mechanical energy for rotation -- Angular momentum -- Conservation of angular momentum

Final Exam Duing class (1-3:55 pm) on 6/7, Mon Room: 41 FMH (classoom) Bing scientific calculatos No smat phone calculatos l ae allowed. Exam coves eveything leaned in this couse. Review session: Thusday

### Solutions to Homework Set #5 Phys2414 Fall 2005

Solution Set #5 1 Solutions to Homewok Set #5 Phys414 Fall 005 Note: The numbes in the boxes coespond to those that ae geneated by WebAssign. The numbes on you individual assignment will vay. Any calculated

### Chapter 8, Rotational Kinematics. Angular Displacement

Chapte 8, Rotational Kinematics Sections 1 3 only Rotational motion and angula displacement Angula velocity and angula acceleation Equations of otational kinematics 1 Angula Displacement! B l A The length

### Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the

### Cubic Spline Interpolation by Solving a Recurrence Equation Instead of a Tridiagonal Matrix

Matematical Metods in Science and Engineeing Cubic Spline Intepolation by Solving a Recuence Equation Instead of a Tidiagonal Matix Pete Z Revesz Depatment of Compute Science and Engineeing Univesity of

### MATH 31 UNIT 5: LESSON 6A GEOMETRIC APPLICATIONS: MAXIMIZING AREA AND VOLUME. NAME ANSWERS Page 1 of 9

NAME ANSWERS Page 1 of 9 EXAMPLE 1: The Starks have 60 metres of fencing with which to make a rectangular dog run. If they use a side of the shed as one side of the run, what dimensions will give the maimum

### PY1052 Problem Set 8 Autumn 2004 Solutions

PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

### Revision Guide for Chapter 11

Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams

### Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of

Oiginall TRIGONOMETRY was that banch of mathematics concened with solving tiangles using tigonometic atios which wee seen as popeties of tiangles athe than of angles. The wod Tigonomet comes fom the Geek

### Gauss Law in dielectrics

Gauss Law in dielectics We fist deive the diffeential fom of Gauss s law in the pesence of a dielectic. Recall, the diffeential fom of Gauss Law is This law is always tue. E In the pesence of dielectics,

### 1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

### TECHNICAL DATA. JIS (Japanese Industrial Standard) Screw Thread. Specifications

JIS (Japanese Industial Standad) Scew Thead Specifications TECNICAL DATA Note: Although these specifications ae based on JIS they also apply to and DIN s. Some comments added by Mayland Metics Coutesy

### 9.3 Surface Area of Pyramids

Page 1 of 9 9.3 Suface Aea of Pyamids and Cones Goa Find the suface aeas of pyamids and cones. Key Wods pyamid height of a pyamid sant height of a pyamid cone height of a cone sant height of a cone The

### 1. CIRCULAR MOTION. ω =

1. CIRCULAR MOION 1. Calculate the angula elocity and linea elocity of a tip of minute hand of length 1 cm. 6 min. 6 6 s 36 s l 1 cm.1 m ω?? Fomula : ω π ω ω π 3.14 36 ω 1.744 1 3 ad/s ω.1 1.74 1 3 1.74

### Voltage ( = Electric Potential )

V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

### Forces & Magnetic Dipoles. r r τ = μ B r

Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

### EAS Groundwater Hydrology Lecture 13: Well Hydraulics 2 Dr. Pengfei Zhang

EAS 44600 Goundwate Hydology Lectue 3: Well Hydaulics D. Pengfei Zhang Detemining Aquife Paametes fom Time-Dawdown Data In the past lectue we discussed how to calculate dawdown if we know the hydologic

### Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6

Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe

### 2008 Quarter-Final Exam Solutions

2008 Quate-final Exam - Solutions 1 2008 Quate-Final Exam Solutions 1 A chaged paticle with chage q and mass m stats with an initial kinetic enegy K at the middle of a unifomly chaged spheical egion of

### Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

### Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

### Continuous Compounding and Annualization

Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

### 4.1 Cylindrical and Polar Coordinates

4.1 Cylindical and Pola Coodinates 4.1.1 Geometical Axisymmety A lage numbe of pactical engineeing poblems involve geometical featues which have a natual axis of symmety, such as the solid cylinde, shown

### Determining solar characteristics using planetary data

Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation