PAP Pre-Calculus Lesson Plans Unit 11 – Sem 2 – 3rd term – Johnston (C114) and Noonan (C116)

February 24th to March 9th 2012 - Vectors

Date Lesson Assignment Did it grade

Friday Feb.24

Law of Sines/Cosines, Area of a Triangle, Polynomial and Rational Functions

WS on Law of Sines/Cosines, Area of a Triangle, Polynomial and Rational Functions

Monday Feb. 27

Sect 6.3: Perform basic vector operations and represent them graphically TEXT/MATERIAL: Larson’s Pre-calculus with Limits

Pg 456: 3,6,9,11,15,20-21,25, 27,32,34,39 and

WS – 2D Vectors

Tuesday Feb. 28

Precal - DA Re-read Section 6.3 – be prepared to use vectors in i-j form

Wednesday Feb. 29

Sect 6.3: Perform basic vector operations and represent them graphically TEXT/MATERIAL: Larson’s Pre-calculus with Limits

Pg 457: 43-63 odds, 66, 67

Thursday Mar. 1

Sect 6.3: Perform basic vector operations and represent them graphically Quiz #7 – Basic Vector operations and Triangle Apps

WS – 6.3

Friday Mar. 2

Sect 6.4: Dot Product of Vectors, Sum of component vectors TEXT/MATERIAL: Larson’s Pre-calculus with Limits

Pg. 467: 1 – 55 odds

Monday Mar. 5

Vectors to solve Navigation Problems

Navigation WS #1

Tuesday Mar. 6

Vectors to solve Navigation Problems Quiz #8 – Geometric Vectors and Dot Products

Navigation WS #2 and Supplemental problems for review

Wednesday Mar. 7

TAKS - ELA Complete Navigation WSs

Thursday Mar. 8

Review for Test #4 - Vectors Study for Test

Friday Mar. 9

Test #4 - Vectors Review for Test #3.4 Unit 11

Area of a Triangle

1sin

2K bc A

2 sin sin

2sin

a B CK

A

( )( )( )

2

K s s a s b s c

a b cwhere s

Right Triangle Trig

sinopposite

hypotenuese

cosadjacent

hypotenuese

tanopposite

adjacent

Triangle Formulas

sin sin sinLaw of Sines:

A B C

a b c

2 2 2Law of Cosines: 2 cosa b c bc A

Other Trig

Functions

1sec

cos

1csc

sin

1cot

tan

Arc Length

is in radians

s r

Area of a Sector

21

2

is in radians

sA r

21

2 180

is in degrees

sA r

Friday, February 24, 2012

PRECALCULUS GT/HONORS

REVIEW OF TRIANGLES

Draw a figure for each problem. Show all work, and give your answers correct to three decimal

places.

1. 36, 22, 90c d C . D ______________

__________________________________________________________________________________

2. 44 , 61 , 70.A B b a =_______________

________________________________________________________________________________________

3. 108 , 54, 66.P q r p _______________

__________________________________________________________________________________

4. 87 , 70 , 12.P Q t p = ________________

________________________________________________________________________________________

5. 20, 12, 9.m k l K ________________

_________________________________________________________________________________

6. 118 , 11, 5.D d e E _____________

7. Find the area of ABC given 43 , 25, 32.C a b

__________________________________________________________________________________

8. Find the area of JKM given j = 18, k = 23, m = 27.

__________________________________________________________________________________

9. A surveyor measures the three sides of a triangular field and gets measurements of

110 m, 135 m, and 224 m. What is the measure of the smallest angle of the triangle?

_________________________________________________________________________________

10. Observers at point C and D, 72 km apart, spot an airplane

overhead at angles of elevation of 42° and 56° respectively.

How far is the plane from the observer at point C?

__________________________________________________________________________________

11. A ramp 8 m long makes a 24° angle with the horizontal.

The ramp is to be replaced by a new ramp whose angle

of inclination is only 15°. How long will the new ramp

be?

__________________________________________________________________________________

12. A ship is 115 miles from one radio transmitter and

140 miles from another transmitter. If the angle

between the signals is 132°, how far apart are the

transmitters?

D C

Airplane

Old

Ramp

New

Ramp

15° 24°

Ship

Transmitter Transmitter

Monday, February 27, 2012

Notes 2-D vectors: Find the magnitude and the resultant.

1. 2a

2. 2b

3. 1

4a c

4. b c

5. a c b

a

b c

Precalculus Name ______________________________

2-D Vectors Worksheet #1 Date __________________ Period ______

Draw the resultant and determine its magnitude.

1. 4b ____________

2. w ___________

3. 1

3a ___________

4. 1

4b __________

5. 5v ____________

6. a b __________

7. 3b v _________

8. 2v w __________

9. w b ___________

10. a b v ________

11. w a b ________

12. 1 1

3 2a b ________

a

b

w

a

v

a

Wednesday, February 29, 2012 Notes: Vectors - Finding the Magnitude and Direction

Notes: Algebraic Representation

Given:

v means vector

Find the magnitude and direction of each resultant below: (Round answers to the nearest thousandth of a degree):

1.) v w 2.) w u

3) v u 4.) 2w u

4cmvv

60

10cmw135

u6cmu

_____

______

v w ______

_______

w u

_______

_______

v u 2 _____

_______

w u

w

75

Thursday, March 1, 2012 – Notes/Examples of Geometric Vectors and Parallelograms/Triangles

I. Look at the parallelogram below. Given: BC x

CD y and P is the midpoint of andAC BD , find

each of the following in terms of andx y .

1. AD = 2. AB = 3. BA =

4. BD = 5. CB = 6. BP =

7. CA = 8. PA = 9. PC =

II: In ∆ABC, 3 1

and .1 4

AP PQ

PB QC If andPQ x QB y , express the following in terms of

andx y .

10. BP = 11. PC = 12. BC = 13. AP =

14. PA = 15. AB = 16. AC = 17. BA =

A

B C

D

P

A

C B

P Q

Monday, March 5, 2012

Navigation Examples

Ex. A ship sails 100 km east, followed by 40 km along a bearing of 120°. How far is the

ship from its starting point? What is the bearing of the ship from its starting point?

Ex. An airplane flies 240 miles on a bearing of 25° and then turns and flies 160 miles along a

bearing of 130°. How far is the plane from its starting point? What is the bearing of the plane

from its starting point?

Ex. An airplane is traveling in the direction 200° with an airspeed of 250 mi/h. There is a

35 mi-per-hour wind with a direction 285°. Find the plane’s ground speed and course.

Ex. A plane is traveling in the direction 160° with an airspeed of 400 mi/h. Its course and

ground speed are 145° and 385 mi/h respectively. What is the direction and speed of

the wind?

Airport

PRECALCULUS GTIHVECTOR WORKSHEET

_/ Work the following on notebook paper. Give decimal answers correct to three decimal places.

1. Forces of 34 pounds and 46 pounds make an angle of 42° with each other and are applied toan object at the same point. Find the magnitude of the resultant force.

2. Joe Jamoke and Ivan Hoe are pulling up a tree stump. Joe can pull with a force of200 poundsand Ivan with a force 0[250 pounds. A total force of 400 pounds is sufficient to pull up thestump.(a) If they pull at an angle of 25° to each other, will the sum of their force vectors be enough

to pull up the stump? What is the sum of their force vectors?(b) At what angle must they pull in order to exert exactly the 400 pounds needed to pull up

the stump? ..

3. Freda Pulliam and Yank Hardy are on opposite sidesof a canal, pulling a barge with tow ropes. Fredaexerts a force of 50 pounds at an angle of 20° to thecanal, and Yank pulls at an angle of 15° with justenough force so that the resultant force vector isdirectly along the canal. Find the number of poundswith which Yank must pull and the magnitude of theresultant vector.

Freda

Yank

4. Aloha Airlines Flight 007 is approaching Kahului Airportat an altitude of 5 km. The angle of depression from theplane to the airport is 9°32'. .(a) What is the plane's ground distance from the airport?(b) If the plane descends directly along the line of sight,

how far will it travel along this line in reaching theairport?

Flight007

~ -==:---1r-::--;-~

5. An airplane has an airspeed of 450 rni/h and a heading of 110°. The wind is blowing fromthe east at 23 miIh. Find the ground speed of the plane and its course.

6. A boat travels at 15 milh on a compass heading of 200°. The velocity of the current is3 mJh toward the north. Find the speed of the boat relative to land, and findits course.

7. An airplane is flying through the air at a speed of 500 kmIh. At the same time, the air ismoving with respect to the ground at an angle of 23° to the plane's path through the airwith a speed of 40 km/h (i.e., the wind speed is 40 km/h). The plane's ground speed is themagnitude of the vector sum of the plane's velocity and the air's velocity with respect tothe ground. Find the plane's ground speed ifit is flying:(a) Against the wind(b) With the wind

TURN-»>

8. A ship is sailing through the water in the EnglishChannel with a velocity of 22 knots along a bearingof 157°, as shown in the figure. (A knot is a nauticalmile per hour, or slightly faster than a regular mile perhour.) The current has a velocity of 5 knots along abearing of 213°. The actual velocity of the ship is thevector sum ofthe ship's velocity and thewater's velocity. Find the actual velocity.

North

s

9. A navigator on an airplane knows that the plane's velocity through the air is 250 kmlhon a bearing of 237°. By observing the motion of the plane's shadow across theground, she finds to her surprise that the plane's ground speed is only 52 kmIh, and itsdirection is along a bearing of 15°. She realizes that ground velocity is the vector sumof the plane's air velocity and the velocity of the wind. What wind velocity would accountfor the observed ground velocity?

PRECALCULUS GTIHVECTOR WORKSHEET .Jf.

"--.../1. A ship steams 100 miles east, and then 40 miles on a heading of 120°,How far is the ship and how does it bear from its starting point?

2. A ship sails 150 miles on a heading of 220° and then turns and sailsdirectly east 'for 50 miles. How far is the ship and how does it bear fromits starting point?

3. An airplane flies on a compass heading of 90° at 200 miles per hour. Thewind affecting the plane is blowing from 300° at 30 miles per hour. Whatis the true course and ground speed of the airplane? .

4. Let the airplane in Exercise 3 fly 250 miles per hour on a heading of 180°.lf the wind direction and speed are the same as given, what are the truecourse and ground speed of the airplane:

6, At what compass heading and air speed should an aircraft fly if a wind of40 miles per hour is blowing from the north, and the pilot wants tomaintain a ground speed of 200 miles per hour on a true course of 90°?

6. Let the wind in Exercise 5 be blowing at 40 miles per hour from 305°,while the pilot still maintains the same true course and ground speed.What should be his compass heading and air speed?

7. A ship is moving through the water on a compass heading of 30cat a

speed of 20 knots (nautical miles per hour). It is traveling in a currentthat causes the ship to move on a path with a heading of 45~. Find thespeed of the current if it is flowing directly from the north.

8. A plane is flying with a compass heading of 300~ at an air speed of 300miles per hour. If its true course is observed to be 330°, and if the windis blowing from 245°, what is the speed of the wind?

9. Two ships leave a harbor, one traveling at 20 knots on a course of SOc andthe other at 24 knots on a course of 140°. How far apart are the shipsafter two hours? What is the bearing from the first' ship to the second at

that time?10. Two airplanes leave an airport at noon. one flying on a true course of

345= and the other on a true course of .f5~. If the first airplane averages240 miles per hour ground speed and the other 200 miles per hour groundspeed, how far apart are the airplanes after one hour? How does thesecond airplane bear from the first ')

11. A pilot makes a flight plan that will take him from city A to city B, adistance of 400 miles. City B bears 60° from city A. and the wind at theplanned flight altitude is 30 miles per hour from 160°. If the airplanecruises at 320 miles per hour air speed, and if the pilot takes off at noon.what will be his compass heading and what is his ETA (estimated timeof arrival) at city B?

12. When the pilot in Exercise 11 decides to return to city A, the wind hasshifted to 40 miles per hour from 90°, What must be his compass headingfor the return trip, and, if he takes off at 6 :00 P.M., what will be his ETA

at city A7

Tuesday-Wednesday, March 6 and 7

Sample Problems for Test Review

I. Look at the parallelogram below. Given: AB x

AD y and P is the midpoint of andAC BD , find

each of the following in terms of andx y .

1. BC = 2. AB = 3. DC =

4. DB = 5. CB = 6. BP =

7. CA = 8. PA = 9. PC =

II: In ∆ABC, 2 1

and .1 3

AP PQ

PB QC If andPQ v QB w , express the following in terms of

andv w .

10. BP = 11. PC = 12. BC = 13. AP =

14. PA = 15. BA = 16. AC = 17. AB =

A

B C

D

P

A

P

C B

Q

III. The initial point and terminal point are given. Find the component form and the magnitude of the vector.

initial point terminal point

____________ 18. (1, 11) (9,3)

____________ 19. (-3, -5) ( 5, 1)

____________ 20. (-2, 7) (5,-17)

____________ 21. (-3, 1) (5,6)

____________ 22. (0, -2) (3,6)

____________ 23. (-6, 4) (0,1)

IV. Find the resultant vector given 3,9 4, 6 2,1u v w

____________ 24. u - v ____________ 25. 1 1

3 4u v

____________ 26. 2u - 3v ____________ 27. 1

22

u v w