Math 170 Supplement Fall2011

8/3/2019 Math 170 Supplement Fall2011

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TECHNOLOGY MATHEMATICS 2

TABLE OF CONTENTS

General Strategy for Factoring ............................................................1

Factoring ...............................................................................................2

Applications of Factoring .....................................................................4

Inverse Trig Functions ..........................................................................6

Trigonometric Graphs ...........................................................................7Quadratics Applications…………………………………………....... 9

Reviews for Term Tests ........................................................................ 12

Answers ........................................................................................17

Review for the Final Examination ......................................................20

Answers ........................................................................................23

Test/Examination Instructions

Guidelines for Writing Math Exams in the Gym

Math 170 Prerequisites ........................................................... Appendix

Graph Paper



Technology Mathematics 2 Supplemental Package Page 1 of 26

General Strategy for Factoring

Any polynomial

Common factor

2 terms

(binomial)

3 terms

(trinomial)4 terms

Check for a

Perfect Square

Trinomial

Factor by

Grouping

2/2 split

Factor by

Grouping

3/1 or 1/3

Difference

of Squares

Difference of

CubesSum of

Cubes

ax2+bx+c

a=1

ax2+bx+c

a1

Factor using:

1) “Product/SumMethod” ( ac/ b )

or

2) “Inspection”

Factor using:

1) “Product/Sum Method”(ac/b)

and Decomposition / Grouping

or

2) “Inspection”

Factor as a

Difference

Squares




MATH 170 Factoring

Factoring

Factor completely:

1. 6 x2+7 x3

2. 5b235b+60

3. 54r 224s

2

4. 12s260s+75

5. 7 y28 y3

6. 7 R2

16 R+4

7. 2ak 3bm6bk +am

8. p2q+3 p2

16q48

9. 27a38

10. 49 x2112 xy+64 y2

11. (2a+3b)249b

2

12. ( x2)3+125

13. 2 x414 x2

36

14. 15a2+32a+9

15. (m2n

2)2(mn)

16. 40 x2+7 xy-3 y

2

17. 8 x3 y

3

18. 8 x

3

12 x

2

y+6 xy

2

9 y

3

19.2

25309

x x

20. 3 x624 x3

y3

21. 9 x424 x

2+16

22. 64a6




MATH 170 Factoring Answers

Answers:

1. (2 x+3)(3 x1)

2. 5(b4)(b3)

3. 6(3r 2s)(3r +2s)

4. 3(2s5)2

5. 7 y(12 y)(1+2 y)

6. (7r 2)(r 2)

7. (2k +m)(a3b)

8. ( p4)( p+4)(q+3)

9. (3a2)(9a2+6a+4)

10. (7 x8 y)2

11. 4(a2b)(a+5b)

12. ( x+3)( x

2

9 x+39)

13 2( x3)( x+3)( x2

+2)

14. (3a+1)(5a+9)

15. (mn)(m+n2)

16. (8 x+3 y)(5 x y)

17. (2 x y)(4 x2+2 xy+ y2)

18. (2 x3 y)(4 x2+3 y

2)

19. 2

25

3 53

x

xor

x

20. 3 x3( x2 y)( x2+2 xy+4 y2)

21. (3 x24)

2

22. (2a)(2+a)(4+2a+a2)(42a+a

2)




MATH 170 Applications of Factoring

Applications of Factoring

Perform the indicated operations and express each answer in its simplest form.

1.3)52(

3

x x

x

2.)12(2)12(3

12

E E E

E

3.22

33

34 r Rr R

r R

4. x

x x x

x x

7352

251572

2

2

2

5.bb

b

bb

bb

63

142

166

42122

2

6.102

4

4

5

25

162

2

x

x

x

x

x

x

7. 32 8

7

4

3

6

1

xy x y x

8.

122

2

ba

a

ab

aba

9.223

6

32

3

3

2

23

dc

d c

cd

d c

d

d c

10.127

4

9

6

3

2222

cc

c

ccc

c

11.

2

2

352

6112

y y

y y

12.9

6

183

6

3

3222

x x x

x

x x

x

13.

24

2

42

x

x

x

14.

2

23

2

22

x

x x x

x




MATH 170 Applications of Factoring

Answers:

1.12

1

x

2.23

1

E

3.r3R

rRrR22

4. 2 x3

5.)8(2)3(3

bbb

6. 2

7.32

32

24

21184

y x

x xy y

8. 2

2)(

b

baa

9.23

33

2

3

cd

d c

10.3

1

c

11.3

6

y

y

12.3

4

x

13. 2(2 x)

14. x

1




MATH 170 Inverse Trigonometric Functions

Inverse Trigonometric Functions

Find all values of , as a fractional multiple of , such that 0 < 2.

1. cos =2

2

2. cot = 3

3. csc = - 2 and cos < 0

4. sin =3

2

5. sec = -2

6. tan = 1 and sin < 0

Answers:

1. rad 4

3 and rad

4

5

2.6

rad and 76

rad

3. rad 6

7

4.4

3rad

and

5

3rad

5.2

3rad

and

4

3rad

6.5

4

rad




MATH 170 Trigonometric Graphs

Trigonometric Graphs

Each of the following graphs represents the initial cycle of a trigonometricfunction. Identify the function and express it in the form

sin cos y a bx c or y a bx c .

1. 2. y y

x x0

10

3

0

2

-3 - 4

3. 4. y y

1 5

x x

3

7

3

6

1

6

5

5. 6. y y

3

1

x x

2

9

2

8

8

5

-2




Math 170 Quadratic Equations: Applications

Answers

1. x y53sin3

2. x y 4cos4

3.

44

3cos

x y

4.

32sin5

x y

5.

42

1sin

3

1 x y

6.

24cos2

x y





Quadratic Equations: Applications

1. The area of a rectangle is 40 cm2. Find the dimensions of the rectangle if one side is 3 cm

longer than the other.

2. The length of a classroom exceeds its width by 4 m. Find the length and width if the areais 60 m

2.

3. Find the value of x in the diagram if the shaded part has an area of 91 cm2.

4. In a trapezium the longer parallel side (a) is twice the length of the shorter side (b). The

distance (h) between the parallel sides is 5 cm more than the shorter side. Find a, b, and h

if the area is 36 cm2.

Hint: Area of a trapezium is A ½h(a+b)

5. The outside surface area of a tumbler is S 2rh + r 2.

If S 44 cm2 and h 10 cm then find the value of r .

6. The height of a projectile at t seconds is given by the formula: h 2100t 490t 2.

How long can it stay up in the air (correct to one decimal place)?

7. A driver could have saved an hour on a 500 km trip by driving 25 km/h faster. How fast

was he going?

8. One side of a rectangle is twice as long as the other side. When both sides are decreased

by 3.00 cm, the area of the reduced rectangle is 53.0 cm2. Calculate the original

dimensions.

b

h

x cm

4 cm





9. The area of a rectangle is 224 m2

and its perimeter is 62.5 m. Determine the lengths of

the sides.

10. A 12 m by 34 cm rectangular metal sheet will have two opposite sides folded up to form a

section of an open rain gutter. What will be the height of the gutter if its volume will be0.168 m

3?

11. A rectangular piece of corrugated cardboard 1.00 m by 70.0 cm is folded to make an open

box. The bottom of the box has an area of 3840 cm2. Calculate the height of the box.

12. A projectile is shot with a vertical velocity of v = 83 m/s. Its height at t seconds is given

by s vt + ½gt 2 . The value of g is -9.8 m/s

2. Determine the time when the altitude is

(a) 345 m (b) 397 m (c) 0 m.

13. An airplane flew from city A to B, which was 391 km away, and then continued to city C,

which was another 332 km farther. On the second leg of the trip it flew 45 km/h slower

than on the first. The total flying time was 1 hour and 39 min. How fast did the plane flyon each part of the trip?

14. Find the lengths of the edges of the two adjacent cubes in the diagram, given that theircombined volume is 1395 cm

3.

15. A metal cube is to be covered uniformly by 4 cm thick insulation, which will form a

larger cube. Calculate the size of the edges of the metal cube, if 6272 cm3

of insulating

material will be required to cover it.

h h h

h

h

h h

34 cm h

15 cm





16. Determine the length of the radius R of the circle in the diagram below, if the dimensionsof the shaded rectangle are 9 cm by 2 cm.

Answers:

1. 5 cm by 8 cm

2. 6 m by 10 m

3. 11 cm

4. a = 6 cm, b = 3 cm,

and h = 8 cm

5. 2 cm

6. 4.3 s

7. 100 km/h

8. 7.45 cm by 14.9 cm

9. 11.1 m and 20.1 m

10. 7 cm or 10 cm

11. 10.6 cm

12. (a) 7.3 s and 9.6 s

(b) it won’t reach it (c) 0 s and 17 s

13. 460 km/h and 415 km/h

14. 4 cm and 11 cm

15. 12 cm

16. 17 cm

R




MATH 170 Reviews for Term Tests

REVIEW QUESTIONS FOR TERM TESTS

1. Convert to decimal form correct to 2 decimal places.

a) 3617’25”

b) 16958’33”

c) 31846’

2. Convert to degrees, minutes, and seconds.

a) 29.17 to the nearest second

b) 220.36 to the nearest minute

3. Solve the following right triangles.

a) ABC has a right angle at B, c10.49 m and b17.82 m .

b) PQR is right angled at R, P32.1 and r43.4.

c) In KLM: m128.6 cm , K90 and L25.01

4. Solve the following oblique triangles.

a) b 3.07 A 26.0 C 120.0

b) b 3.625 c 2.946 B 69.37

c) b 89.45 c 37.36 C 15.62

d) a 0.0845 c 0.116 B 85.0

e) k 1.78 cm m 6.04 cm n 4.80 cm

5. Determine sin and sec in exact form, given that is in standard position, and itsterminal arm passes through the following points.

a) (5, 12)

b) (16, 30)

6. Given sec = 4/3 and 270< < 360, express tan and csc in exact form.

7. Evaluate each of the following correct to four decimal places.

a) tan 278.4

b) sec 5726’ c) cos (12.456)

d) csc 934.67

8. Find all values of correct to 2 decimal places, given that 0 < 360 and

a) tan = 1.830

b) sin = 0.6374

c) csc = 1.090





9. Convert to radians in terms of .

a) 220

b) 13.5

10. Convert to radians correct to 3 decimal places.

a) 57.6

b) 19234’

11. Convert to degrees. Express non-exact answers correct to 2 decimal places.

a) /6

b) 5 /4c) 4.375

12. Calculate each of the following correct to four decimal places.

a) sin 2.5491

b) cot (0.472)

13. Find all values of correct to three decimal places, for 0 < 2 if

a) cos = 0.9135

b) sec = 1.307

c) tan = 1.860

14. State the amplitude, period, phase shift, and y -intercept for each of the followingtrigonometric functions. Sketch the initial cycle of each function. Label both axes,and show each quarter cycle and the phase shift on the x -axis.

a) y 4 sin 5x

b) y 5 cos (2x - /4)

c) y 2 sin (3x + /3)

d) y -3 cos (2x - /5)





15. Factor completely the following expressions.

a) 28x 227x 10

b) 18a

3

+3a

26a

c) 20x 245

d) 36a 260ab +25b 2

e) 16x 4b4

f) 27a 3+8

g) 25x 2+9y 2

h) 24ax 2 +32ax +60bx 2 + 80bx

i) 75x 3 50x 2 12x +8

16. Perform the indicated operations and simplify.

a)384

3762

2

x x

x x

b)22

22

22

22

23

164

2

6

y xy x

y x

y xy x

y xy x

c)

y

y y

22

44

d)

205

2

4

12

x

x

x

x

e) y y

y

y y

y

y y y 4

3

2

1

82

322234





f)

22

22

22

44

33

232

33

y xy x

y x

y xy x

y x

g)

61212436

2

2 x x x

x x

17. Solve the following.

a)3

12

3

32

x x x x

b)

1

1

1

2

1

52

x x x

18. Solve for1

p .

321

1111

p p pn

p

19. Solve for E3.

3

3

2

21

2

3412 M E

RL

M E

RLx

M E

Rx

20. Solve using the quadratic formula. Express non-exact answers correct to 3 decimalplaces.

a) 24x 2 25x + 20 Answer : 1.572 & 0.530

b) 5x 2 + 7x 2 0 Answer : 0.243 & 1.643

c) x 2 + 3x = 18 Answer : 3 & 6

21. Solve by factoring. Express your answers in exact form.

a) x 2 +3x 10 0 Answer : 2 & 5

b) 6x 2 8 47x Answer : 8 & 1/6

c) 3x 2 + 5x + 2 0 Answer : 1 & 2/3





22. For the equation 4x 2 12x 7 0

a) State the value of the discriminant (D b 2 - 4ac ) Answer : D 256

b) Find the roots Answer : x = ½ & x = 3½

c) Graph the corresponding quadratic function: y 4x 2 12x 7

Answer :

23. Repeat the steps in #38 for the equation: 4x 2

12x + 9 0

Answer : a) D 0 b) x 1½ c)

24. Repeat the steps in #38 for the equation: 4x 2

12x + 11 0

Answer : a) D 32 b) no real roots c)

25. A rectangular room is 4 m wider than it is high, and 8 m longer than it is wide. Thetotal area of the four walls is 512 m 2. Find the width of the room.

Answer : 12 m

y

10 x

2 2 4

y

15

5 x

1 1 2 3 4

y

15

x

-1 1 2 3 4





ANSWERS:

1. a) 36.29 b) 169.98 c) 318.77

2. a) 2910’12” b) 22022’

3. a) a 14.41 m , C36.06, A53.94 b) p23.1, q36.8, Q57.9

c) k141.9 cm , l59.99 cm , M64.99

4. a) a 2.41 B 34.0 c 4.75

b) a 3.391 A 61.11 C 49.52

c) a1 114.7 B1 40.14 A1 124.24

or a2 57.58 B2 139.86 A2 24.52

d) A 37.8 b 0.137 C 57.2

e) K 13.6 M 127.0 N 39.4

5. a) sin 12/13 sec 13/5 b) sin 15/17 sec 17/8

6.3

7tan

7

4csc

7. a) 6.7720 b) 1.8578 c) 0.9765 d) 1.7579

8. a) 118.65 or 298.65 b) 39.60 or 140.40 c) 246.55 or 293.45

9. a) 11 /9 b) 3 /40

10. a) 1.005 b) 3.361

11. a) 30 b) 225 c) 250.67

12. a) 0.5584 b) 1.9589

13. a) 2.723 or 3.561 b) 2.442 or 3.841 c) 1.077 or 4.219





14. Amplitude Period Phase shift y -intercept

a) 4 2 /5 0 (0, 0)

b) 5 /8 (0, 3.54)

c) 2 2 /3 /9 (0, 1.732)

d) 3 1 1/10 (0, -2.427)

y

4

a) y -4 sin 5x x

0

10

5

10

3

5

2

y

5

b) y 5 cos(2x - /4) x

8

8

3

8

5

8

7

8

9

y

c) y 2 sin(3x + /3)2

x

9

18

9

2

18

7

9

5

d) y -3 cos(2x - /5) y

x

10

1

20

7

5

3

20

17

10

11



Technology Mathematics 2 Supplemental Package page 19

MATH 170 Review for the Final Examination

15. a) (7x +2)(4x 5) b) 3a (3a +2)(2a 1) c) 5(2x 3)(2x +3)

d) (6a 5b )2 e) (4x 2+b 2)(2x +b )(2x b ) f) (3a +2)(9a 26a +4)

g) cannot be factored h) 4x (2a +5b )(3x +4) i) (3x

2)(5x

2)(5x +2)

16. a)12

13

x

xb)

3

4( 2 )

x y

x y

c) 2(y +1) d)

)4(5

97

x

x

e))4)(2(

41032

2

y y y

y yf)

)2)((

2

y x y x

y x

g)

x

x )3(2

17.a) x 1 b) x 2

18.233232

321

pp pp pnpp p p p pp p

19.)3(

4

21

21

xME LE xM

E LE

Answers for #20 – 25 are given next to the questions.





REVIEW FOR THE FINAL EXAMINATION

1. Factor each of the following:

a) 81 16x 4

b) 25a

6

b

8

c) 125x 3 + 1

d) 36mx + 10y 24x 15my

e) 20kx 2 45k 12x 2 + 27

f) 12x 2 19xy 21y 2

g) 24a 4 + 4a 3 –140a 2

h) 8x 3 60x 2y +150xy 2 125y 3

i) 49x 2 36y 2 + 60y 25

2. Perform the indicated operations and simplify:

a)2910

3142

2

x x

x x

b) x x x

x62

1952

22

c)22

22

22

22

6

352

2115

310

baba

baba

baba

baba

d)22

22

22

22

6

352

2115

310

baba

baba

baba

baba

e)

h

h

h

h

315

12

102

23

f)

3

234

43

1

x

x

g)273

10

4

8

253

3232

x x x x

x

x x

h)

412

62

545

232

2

2

2

k

k k

k

k k

3. Solve the following equations:

a)32

25

32

92

x x x x

b) 03

3

3114

7

4

53222

x x

x

x x

x

x x

x

c) w

w

ww

w

210

10

35

2

2

2

d)35

5

62

5

13

3

y

y

y y

y





4. Solve for the indicated variable:

a)1

X LC

for C

b)1

V

I

for

5. Solve the following right-angled triangles:

a) In ABC A=90, B=34.3, c=5.72 cm .

b) PQR is right-angled at R. Side p=45.1 cm and r=60.2 cm .

6. Solve the following triangles:

a) In ABC a=5.781”, b=3.952”, and A=63.10

b) In QRS q=47.2 mm , r=32.6 mm , and R=36.0.c) In KLM k=14.82 cm , m=17.82 cm , and L=42.70.

d) In EFG e=105.23 mm , f=87.42 mm , g=119.53 mm .

7. Without calculating the size of angle , sketch all values of in standard position,

and show the coordinates of a point on the terminal arm of each angle, given that 0

< 360:

a)89

8sin

b)17

12cot and 0sin

c)4

65sec

d)2csc

and0cos

8. Calculate the size of each angle in the previous question correct to one decimalplace.

9. Give two positive and two negative co-terminal angles in standard position for an

angle of 143.

10. Express each of the following angles as a fractional multiple of radians.

a) 315

b) 120

c) 202.5

11. Express each of the angles in the previous question in radians correct to 3decimal places.

12. Express 2.913 radians in degrees, minutes and seconds, correct to the nearestsecond.





13. Evaluate each of the following, correct to 5 decimal places:

a) cos 7.8239 b) csc (36.821) c) cot(0.49282)

d) sec 73917’

14. Find all values of , correct to three decimal places, such that

360θ0 .

a) 1.3652θsec

b) .634210θcot and 0θsin c) .617350θcos

15. Find all values of , as a fractional multiple of , such that , 2θ0 .

a) sin =2

1

b) tan =3

1

c) csc =

3

2and tan < 0

16. The graph below represents y = 2 sin(3x ). Graph y = 2 sin(3x + /3) on the sameset of axes. Label every quarter cycle on the x- axis. Label the y -axisappropriately.

17. State the equation of the graph below as

a) y = a sin(bx+c), with a<0 b) y = a sin(bx+c), with a>0

c) y = a cos(bx+c), with a<0

Each tick-mark represents 2 on the y-axis, and8

on the x-axis





18. State the amplitude, frequency, period, and phase shift of each of the followinggraphs:

a) y = 6 sin(3x )

b) y = 7 cos(x /4)

c) y = 5 sin(2t + /3)d) y = cos(4x /5)

19. Sketch the initial cycle for each of the functions in the previous question andmark each quarter period on the horizontal axis.

20. Solve by factoring and express your answers in exact form.

a) 2x 2 6 = 11x Answer: 6 &2

1

b) 6x 2 + 11x = 7 Answer: 2

1&

3

7

c) 6x 2 + 17x +10 = 0 Answer: 2 &6

5

21. Solve for x , correct to three decimal places:

a) 5x 2 + 2 = 9x Answer: 1.540, 0.260

b) 2.25x 2 6.6x + 4.84 = 0 Answer: 1.467

c) 4.2x 2 7.3x = 5.18 Answer: 2.279, 0.541

d)62

5

23

127

x

x

x

x Answer: 0.560, 4.495

e)52815

41

15

52

x x

x

x Answer:

3

2&

5

8

22. Solve for x correct to 5 decimal places, and verify each root.

a) x 6 + 2x 3 = 12 Answer: 1.37605, 1.66377

b) 2x 4 – 7x 2 + 4 = 0 Answer: ±1.66757, ±0.84807

c) 0661

31

x x Answer: 1

64(reject 0.001372)

23. A picture will be inserted into a frame of uniform width. Find the width of theframe, if the area of the frame is three quarters of the area of the picture,and the outside dimensions of the frame are 140 cm by 200 cm .

Answer: 20 cm




MATH 170 Answers to Final Exam Review

REVIEW FOR THE FINAL EXAMINATION: ANSWERS

1. a) (9+4 x2)(3+2 x)(32 x)

b) (5a3b

4)(5a

3+b

4)

c) (5 x+1)(25 x25 x+1)

d) (3m2)(12 x5 y)

e) (5k 3)(2 x+3)(2 x3)

f) (3 x7 y)(4 x+3 y)

g) 4a2(3a7)(2a+5)

h) (2 x5 y)3

i) (7 x+6 y5)(7 x6 y+5)

2. a)25

37

x

xb)

)3(2

14

x x

xc)

ba

ba

2

2d)

ba

b

2

2

e) )5(6

85

h

h

f) 3

3

x

x

g) )13)(2(

11

x x

h))13)(32(5

)12(4

k k

k

3. a) 2 b) 18 c) 1.5 d) 5

4. a)

1C

L X

b)

V

I V

5. a) C=55.7 b=3.90 cm a=6.92 cm

b) P=48.5 q=39.9 cm Q=41.5

6. a) B=37.56 C=79.34 c=6.370”

b) Case 1: Q=58.3 S=85.7 s=55.3 mm.

Case 2: Q=121.7 S=22.3 s=21.1 mm.

c) K=55.42 M=81.88 l=12.21 cm.

d) G=76.073 E=58.703 F=45.224

7. Coordinates of the points are:

a) (5,8) and (5,8) b) (12, 17) c) (4, 7) and (4, 7) d) 1,3





8. a) 302.0 and 238.0 b) 234.8 c) 119.7 and 240.3 d) 150.0

9. 503, 863, 217, 577

10. a) 7 /4 b)2 /3 c) 9 /8

11. a) 5.498 b) 2.094 c) 3.534

12. 16654’9”

13. a) 0.03008 b) 1.66857 c) 1.86214 d) 1.05944

14. a) 42.904o, 317.096

ob) 237.617

oc) 128.123

o, 231.877

o

15. a) rad rad

4

3,

4

b) rad rad

6

11,

6

5 c) rad

3

2

16. 2

1

9

18

9

2

18

7

9

5

17. a) y = 5 sin(2 x + /4) b) y = 5 sin(2 x - 3 /4) c) y = 5 cos(2 x - /4)

18. a) ampl=6 freq=3/2 period=2 /3 phase shift=0

b) ampl= 7 freq=1/2 period=2 phase shift= /4

c) ampl= 5 freq=1/ period= phase shift= /6

d) ampl= 1 freq=2 period=1/2 phase shift= 1/20





y

6

19.a) y -6 sin 3 x x

0

6

3

2

3

2

y

7

b) y 7 cos( x - /4) x

4

4

3 4

5 4

7 4

9

y

c) y 5 sin(2t + /3)5

t

6

12

3

12

7

6

5

y

d) y - cos(4 x - /5) 1

x

20

1

40

7

10

3

40

17

20

11

Answers to # 20-23 are next to the questions.



MATH 170TECHNOLOGY MATHEMATICS 2

TEST #X / FINAL EXAMINATIONXXX MINUTES, XXX MARKS

FALL / WINTER / SUMMER 20XX

NAME (PRINT):

SIGNATURE:

STUDENT NO:

PROFESSOR:

DATE: MARK:

DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO.

INFORMATION & INSTRUCTIONS:

1. This is a closed-book, xxx-minute test/examination. You may use your own calculator as described in thecourse outline. No other aids are permitted.

2. All communication devices (cell phones, pagers, Palm’s, Blackberry’sTM

, iPAQ’s etc.) must be switched off and removed from your desk. The use of any communication device or electronic aid (other than anapproved calculator) is strictly prohibited during the test.

3. The test consists of xx questions on xx pages excluding this cover page. The marks for each questionare indicated in the left margin. The marks total xxx.

4. For each question, a complete solution (all work) must be provided in the space allocated on the testpaper.

5. The use of scrap paper is NOT permitted. The backs of the test pages may be used for rough work whichwill not be marked. Pages must NOT be detached from the test.

6. Students must put their official photo-ID, the Centennial Student Card, in full view when writing a test orexamination.

THIS EXAMINATION REMAINS THE PROPERTY OF CENTENNIAL COLLEGE



GUIDELINES FOR WRITING MATH EXAMS IN THE GYM

1. Plan on arriving at the gym at least 15 minutes prior to the start of the examination. Typically,

students will be admitted to the gym for seating 10 – 15 minutes prior to the scheduled start time;however, do not enter until instructed to do so. No additional time will be allocated to any student

who arrives late.

2. Only bring to the gym those items that you will require to write the examination (pens/pencils,

calculator with fresh battery, ruler, eraser) and your student photo ID card. Textbooks,

notebooks, knapsacks, scrap paper and the like should be left at home or stored securely in

your locker. If you do bring these items, you will NOT be permitted to take them with you toyour desk in the gym. You will be required to leave these items along the gym wall inside the

main doors, and the College will not be responsible should they go missing during the course of

the examination.

3. All cell phones and pagers should be switched-off prior to entry into the gym and shouldremain switched-off until you have left the gym at the conclusion of the examination. The use of

these devices during the course of the examination is strictly prohibited.

4. When you enter the gym, you will see desks arranged in rows on both sides of a centre aisle. On

the desk at the front of each row will be two signs: one indicating the course number and the otherindicating the Professor’s name. You are required to seat yourself in a row designated for your

course and your Professor.

5. Once students are seated in the gym, copies of examinations will be distributed. Do NOT write

anything on the examination paper and do NOT turn the cover page until instructed to do so .

The start of the examination will be formally announced.

6. At some point during the first hour of the examination, your Professor (or his/her designate) will

circulate with an examination attendance sheet. You will be asked to print your name on the sheetand sign beside it. You should be prepared to produce your student photo ID card if

requested.

7. If you have a question during the examination, please raise your hand and wait for a Professor to

attend at your desk. Similarly, should you need to use washroom facilities during the examination,

identify this to one of the Professors in attendance and you will be escorted to the washroom by a

same sex faculty member.

8. If you finish writing the examination early, quietly bring your exam to the tables set up inside the

main gym doors and place it on the pile of exam papers designated for your Professor.

9. When the end of examination is announced, all students are required to immediately submit theirpapers as indicated above.



Appendix Math170 Prerequisites i

Review

1. The temperature in a building rose from 19.0oC to 21.0oC during the day. Find the

percent change in temperature.

2. A solution is made by mixing 75.0 litres of alcohol with 125 litres of water. Find the

percent concentration of alcohol in the mixture.

3. How many litres of gasoline are there in 155 litres of a methanol-gasoline blend that is

10.0% methanol by volume?

4. The air we breathe is composed of various gases. Approximately 20% of air is oxygen.

What volume of air would contain 4800 cm3 of oxygen?

5. Simplify the following expressions.

a) nmmnmnm 76439

b) 92635874 aaaaaaa

c) 2924 x y y x y

6. Perform the indicated operations. Express your answers without negative exponents.

a)

2

3

3

2

b

ab)

541

132

4

5

x zw

z y x

c)

3

3

2

2

3

wz

y xd) 332

326 babaab

e) 232 xz xyz x f) 123 x x g) 22

3234 q pq p pq

h) y x y x 23 i) 121

2 mmm j) 2220 y

k) x

x x

3

2392

l)32

1283223

x

x x xm)

3

352

4

436

xy

xy y x

n) 1

2424

x

x x

7. Solve the following equations and verify your answers.

a)

43

12

2

53

x xb)

8

56

16

33

2

53

t t t



Appendix Math170 Prerequisites ii

8. A certain paint mixture weighing 300 kg contains 20% solids suspended in water. Howmany kilograms of water must be allowed to evaporate to raise the concentration of

solids to 25%? Round your answer to the nearest tenth of a kilogram.

9. How many litres of a solution containing 18% sulfuric acid and how many litres of another

solution containing 25% sulfuric acid must be mixed together to make 550 of solution

containing 23% sulfuric acid? ( All percentages are by volume. )

10. Fifteen litres of fuel containing 3.2% oil is available for a certain two-cycle engine. This

fuel is to be used for another engine requiring a 5.5% oil mixture. How many litres of oilmust be added?

11. Find the domain of each of the following functions.

a)7

3)(

x x f b) t S 36 c) r F 4

12. Graph each of the following functions. Your graphs must be large, neat and adequatelylabeled.

a)4

5

x y b) 13 x y c) 2

22 x x y

13. Given the functions x

x x f

13)(

and 2

1)( x xg , find

a) )0(g b) )0( f c) )2(g d) )( xg f

14. Rearrange the following formulae for the indicated variable.

a) R T r r 11

( ) for r b)

R

M V IR( )for V

c) E P R ( ) for R d)gM

m f F

for f

15. Solve the following systems of equations using Cramer’s Rule.

a)2254

1823

y x

y xb)

643

1952

y x

y x

c)

2343

1022

1432

z y x

z y x

z y x

d)

2

6

5

z x

z y

y x



Appendix Math170 Prerequisites iii

16. Determine one positive angle and one negative angle, of least magnitude, coterminalwith

a) o3.875 b) o

8.975

17. Convert each of the following angles to degrees, minutes and seconds rounded to the

nearest second.

a) 15.2542o

b) 94.8o

c) 123.3083o

18. Convert each of the following to degrees rounded to two decimal places.

a) 059137 b) 45217 c) 740171

19. Calculate the value for each of the following trigonometric functions.

a) 05310sec b) cot 112 c) csc 212

20. Find for each of the following functions. Round your answer to two decimal places.

a) 1151.0sin θ b) 7588.5sec θ c) 2617.1cot θ

d) 0480.1csc θ e) 1151.0cos θ f) 1151.0tan θ

21. In the following triangle write the ratio for each of the six trigonometric functions.

a) sin A

b) cos

c) tan

14.7 33.6 d) sec

e) cot

θ f) csc C 30.2 B



Appendix Math170 Prerequisites iv

22. The terminal side of an angle in standard position passes through the point (3, 4). Writethe ratio of each of the six trigonometric functions as a fraction.

23. You have 165 ℓ of solution containing 17% alcohol. How many litres of a solutioncontaining 25% alcohol should you add in order to produce a solution that is 20% alcohol?

24. A car’s cooling system is full with 19.3 of coolant, which is 11.2% methanol. For thewinter, the methanol concentration must be increased to 21.6%. How much coolant

must be replaced with pure methanol to get the desired concentration of 21.6%?

Answers

1. The temperature has increased by approximately 10.5%.

2. The concentration of alcohol in the solution is 37.5%.

3. There are 139.5 of gasoline in the methanol-gasoline blend .

4. Approximately 24,000 cm3 of air would contain 4800 cm3 of oxygen.

5. a) n5 b) 12a c) x57

6. a)2

6

4

9

a

bb)

7

33

4

5

x

zwyc)

93

36

27

8

z y

w x d) 56

36 ba

e) 336 yz x f) 2 x g) 3223 12812 pqq pq p

h) 22333 y y x xy x i) 13

23 mmm

j) 4004024 y y k)

x x

3

213 l) 4

2 x

m) 192 xy n)

1

35544

23

x x x x

7. a) 7 x b) 3t

8. 60 kg of water must be allowed to evaporate.



Appendix Math170 Prerequisites v

9. 157 of 18% sulfuric acid solution must be added to 393 of 25% sulfuric acid

solution.

10. 0.37 of oil must be added.

11. a) The domain is all real numbers except for 7 x .

b) The domain is all real number such that 2t .

c) The domain is all real numbers such that 4r .

12.

(a) (b) (c)

13. a) 1)0( g b) undefined is f )0( c) 3)2( g

d) 2

2

1

32)(

x

x xg f

14. a) T

r T Rr

11 b)

M

MI RV

c)

E P R

d)

mgFM f

y = 5/(x + 4)

-25

-20

-15

-10

-5

0

5

10

15

20

25

-12 -10 -8 -6 -4 -2 0 2 4 6

y = 3x - 1

-12

-9

-6

-3

0

3

6

9

12

-4 -3 -2 -1 0 1 2 3 4 5

y = 2 - x - 2x2

-14

-12

-10

-8

-6

-4

-2

0

2

4

-4 -3 -2 -1 0 1 2 3



Appendix Math170 Prerequisites vi

15. a) 62 y x b) 32 y x c) 321 z y x

d) no solution

16. a)

7.204&3.155 b)

8.255&2.104

17. a) 515115 b) 8494 c) 0381123

18. a) 37.33 b) 217.9 c) 71.18

19. a) 1.5294 b) -0.4040 c) -1.8871

20. a) 6.61

b) 80.00

c) 38.40

d) 72.59 e) 83.39 f) 6.57

21. a)6.33

7.14sin b)

6.33

2.30cos c)

2.30

7.14tan

d)2.30

6.33sec e)

7.14

2.30cot f)

7.14

6.33csc

22. a)5

4sin b)

5

3cos c)

3

4tan

d)45csc e)

35sec f)

43cot

23. You should add 99 of a 25% alcohol solution to obtain a solution that is 20% alcohol.

24. Approximately 2.26 of coolant must be drained and 2.26 of methanol must be added to

raise the methanol content of the system to 21.6%.

Math 170 Supplement Fall2011

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Transcript of Math 170 Supplement Fall2011