Math 11A FINAL EXAM REVIEW Chapter 3 – Sinusoidal Functions

1. Identify the transformations on each of the following equations:

a. xy 3sin2 = b. )45(3cos)1(2

1°−=+ xy

c. )30(4

1sin)5(3 °+=−− xy d. xy

3

2sin)1( =−−

e. )10(3

1cos6 °−=− xy f. xy 12cos1 =+

2. Given that )54,103(),( −−+→ yxyx is the mapping rule for transformations on y=cos x,

a. State the resulting equation b. Identify the amplitude, period, and equation of the sinusoidal axis

3. If the transformations on y=sin x give a sinusoidal axis of y = -7, amplitude of ½ and a

period of 90°, write its equation. 4. Determine the equation of the graphs shown below in terms of

a. y=sin x b. y=cos x

-360 -300 -240 -180 -120 -60 60 120 180 240 300 360

-4

-3

-2

-1

1

2

3

4

x

y

-360 -270 -180 -90 90 180 270 360

1

2

3

4

x

y

5. A Ferris wheel with a radius of 20m can make 2 full turns in 60 seconds. The maximum height reached by a rider from the bottom is 42m.

If the x coordinates represent time, in seconds, and the y coordinates represent the height of the Ferris wheel, in meters, determine the ordered pairs that correspond to each of the 5 variables labeled on the Ferris wheel diagram below, for one complete rotation.

Point A: _______ Point B: _______ Point C: _______ Point D: _______ Point A: _______

6. The graph below indicates the pattern of motion of a windmill, measuring the height of the

tip of one of the blades above the ground at any given point in time.

a. What is the period of the windmill? b. What is the amplitude? c. How high is the axle of the spinning blades above the ground? d. List the transformations that have occurred and provide an equation for this relationship.

7. In one area of the Bay of Fundy, the tides cause the water level to rise 4.5m above average sea level and to drop to 4.5m below average sea level. One cycle is completed approximately every 12 hours. Assume that changes in the depth of the water over time can be modeled by a sine function.

a. If the water is at average sea level at 5am and the tide is coming in, draw a graph to

show how the depth of the water changes over the next 24 hours. Assume that at low tide the depth of the water is 2.5m.

b. Write an equation for the graph.

c. What is the depth of the water at 11:30am? (round to the nearest hundredth)

D

C

B

A

0123456789

101112

0 30 60 90 120 150

Time (s)

Heig

ht

(m)

–360–270–180 –90 90 180 270 360 450 540 630 720

–4

–2

2

4

x

y

–360–270–180 –90 90 180 270 360 450 540 630 720

–4

–2

2

4

x

y

8. Determine the period, Amplitude and Sinusoidal axis, without graphing.

)45(3cos)3(2

1−=+− xy

9. Mathman got his cape caught on the blade of a windmill and was hoisted up into the air, around and around he went. The graph shows his height above the ground in meters versus time in seconds.

a) What is the amplitude of the function? b) What does the amplitude represent? c) What is the period? d) What does the period represent? e) What is the height intercept? f) What does the height intercept represent? g) What is the equation of the sinusoidal axis? h) What does the sinusoidal axis represent?

10. a) Graph: )45sin()2(3

1 o−=− xy b) Graph: )30(3cos)4( o+=−− xy

11. For each graph below, list the transformations that are occurring, and then determine the equation of the line. a) b) b)

Transformations: Rx = ________ Transformations: Rx = ________

VS = ________ HS = ________ VS = _______ HS = _______ VT = ________ HT = ________ VT = _______ HT = _______

Cosine Equation: _________________ Sine Equation: __________________

12. Find the sine equation for the curve in #5a

–360 –270 –180 –90 90 180 270 360 450 540 630 720

–4

–2

2

4

x

y

–360–270–180 –90 90 180 270 360 450 540 630 720

–4

–2

2

4

x

y

13. A spring is oscillating sinusoidally back and forth from a motion sensor. At t=3 sec, the spring is at its maximum distance from the sensor, 12 cm. At t=7 sec, the spring is at its minimum distance from the sensor, 3 cm. a) Find an equation to represent the relationship. b) How far away is the end of the spring at t=10 sec?

14. A Ferris wheel makes a complete revolution every 15 seconds. At the top, a seat is 22m above the ground. At the bottom it is 1m above the ground.

a) Sketch a graph of the seats movement. b) What is the radius of the Ferris wheel? _________ c) How high is the axle above the ground? _________ d) What is the equation of the sinusoidal axis? _________

Chapter 3 Review Answers: 1. a). VS½ HS 1/3 b). VS2 VT-1 HS 1/3 HT45 c). Rx VS 1/3 VT5 HS4

HT-30 d). Rx VT1 HS 3/2 e). VT6 HS3 HT-10 f). VT-1 HS 1/12

2. a). )10(3

1cos)5(

4

1−=+− xy b). amp=4, period=1080°, sin axis y = -5

3. ( ) xy 4sin72 =+

4. a). Graph 1: xy 2sin3

1= Graph 2: ( ) ( )45sin32 +=−− xy

b). Graph 1: ( )452sin3

1−= xy Graph 2: ( ) ( )135cos32 +=− xy

5. A: (0,2) B: (7.5,22) C: (15,42) D: (22.5,22) A: (30,2)

6(a). 60s 7(b). 5m 7(c). 6m 7(d). ( ) ( )156sin65

1−=−− xy or ( ) xy 6cos6

5

1=−

7(b). ( ) xy 30sin79

2=− or ( ) ( )330sin7

9

2−=− xy

8. Period: 120 Amplitude: 2 Sinusoidal Axis: y = -3 9. (a) 6 m (b) length of a blade (c) 18 s (d) time it takes a blade to make 1 rev. (e) 2 m (f) height of blade when graphing begin (g) y = 8 (h) height of axle 10. (a) (b)

11. (a) Rx, VS: 4, HS: none, VT: 5, HT: none

xy cos)5(4

1=−− or )180cos()5(

4

1+=− xy or )180cos()5(

4

1−=− xy

(b) Rx: No, VS: 6, VT: none, HS: 1/30, HT: 2

)2(30sin6

1−= xy or )4(30sin

6

1+=− xy or )8(30sin

6

1−=− xy

12. )90sin()5(4

1+=−− xy or )90sin()5(

4

1−=− xy

13. (a) )5(45sin)5.7(5.4

1−=−− xy or )3(45cos)5.7(

5.4

1−=− xy (b) 11.5

14. (a) (b) 10.5 m (b) 11.5 m (c) y = 11.5

Chapter 4 – Trigonometric Equations

1. Give two co- terminal angles for: a) 75º b) -130º 2. Given the following angle measurements, express in radians: a) -25° b) 625°

3. Given the following angle measurements, express in degrees: a) 5

4π b)

9

π

4. Simplify the following radicals:

a) 2

106 b)

53

22− c) 9624 − d) )85)(37( −−

e) 63 f) ¼ 24 g) )123)(64(− h) )65(32 −−

i) 10

60− j) 50218483 −+ k)

55

32

5. Determine the exact value for each of the following expressions:

a) oo 135cos210cos b) sin2120° + cos(-225°) c) °

°−

210cos

240sin3

d) o

o

o

o

150sin

45cos

30cos

60sin+ e)

o

oo

150sin

135cos180cos2 + f)

o

o

225sin

210cos30sin −°

g) ooo 30sin)180cos(240cos2 2 −− h) 1- sin²(-45º)

6. Solve the following trigonometric equations for the interval(s) indicated.

a. 2 sin x + √2 = 0 (0°≤ x ≤360°) b. cos x = - ½ (all solutions) c. sin x = 4/5 (all solutions) d. 3 cos x – 1 = 0 (0°≤ x ≤360°) e. cos x = -0.7660 (all solutions) f. csc x = 2 (0°≤ x ≤180°) g. 4 sec x + 7 = 0 (0°≤ x ≤360°) h.

7. Convert the following from degrees to radians, or from radians to degrees:

a. radians8

3π b. 145°

c. radians5

2π d. 280°

e. 2.83 radians

8. Name the 8 basic trigonometric identities studied. 9. Solve this system of equations algebraically y x y x= − =2 2 2sin , sin using degree measure and

then illustrate the solution graphically. 10. Solve for x (in degree mode, unless otherwise indicated), remembering to check for any possible restrictions on x:

a) ( )2 3cos , ,x x= − ∈ −∞ ∞

b) [ ]cos sin , ,x x x− + = ∈ ° °2 1 0 0 3602

c) [ ]sin cos , ,x x x= ∈ 0 π

d) ( )cos

sin, ,

2

11

x

xx

+= ∈ −∞ ∞

e) [ ]cot csc , ,2 1 0 0 2x x x− − = ∈ π

f) [ ]sin cos cot , ,x x x x+ = ∈ − ° °2 360 360

g) [ ]sec cos tan , ,x x x x− − = ∈ ° °2 0 0 360

h) 3 4 42tan sec ,x x= − − all solns radian mode

i) [ ]cos , ,32

260 60x x= − ∈ − ° °

j) ( ) ( )sin , ,2 1x x+ = − ∈ −∞ ∞π

k) ( )cos , ,2

2

3

4

xx

= ∈ −∞ ∞

l) [ ]cos . cos , ,2 0 63 0 360 360x x x− = ∈ − ° °

m) sin . ,x − = −2 2 45 all solns radian mode

n) ( ) ( )tan . , ,4 60 0 6x x− ° = ∈ −∞ ∞

11 Solve for x (in degree mode, unless otherwise indicated): a) ( )sin log ,−

=1

2 1 x

b) ( ) ( )log sin log , ,x x+ = ∈ −∞ ∞2 1 c) ( )sin−

+ =1 2 2

3

2x x

π

d) [ ]2 2 2 0 4 42cos cos , ,x x x− − = ∈ − π π

12. Verify the following identities: a) tan cot sec cscx x x x+ = b) csc cos csc2 2 2 1x x x− = c) sin cos sec sin sin tanx x x x x x− = −

2 d) 5 3 3 22 2 2sin cos sinx x x+ = +

e) cos sin

cos sincos sin

4 4x x

x xx x

−

+= −

f) sec

sin

sin

coscot

x

x

x

xx− =

g) sec csc sec csc2 2 2 2x x x x= +

h) cos sec

tancot

x x

xx=

i) cos

sin

sin

cossec

x

x

x

xx

1

12

++

+=

j) tan

tansin

2

2

2

1

x

xx

+=

k) ( )cos

seccot cos

x

xx x

+= −

112

13. Convert to radian measure: °25

14. Express in degrees: π

7

15. Evaluate, leaving in simplest radical form:

a) sin tan10

3

3

4

π π

+

b) 3

1 64

+

cos

π

16. A railway company is preparing to build a new line through Rolling Mountains. You set up a Cartesian coordinate system with its origin at the entrance to the tunnel through Bald Mountain. Your survey crew finds that the mountain rises 250 metres above the level of the track and that the next valley goes down 50 metres below the level of the track. The cross-section of the mountain and valley is roughly sinusoidal with a horizontal distance of 700 metres from the top of the mountain to the bottom of the valley.

a) Write the equation expressing the vertical distance from the track versus the horizontal distance from the tunnel entrance. (Hint: you will have to use substitution to find the missing variable(s).)

b) How long will the tunnel be? c) How long will the bridge be?

17. Solve for all possible values of x.

a) cos4x = 2

1 b) 2sinx - 3 = 0 c) 2 cos²x - 1 = 0

d) 4sin6x + 2 = 3 e) 2sinx + 3 = 4 f) 3cosx = -2 18. Solve for the values of x, where oo 3600 ≤≤ x

a) 2cosx - 3 = 0 b) 2 sinx + 3 = 4

19. Prove each of the following trig identities: a) 2tan²θcosθcotθ = 2sinθ b) θθθθ 32 coscossincos =−

c) secθ (cosθ + sinθ) = 1 + tanθ d) θθ

θcos

tan

sin=

20. Tarzan is swinging on a vine in the jungle. His height above the ground with respect to time is modeled by a sinusoidal function. He is 6 m above the ground at 2 sec and 10 m above the ground at 6 sec.

a) Find an equation to represent the situation. b) At what times is his distance 9 m? (all possible values)

Solutions – Chapter 4 1. (a) 435o, 795o, -285o, -645o, etc... (b) -490o, -850o, 230o, 590o, etc...

2. (a) 36

5π− (b)

36

125π 3. (a) 144o (b) 20o

4. a) 56 b) 15

102− c) 62− d) 670 e) 73 f) 6

2

1

g) 272− h) 312152 +− i) - 6 j) 28 k) 25

152

5. a) 4

6 b)

4

223 − c) -3 d) 21+ e) 24 −− f)

2

62 −−

g) 1 h) 2

1

6. a) x=225° b)x=120°+360°k, kεI c)x=53.1°+360°k, kεI x=315° x=240°+360°k, kεI x=126.9°+360°k, kεI d) x=70.5° e)x=220°+360°k, kεI f) x=30° g) x=270° x=289.5° x=140°+360°k, kεI x=150° x=30° x=150° 7. a) 67.5° b) 0.8π radians or c) 72° 8 d) 1.56π radians or e) 162.1° 2.53 radians 4.89 radians

8. sincsc

, cossec

, tancot

, tansin

cos, cot

cos

sin, sin cos , cot csc , tan secx

xx

xx

xx

x

xx

x

xx x x x x x= = = = = + = + = + =

1 1 11 1 1

2 2 2 2 2 2

9. x k k k I= °+ ° °+ ° ∈30 360 150 360, ,

10. a) x k k k I= °+ ° °+ ° ∈150 360 210 360, ,

b) x = ° ° °60 120 180, ,

c) x =π π

4

5

4,

d) x k k I= ° ∈180 ,

e) x =π π π

6

5

6

3

2, ,

f) x = − ° − ° ° °330 20 30 150, , ,

g) x = ° °210 330,

h) x k k I= + ∈π π2 ,

i) x = ± °45

j) x k k I= + ∈π

π4

,

k) x k k k I= ± °+ ° ± °+ ° ∈60 720 300 720, ,

l) x = ± ° ± ° ± ° ± °50 95 90 270 309 05. , , , .

m) x k k k I= + + ∈58 2 36 2. , . ,π π

n) x k k I= °+ ° ∈22 74 45. ,

11. a) x = °0 b) { }x ∈ ∅

c) x = −1

d) x = ± ±0 2 4, ,π π

13. 0.436 14. 2571. °

15. − −3 2

2 b)

− +3 9 2

17

16. a)

b) ( ) ( )[ ]1

76 36 2d x− = ° −cos

c) 1886. m

d) 5 61 10 8 39 10. , . ,+ + ∈k k k I

17. a) ( ) ( )1

150100

9

35512 6y x− = ° −

cos .

b) 10252. m

c) 374 8. m

18. a) 15o + 90k, k ε I; 75o + 90k, k ε I b) 60o + 360k, k ε I; 120o + 360k, k ε I c) 45o + 360k, k ε I; 315o + 360k, k ε I; 135o + 360k, k ε I; 225o + 360k, k ε I d) 2.4o + 60k, k ε I; 27.6o + 60k, k ε I e) 30o + 360k, k ε I; 150o + 360k, k ε I f) 131.8o + 360k, k ε I; 228.2o + 360k, k ε I 19. a) 30o, 330o b) 45o, 135o

20. a) )4(45sin)8(2

1−=− xy b) 4.7+8k and 7.3+8k

Chapter 6 – Trigonometry and its Applications

1. Find the area of ∆ABC if a = 7.5 cm, b = 9 cm, and C = 100o. 2. State the Law of Sines and the Law of Cosines. 3. Find the smallest angle of the triangle with sides 7, 9, and 12. 4. Solve ∆ABC (that is, find the measures of all sides and angles) given: A = 35o, b = 32, c = 41 5. Find ∠B in ∆ABC if: A = 40o, a = 15, and b = 30 6. In ∆ABC, a = 8 m, b = 10 m, and c = 12 m. Determine, to the nearest degree, the measure of the

largest angle. 7. In ∆ABC, a = 8 m, b = 10 m, and c = 12 m. Determine, to the nearest degree, the measure of the

largest angle. 8. Find the value of the unknown in each triangle:

a) b) x 15

35°

c) d) 75° 48° 1.6 2.8 x 53 52° X

8

40°

7

x

9. A soccer net is 4.6 m wide. When Beckham gets possession of the ball he immediately notices that he is 12 m from one goal post and 16 m from the other goal post. Within what angle must he shoot in order to score a goal? 10. Find the area of the following triangle a = 27, b = 25 and c = 18 11. Firefighters are called to the scene of a fire that has broken out on the fourth floor of an apartment building. Their only access to one of the apartments is through a window that is located at a height of 12m.

a) If the base of the ladder must be 5 m from the wall, how long must the ladder be in order to reach the required height? b) What is the angle of elevation formed by the ladder with the ground?

12. Find the perimeter of ∆LMN, where M = 64°, N = 47° and l = 50km

13. Find <C in ∆ABC, where AB = 10cm, AC = 8cm, and <B = 53°. 14. The longest car ferry is the 173 metre Norland operating on the North Sea. What angle would her length subtend when viewed from a point at sea which is 300 m from her bow and 220 m from her stern? 15. An oil well is to be located on a hillside that slopes at 10o. The desired rock formation has a dip of 27o to the horizontal in the same direction as the hill slope. The well is located 3200 feet downhill from the nearest edge of the outcropping rock formation. How deep will the driller have to go to reach the top of the formation? 16. Find the area of the shaded region, given that the circle has a radius of 5 cm and that the triangle is equilateral:

Solutions – Chapter 6

1. 332 2. cm

2. a

A

b

B

c

Ca b c bc A

sin sin sin, cos= = = + −

2 2 22

3. 354. ° 4. a B or128 C or16= = ° ° = ° °23 6 511 9 939 1. , . . , . .

6. 82.83o

7. same question as #1 8. (a) 41o (b) 26.15o (c) 2.84 (d) 66.24 9. 218.17 unit2 10. 11. (a) 13 (b) 67.38o 12. 137.31 km 13. 86.65o or 93.35o 14. 34.74 m 15. 1050 m 16. 461 2. cm

CHAPTER 1 – Investigations in 3 Space

1) Given the following matrices, find the following without the calculator:

A =

23

813 B =

−

−

748

264

601

C =

− 61752

30413 D =

−− 43

12

a) The dimensions of matrix C _______ b) A

c) A •C d) A 1− e) D f) B •C

g) IA h) AA •−1 i) 2A + 3B j) D 2

2) Use the inverse method to solve each system.

a. y – 2 = 3x b. y + 2x = 8 x + y = 6 3y + 2x = 12

3. Given the following matrices, find:

A =

−

63

12

04

B =

243

512 C = [ ]75− D =

30

15

63

E =

−

05

31

b. BA b. CB c. AC d. A + 3D e. 2D – A f. C2 g. E2

4. Paul has a bike repair shop. His inventory is shown in the matrix

[ ]2635# itemsof

FramesPedalsHandlebarsTires

The value of each type of item is given by: Value($) Handlebars 30 Pedals 25 Frames 300

How would you multiply these matrices to find the total value of all items? Calculate this total.

5. Find the determinants and inverses for:

c.

21

45 b.

−

−

212

14 c.

− 40

27

6. Solve the following systems of equations using matrices:

d. 3x + y = 10 b. 6x + 7y = 3 x + 5y = -6 2x – 10y = 1

7. A cannonball is shot from a canon and hits a target 300 m away and 9 m above the ground. The canon is 2.5 m above the ground. The ball’s path forms a parabola and the ball passes just over a tree 12 m high, 120 m from the target. What is the equation of the parabola?

Anton found that if he studied for tests, he did better, but if he studied too much, his improvement was minimal. The table shows his latest results. Use a quadratic function to estimate what Anton’s grade would be if he studied for only one hour

8. Lauren and Hilary were in charge of ticket sales for the school concert. Lauren sold 3 adult tickets and 20 student tickets. She collected $59.75. Hilary sold 4 adult tickets and 27 student tickets. Hilary collected $80.50. Determine the cost of each adult ticket and each student ticket.

9. If you multiply a 3x5 matrix by a 5x2 matrix what are the dimensions of the solution matrix?

10. a) Solve the following system of equations by substitution

85109

812

−=−

=+

yx

xy

11. Justin and Lucy are a young couple working on the assembly line of a major automotive company. Two weeks ago Justin worked 37 hours, Lucy worked 41 hours, and their combined salary prior to deductions was $2307. Last week Justin worked 42 hours, Lucy worked 39 hours, and their combined salary prior to any deductions was $2385. How much is each making per hour?

12. Graph the plane which contains 13) Determine the intercepts of the following the three points (4, 0, 2), (-2, 0, 0), plane and draw a sketch.

and (0, 3, -4) 2x + y +3z = 6

14) Give the equation of the plane containing the following 3 points.

( 0, 0, -4 ) ( 2, 0, 0 ) ( 0, 5, 0 ) 15) A plane is given by the equation 2x + 3y – 4z = 12. Which of the following points are on the plane? (Circle) (1, 0, 3) (0, 1, -2) (6, 0, 0) (0, 2, 1) (3, 2, 0)

Study Time (hrs)

Grade (%)

2 85

3 95

5 96

b) Solve the following systems of equations by elimination.

245

1332

−=+

=−

yx

yx

032

1625

4323

=+−

−=++

−=−+

zyx

zyx

zyx

Chapter 1 Review Answers:

1. (a) 2 x 5 (b) 2 (c)

−

−

212261313

8781085355 (d)

−

−

5.65.1

41 (e) -5 (f) not possible

(g)

23

813 (h) I (j)

−

136

21 2. (a) x = 1, y = 5 (b) x = 3, y = 2

3(a).

826

2925 (b).

−11

23

11

(c). cannot do (d).

153

217

1813

(e).

− 03

38

122

(f). cannot do (g).

−

−−

155

314

4. $940

5. a). det(A) = 6 A-1=

−

−

65

61

32

31

b). det(B) = 4 B-1=

−

−

134

12

1 c). det(C) = -30 C-1=

−30

70

301

152

6. a).

−=

2

4

y

x b).

=

02

1

y

x

7. 5.21175.01019.3 24 ++×−= − xxy

8. 68.67

9. adults $3.25, students $2.50 4. 3 x 2 10. (a) x = 25, y = 31 (b) x = 2, y = -3 (b) x = -2, y = -2, z = -2 11. Justin $28/hr, Lucy $31/hr 12. 14. 10x + 4y – 5z = 20 15. (6,0,0) & (3,2,0)

Chapter 5 – Statistics

1. Given the following data:

45 68 97 68 73 80 39 51 98 72 79 64 68 89 81 33 48 51 66 73 78 85 83 77 68 65 72 77 90 60

a) Find the mean, median, mode, and range b) Create a box-and-whisker plot c) Create a histogram

2. Find the standard deviation for the following: 15, 14, 18, 12, 14, 18, 20, 17

3. The January exams for Math 11 were all piled together and found to have a mean of 68% with a standard deviation of 6. Assuming normal distribution, draw the normal curve and answer the following questions.

a) What percent of the students had marks between 68% and 74%? b) What percent had marks between 56% and 74%? c) If the results are for 240 students, how many had a mark over 74%?

4. Some common sources of bias in surveys are: a sample selection process that is not random, leading questions, untruthful replies to questions, and author bias. Explain why these can create bias. 5. Classify each sample selection method(s):

a) You survey every fifth person on the potential graduate list for the school. b) You survey 0.01% of the population from each province. c) The school board wants to determine student opinion on the IB program. They decide to

survey all students from Citadel High School. d) A local radio station wants to know if people prefer eating fried scorpions or fried tarantulas.

They ask listeners to call/email their preference. e) To determine the public’s view of their store, Mall-Mart surveys the first 100 people who

enter the store on a given day. f) A credit card company trying to enrol new clients by selecting every 5th person in the phone

book. g) The lotto 6-49 draw gets its name from having to select 6 numbers from a population of 49,

whereby each number has an equal chance of being selected. h) At the YMCA, there are approximately 300 men and 600 women who use the facility. To

determine the satisfaction with the equipment, 20 men and 40 women were randomly selected and asked to complete a survey.

i) An athletic organization wishes to determine which sports Grade 11 students are participating in across Canada. 50 schools are randomly selected from all over Canada. Then every Grade 11 student in all 50 schools is surveyed.

j) To determine how pleased people are with the prime minister, a survey was conducted in each province. Since Ontario has the largest population then most people surveyed should be from that province. Similarly, PEI has the smallest population, therefore the smallest number should from that province.

6. You shuffle a deck of cards and ask 100 people to close their eyes and pick any six cards. The cards were then replaced and reshuffled. You record how many reds were picked by each person.

a) Explain why this is a binomial situation. b) How might the results have differed if you asked only 40 people?

7. Forty randomly chosen people were asked, "Do you get enough sleep?". Seventeen replied "yes".

a) Find the point estimate for this survey. b) Write an approximate 90% confidence interval for the population proportion using the diagram

on page 203. 8. Suppose that out of 401 people polled, 286 said that they preferred watching movies in a theatre to watching them at home. Write a 95% confidence interval for the population proportion. 9. In a press release dated February 26, 1999, Angus Reid reported that 42% of Canadians said that they were concerned about the effects of the Y2K bug on their investments. The reported margin of error for the study was ±2.5%, 19 times out of 20. Approximately what was the sample size? 10. Data concerning the conviction rate in criminal trials was kept in the Seine Department in France between 1825 and 1830. In 1825 there were 567 convictions of the 802 accused. In 1830 there were 484 convictions of the 804 accused. Explain whether or not there was a statistically significant change in the five year period?

Solutions – Chapter 5

1. (a) mean=69.9, median=72, mode=68, range=65 (b) values for box-and-whisker: LE=33, Q1=64, Q3=80, UE=98, Med=72, Mean=69.9 (c) use the following frequency table for histogram:

Bin Frequency

30-40 40-50 50-60 60-70 70-80 80-90 90-100

2 2 2 8 8 5 3

2. 2.5 3. (a) 34% (b) 81.5% (c) 38 4. unfair representation of population, misguides readers 5. a) systematic b) stratified c) cluster d) voluntary e) convenience f) systematic g) simple random h) stratified i) cluster j) stratified 6. a) two options: red and black

b) varies – should look less bell-shaped or clustered 7. a) 0.425 8. 66.8% - 75.8% 9. 1559 10. significant change – 1830 not within 95% confidence interval established for 1825

Exponential and Logarithmic Functions Review: 1. (a) Indicate whether or not xy )75.0(1.2= represents a growth curve and explain your answer.

2. Algebraically solve for ‘x’. (a) 1

2

819

1 +

+

=

x

x

(b) 2)4(log)4(log 33 =−++ xx

3. Suppose that the number of bacteria in a petri dish at ‘noon’ is 600. It is known that the bacteria population doubles every 20 minutes. (a) Determine the number of bacteria in the petri dish 3 hours later. (b) How long after ‘noon’ will the bacteria population increase to 1 228 800? (provide units with you answer). 4. The number of cell phone users in Atlantic Canada has increased by 15% every year for the past several years. If this rate continues and there are now 375 000 cell phone users, in how many years will the number of cell phone users reach 1 000 000? Set up and solve an equation that represents this situation.

5. Show that 3log

15log

5

3 = .

6. A recently discovered bacteria doubles in number every 15 minutes. A biologist determined that the initial number of bacteria in a sample was 500. (a) Determine an equation representing this situation. Let ‘t’ represent time in minutes and ‘N’ represent the number of bacteria. (b) How many bacteria will there be after an hour if the growth of the bacteria is not interrupted?

7. Solve for ‘x’ algebraically. (a) 2

4

8

116

+

=

xx

(b) 135 +=

xx

8. The number of cell phone users in a city has increased by 10% every year for the past several years. With this constant rate, if there are 500 000 cell phone users today, in how many years will there by 800 000 cell phone users? Set up and solve an equation(s) that represents this situation.

9. Algebraically solve for ‘x’. (a) 64log120log 4=x (b) 26log)4(log 33 =++x

10. The population of a bacteria culture can be described using the following function, where P is the population in bacteria/cm2 and t is time in minutes.

20)2(300

t

P =

a) What is the initial concentration of the bacteria? b) How long does it take for the bacteria to double? c) What will the concentration of bacteria be in 50 minutes?

11) Solve each system a) 42

162

=

=

−

+

yx

yx

b) 82

813

=

=

−

+

yx

yx

c) 9

1

39

82

−

+

=

=

xy

yx

12) Solve for x a) 025)5(26)5( 2 =+− xx b) 02)2(322 =+− xx

Exponents and Logs Answers: 1. Not a growth curve. 2. (a) x = -4/3 (b) x = 5 3. (a) 307200 bacteria (b) 220 min. 4. 7.02 years 5.

6. (a) 15)2(500

t

N = (b) 8000 bacteria

7. x = -3/2 or -1.5 (b) x = 2.15 8. 4.93 years 9. 9324.41203 ==x (b) x = -5/2 or -2.5

10. a) 300 bacteria/cm2 b) 20 minutes c) 1697 bacteria/cm2

11. a) (3, 1) b)

2

1,

2

7 c) (21, 6) 12. a) x = 0 or x = 2 b) x = 0 or x = -1

REMEMBER: This is a brief (ha) overview of the semester. Besides completing the review, go

over the notes, additional examples, and your tests and quizzes!

GOOD LUCK!!