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Notas de trigonometria
Citation preview
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
1
Trigonometry Final Exam Review Chapters 7 8 9 Note A portion of Exam 3 will cover Chapters 1 ndash 6 so be sure you rework problems from the first and second exams and from the Exam 1 and Exam 2 Reviews Directions Work these problems with no resources other than the departmental formula sheet and a graphing calculator Your book notebook homework solutions manual etc should be closed Read and carefully follow all directions Show all of your work on all problems Anytime you are asked to perform a calculation manually or to give an exact answer you may not use a calculator Chapter 7 Applications of Trigonometry and Vectors
1 Determine the remaining sides and angles of the triangle ABC Show all work and or support your answer
2 Determine the remaining sides and angles of the triangle ABC Show all work and or support your answer
C = 7183deg B = 4257deg a = 2614
3 A ship is sailing due north At a certain point the bearing of a lighthouse 125 km distant is N 388deg E Later on the captain notices that the bearing of the lighthouse has become S 442deg E How far did the ship travel between the two observations of the lighthouse Show all work and or support your answer
4 Mark notices that the bearing of a tree on the opposite bank of a river flowing north is 11545deg Lisa is on the same bank as Mark but 4283 m away She notices that the bearing of the tree is 4547deg The two banks are parallel What is the distance across the river Show all work and or support your answer
5 Find the area of the triangle ABC Show all work and or support your answer
A = 3050deg b = 1300 cm C = 11260deg
6 A real estate agent wants to find the area of a triangular lot A surveyor takes measurements and finds that two sides are 521 m and 213 m and the angle between them is 422deg What is the area of the triangular lot Show all work and or support your answer
7 Determine the number of triangles ABC possible with the given parts Show all work and or support your answer
a) b = 60 a = 82 B = 100deg b) a = 35 b = 30 A = 40deg c) B = 54deg c = 28 b = 23
8 Find the unknown angles in triangle ABC if the triangle exists Show all work and or support your answer
a) C = 41deg20 b = 259 m c = 384 m b) B = 482deg a = 890 cm b = 697 cm
c) B = 743deg a = 859 m b = 783 m
9 Solve the triangle ABC if possible Show all work and or support your answer
C = 523deg a = 325 yd c = 598 yd
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
2
10 Solve the triangle ABC if possible Show all work and or support your answer
A = 38deg40 a = 972 km b = 118 km
11 Solve each triangle and then find the area of the triangle Approximate values to the nearest tenth Show all work and or support your answer
a) b)
12 Solve each triangle and then find the area of the triangle Show all work and or support your answer
a) C = 283deg b = 571 in a = 421 in b) a = 189 yd b = 214 yd c = 325 yd
13 The sides of a parallelogram are 40 cm and 60 cm One angle is 58deg while another is 122deg Find the lengths of the diagonals of the parallelogram Show all work and or support your answer
14 A weight is supported by cables attached to both ends of a balance beam as shown in the figure What angles are formed between the beam and the cables Show all work and or support your answer
15 Using the vectors in the figure below draw a sketch to represent
a) c + d b) d ndash c c) 3c
16 Use the vectors in the adjacent figure to find a) a + b b) a ndash b c) ndash a Show all work and or support your answer
17 For vectors u and w with angle θ between them sketch the resultant u + w 8=u 12=w θ = 20deg
18 Find the magnitude and direction angle for vector v = 8- 15 Show all work and or support your answer
19 Vector v has magnitude 415=v and direction angle 0327α prime= o Find the horizontal (a) and vertical (b) components of v Show all work and or support your answer
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
3
20 Write vector v in component form b a Show all work and or support your answer
21 Two forces act at a point in the plane One is 19 newtons the other is 32 newtons The angle between the forces is 118deg Find the magnitude of the resultant force Show all work and or support your answer
22 Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the adjacent figure Round your answer to the nearest tenth Show all work and or support your answer
23 Given u = 5 2minus and v = 3 4 find ndash2u + 4v Write your answer in the form ai + bj Show all work and or support your answer
24 Find the dot product of 3- 2 and 5 6 Show all work and or support your answer
25 Find the angle between the vectors -5i + 12j and 3i + 2j Show all work and or support your answer
26 Given u = 1 2minus v = 4 3 and w = 12 5minus evaluate ( )wvu minussdot Show all work and or support your answer
27 Determine whether the vectors 2 1 and 3 6minus are orthogonal Show all work and or support your answer
28 Two tugboats are pulling a disabled speedboat into port with forces 1240 lb and 1480 lb The angle between these forces is 282deg Find the magnitude and direction of the equilibrant Show all work and or support your answer
29 Two forces of 128 lb and 253 lb act at a point The equilibrant force is 320 lb Find the angle between the forces Show all work and or support your answer
30 A force of 176 lb makes an angle of 78deg50 with a second force The resultant of the two forces makes an angle of 41deg10 with the first force Find the magnitudes of the second force and the resultant Show all work and or support your answer
31 A 130-lb force keeps a 600-lb object from sliding down an inclined ramp Find the angle of the ramp Show all work and or support your answer
32 Find the magnitude of force necessary to keep a 3500-pound car from sliding down a ramp of 5deg Show all work and or support your answer
33 Two people are carrying a box One person exerts a force of 150 lb at an angle of 624deg with the horizontal The other person exerts a force of 114 lb at an angle of 549deg Find the weight of the box Show all work and or support your answer
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
4
34 A ship leaves port on a bearing of 340deg and travels 104 miles The ship then turns due east and travels 46 miles How far is the ship from port and what is its bearing from port Show all work and or support your answer
35 A pilot is flying at 168 mph She wants her flight path to be on a bearing of 57deg40 A wind is blowing from the south at 271 mph Find the bearing the pilot should fly and find the planersquos groundspeed Show all work and or support your answer
36 A pilot wants to fly on a bearing of 749deg By flying due east he finds a 42-mph wind blowing from the south puts him on course Find the groundspeed and airspeed Show all work and or support your answer
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 Write 288minus as the product of a real number and i Show all work and or support your answer
38 Solve 4x23x 2 minus=+ and express all nonreal complex solutions in terms of i Show all work and or support your answer
39 Perform the indicated operations and then simplify Show all work and or support your answer
a) 40
10minus
b) 8
612 minussdotminus
40 Perform the indicated operation and then simplify Write your answers in standard form Show all work and or support your answer
a) ( ) ( ) ( )iii 54324 +minus++minusminusminus b) ( )( )ii 2432 minus+minus c) ii
5132
minus+
41 Simplify each power of i Show all work and or support your answer
a) 25i b) 13minusi
42 Find the sum of i65minus and i32+minus by graphing each number in the complex plane and finding their resultant Show all work and or support your answer
43 Write the complex number 3[cos 150deg + i sin 150deg] in rectangular form Show all work and or support your answer
44 Write the complex number ndash5 ndash 5i in trigonometric form r(cos θ + i sin θ) with θ in the interval [0deg 360deg) Show all work and or support your answer
45 Find each product and write it in rectangular form On part b round to four decimal places Show all work and or support your answer
a) ( )[ ] ( )[ ]oooo 120sin 120cos5300sin 300cos8 ii +sdot+ b) (12 cis 185deg) (3 cis 125deg)
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
5
46 Find the quotient ( )( )oo
oo
150sin150cos2
120sin120cos4
i
i
+
+ amp write it in rectangular form Show all work and or support your answer
47 Convert the numerator and denominator of 31
2ii
minusminus to trigonometric form Then find the quotient and write it in
rectangular form Show all work and or support your answer
48 Find each power Write each answer in rectangular form Show all work and or support your answer
a) [ ]3100 3 ocis b) ( )522 iminusminus
49 Find all cube roots of each complex number Leave answers in trigonometric form Show all work and or support your answer
a) o300 27 cis b) iminus3
50 Find all complex solutions of each equation Leave answers in trigonometric form Show all work and or support your answer
a) 0x 4 =minus i b) 016x 4 =+
51 For each pair of polar points plot the point give two other pairs of polar coordinates (including one with a negative r) and give the rectangular coordinates for the point Show all work and or support your answer
a) ( )o135 2minus b) ⎟⎠⎞
⎜⎝⎛ minus
35 3 π
52 Given the rectangular coordinates ⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus
21
23 give two pairs of polar coordinates for the point including one with a
negative r Select θ so it is in the interval [0deg 360deg) Show all work and or support your answer
53 For each rectangular equation give its equivalent polar equation and sketch its graph Show all work and or support your answer
a) 3x ndash 2y = 6 b) 9yx 22 =+
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
6
54 Match each equation with its corresponding graph
a r = a cos θ A
b r = a sin θ B
c r = a plusmn b sin θ with 1 ltba C
d r = a plusmn b cos θ with 1 =ba D
e r = a cos 4θ E
f r = a sin 5θ F
g r = θsin b θ acos
cminus
G
h r2 = 4 sin 2θ H
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
7
55 Identify the type of polar equation and draw a complete graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 + 2 cos θ b) r = 4 cos 2θ
56 For each polar equation give its equivalent rectangular equation and sketch its graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 sin θ b) sinθ13r
minus=
Chapter 9 Exponential and Logarithmic Functions
57 Graph the following exponential function Give the equation of the horizontal asymptote a 5-point t-chart with the ldquokey pointrdquo at the center and an accurate graph with one notch on each axis labeled Show all of your work
( ) 12y 4x +minus= minus
58 Solve the following exponential equations algebraically Show all your work and give an exact answer
a 49
32 x
=⎟⎠
⎞⎜⎝
⎛minus
b 6 x 4 x 82 minus+ = c 125x 43=
59 Find the balance A for P dollars invested at rate r for t years and compounded n times
a P = $2000 r = 65 t = 10 years compounded quarterly
b P = $3000 r = 7 t = 8 years compounded continuously
60 Find the required annual interest rate to the nearest tenth of a percent for $48000 to grow to $7818694 when compounded semiannually for 5 yr Round your decimal answer to four decimal places and convert to a percent Show all of your work
61 Solve the following logarithmic equations algebraically Show all your work and give an exact answer
a 2161log x minus= b 5xlog
3= c x4log
16=
62 Graph the following logarithmic function Give the equation of the vertical asymptote the exponential form of the equation a 5-point t-chart with the ldquokeyrdquo point at the center and an accurate graph with one notch on each axis labeled Show all of your work
y = ( ) 42xlog 3 minus+
63 Use the properties of logarithms to expand or condense the following log expressions Show all of your work
a 4
3
bz
xylog b zlog32ylog
31xlog 2 aaa minusminus
64 Find the pH for lye soap which has a hydronium ion concentration of 141032 minustimes moles per liter
65 Find the [ ]+OH3 for drinking water which has a pH of 65
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
8
66 Use the change of base theorem to find an approximation for 15log21 Round your answer to four decimal places
67 Given f(x) = 3 x evaluate the following Show all of your work
a f (log 3 7) b f [log 3 (ln 3)] c f [log 3 (2 ln 3)]
68 a Find the decibel rating of a sound having an intensity of 100000 I0
b If the intensity of the sound is doubled by how much is the decibel rating increased
69 a Find the Richter scale rating for an earthquake having an amplitude of 1000000 I0
b Express the magnitude of a 67 earthquake (on the Richter scale) in terms of I0
70 Solve
a 64 13x =minus b x5x 32 =+ c 1116e 34x =minus+ d 50004
06110004t=⎟
⎠
⎞⎜⎝
⎛ +
71 a log (2x ndash 1) + log 10x = log 10 b ln x + ln (x ndash 1) = 2
c ( ) 12log3xlog4xlog 666 =minusminus d log (11x + 9) = 3 + log (x + 3)
72 Elena McDuff wants to buy an $8000 painting She has saved $6000 Find the number of years (to the nearest tenth) it will take for her savings to grow to $8000 at 57 compounded monthly
73 At what interest rate will $2500 grow to $467104 if invested for 10 years and interest is compounded quarterly
Answers
Chapter 7 Applications of Trigonometry and Vectors
1 A = 372deg a = 178 m c = 244 m
2 A = 6560deg b = 1942 cm c = 2727 cm
3 278 km
4 2934 m
5 6594 cm2
6 373 m2
7 a) 0 b) 1 c) 2
8 a) B asymp 26deg30 A asymp 112deg10 b) A1 asymp 722deg C1 asymp 596deg A2 asymp 1078deg C2 asymp 240deg c) no such triangle
9 A asymp 255deg B asymp 1022deg b asymp 739 yd
10 B1 asymp 49deg20 C1 asymp 92deg00 c1 asymp 155 km B2 asymp 130deg40 C2 asymp 10deg40 c2 asymp 288 km
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
9
11 a) A asymp 223deg B asymp 1082deg C asymp 495deg area = 2units 152231 asymp
b) A asymp 567deg C asymp 683deg b asymp 882 area = 2units 3680613547079 asymp
12 a) c asymp 283 in A asymp 449deg B asymp 1068deg area = 2units 57 7325368851 asymp
b) C asymp 107deg20 B asymp 39deg00 A asymp 33deg40 area = 2units 193040 372645000 asymp
13 52 cm 88 cm
14 264deg and 363deg
15 a) b) c)
16 a) 10 0 b) 2 8 c) 6- 4minus
17
18 17v = θ = 3319deg
19 a asymp 137 b asymp 711
20 v = 34 4
21 29 newtons
22 244 lb
23 20i + 2j
24 ndash3
25 7893deg
26 ndash24
27 orthogonal
28 2640 lb at an angle of 1672deg with the 1480-lb force
29 701deg
30 magnitude of second force 190 lb magnitude of the resultant 283 lb
31 125deg
32 305 lbs
33 226 lb
34 135 mi 504deg
35 bearing 65deg30 groundspeed 181 mph
36 groundspeed 161 mph airspeed 156 mph
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
10
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 i 212
38 i32
32x plusmnminus=
39 a) i21 b) ndash3
40 a) i+minus10 b) i162+minus c) i21
21+minus
41 a) i b) ndash i 42 i33minus
43 i23
233+minus
44 ( )oo sin225 icos225 25 +
45 a) 32020 i+ b) 308580 + 185414 i
46 iminus3
47 i21
23minusminus
48 a) i2
3272
27minus b) 128 + 128i
49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+
( )oo 350sin 350 cos23 i+
50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg
b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg
51 a) ( )o495 2minus (2 315deg) ( )2 2 minus
b) ⎟⎠⎞
⎜⎝⎛
3π 3 ⎟
⎠⎞
⎜⎝⎛minus
34π3 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
233
23
52 (1 210deg) (-1 30deg)
53 a) 2sinθ3cosθ
6rminus
= b) r = 3 or r = ndash3
54 a F b A c D d G e H f B g C h E
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
2
10 Solve the triangle ABC if possible Show all work and or support your answer
A = 38deg40 a = 972 km b = 118 km
11 Solve each triangle and then find the area of the triangle Approximate values to the nearest tenth Show all work and or support your answer
a) b)
12 Solve each triangle and then find the area of the triangle Show all work and or support your answer
a) C = 283deg b = 571 in a = 421 in b) a = 189 yd b = 214 yd c = 325 yd
13 The sides of a parallelogram are 40 cm and 60 cm One angle is 58deg while another is 122deg Find the lengths of the diagonals of the parallelogram Show all work and or support your answer
14 A weight is supported by cables attached to both ends of a balance beam as shown in the figure What angles are formed between the beam and the cables Show all work and or support your answer
15 Using the vectors in the figure below draw a sketch to represent
a) c + d b) d ndash c c) 3c
16 Use the vectors in the adjacent figure to find a) a + b b) a ndash b c) ndash a Show all work and or support your answer
17 For vectors u and w with angle θ between them sketch the resultant u + w 8=u 12=w θ = 20deg
18 Find the magnitude and direction angle for vector v = 8- 15 Show all work and or support your answer
19 Vector v has magnitude 415=v and direction angle 0327α prime= o Find the horizontal (a) and vertical (b) components of v Show all work and or support your answer
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
3
20 Write vector v in component form b a Show all work and or support your answer
21 Two forces act at a point in the plane One is 19 newtons the other is 32 newtons The angle between the forces is 118deg Find the magnitude of the resultant force Show all work and or support your answer
22 Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the adjacent figure Round your answer to the nearest tenth Show all work and or support your answer
23 Given u = 5 2minus and v = 3 4 find ndash2u + 4v Write your answer in the form ai + bj Show all work and or support your answer
24 Find the dot product of 3- 2 and 5 6 Show all work and or support your answer
25 Find the angle between the vectors -5i + 12j and 3i + 2j Show all work and or support your answer
26 Given u = 1 2minus v = 4 3 and w = 12 5minus evaluate ( )wvu minussdot Show all work and or support your answer
27 Determine whether the vectors 2 1 and 3 6minus are orthogonal Show all work and or support your answer
28 Two tugboats are pulling a disabled speedboat into port with forces 1240 lb and 1480 lb The angle between these forces is 282deg Find the magnitude and direction of the equilibrant Show all work and or support your answer
29 Two forces of 128 lb and 253 lb act at a point The equilibrant force is 320 lb Find the angle between the forces Show all work and or support your answer
30 A force of 176 lb makes an angle of 78deg50 with a second force The resultant of the two forces makes an angle of 41deg10 with the first force Find the magnitudes of the second force and the resultant Show all work and or support your answer
31 A 130-lb force keeps a 600-lb object from sliding down an inclined ramp Find the angle of the ramp Show all work and or support your answer
32 Find the magnitude of force necessary to keep a 3500-pound car from sliding down a ramp of 5deg Show all work and or support your answer
33 Two people are carrying a box One person exerts a force of 150 lb at an angle of 624deg with the horizontal The other person exerts a force of 114 lb at an angle of 549deg Find the weight of the box Show all work and or support your answer
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
4
34 A ship leaves port on a bearing of 340deg and travels 104 miles The ship then turns due east and travels 46 miles How far is the ship from port and what is its bearing from port Show all work and or support your answer
35 A pilot is flying at 168 mph She wants her flight path to be on a bearing of 57deg40 A wind is blowing from the south at 271 mph Find the bearing the pilot should fly and find the planersquos groundspeed Show all work and or support your answer
36 A pilot wants to fly on a bearing of 749deg By flying due east he finds a 42-mph wind blowing from the south puts him on course Find the groundspeed and airspeed Show all work and or support your answer
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 Write 288minus as the product of a real number and i Show all work and or support your answer
38 Solve 4x23x 2 minus=+ and express all nonreal complex solutions in terms of i Show all work and or support your answer
39 Perform the indicated operations and then simplify Show all work and or support your answer
a) 40
10minus
b) 8
612 minussdotminus
40 Perform the indicated operation and then simplify Write your answers in standard form Show all work and or support your answer
a) ( ) ( ) ( )iii 54324 +minus++minusminusminus b) ( )( )ii 2432 minus+minus c) ii
5132
minus+
41 Simplify each power of i Show all work and or support your answer
a) 25i b) 13minusi
42 Find the sum of i65minus and i32+minus by graphing each number in the complex plane and finding their resultant Show all work and or support your answer
43 Write the complex number 3[cos 150deg + i sin 150deg] in rectangular form Show all work and or support your answer
44 Write the complex number ndash5 ndash 5i in trigonometric form r(cos θ + i sin θ) with θ in the interval [0deg 360deg) Show all work and or support your answer
45 Find each product and write it in rectangular form On part b round to four decimal places Show all work and or support your answer
a) ( )[ ] ( )[ ]oooo 120sin 120cos5300sin 300cos8 ii +sdot+ b) (12 cis 185deg) (3 cis 125deg)
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
5
46 Find the quotient ( )( )oo
oo
150sin150cos2
120sin120cos4
i
i
+
+ amp write it in rectangular form Show all work and or support your answer
47 Convert the numerator and denominator of 31
2ii
minusminus to trigonometric form Then find the quotient and write it in
rectangular form Show all work and or support your answer
48 Find each power Write each answer in rectangular form Show all work and or support your answer
a) [ ]3100 3 ocis b) ( )522 iminusminus
49 Find all cube roots of each complex number Leave answers in trigonometric form Show all work and or support your answer
a) o300 27 cis b) iminus3
50 Find all complex solutions of each equation Leave answers in trigonometric form Show all work and or support your answer
a) 0x 4 =minus i b) 016x 4 =+
51 For each pair of polar points plot the point give two other pairs of polar coordinates (including one with a negative r) and give the rectangular coordinates for the point Show all work and or support your answer
a) ( )o135 2minus b) ⎟⎠⎞
⎜⎝⎛ minus
35 3 π
52 Given the rectangular coordinates ⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus
21
23 give two pairs of polar coordinates for the point including one with a
negative r Select θ so it is in the interval [0deg 360deg) Show all work and or support your answer
53 For each rectangular equation give its equivalent polar equation and sketch its graph Show all work and or support your answer
a) 3x ndash 2y = 6 b) 9yx 22 =+
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
6
54 Match each equation with its corresponding graph
a r = a cos θ A
b r = a sin θ B
c r = a plusmn b sin θ with 1 ltba C
d r = a plusmn b cos θ with 1 =ba D
e r = a cos 4θ E
f r = a sin 5θ F
g r = θsin b θ acos
cminus
G
h r2 = 4 sin 2θ H
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
7
55 Identify the type of polar equation and draw a complete graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 + 2 cos θ b) r = 4 cos 2θ
56 For each polar equation give its equivalent rectangular equation and sketch its graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 sin θ b) sinθ13r
minus=
Chapter 9 Exponential and Logarithmic Functions
57 Graph the following exponential function Give the equation of the horizontal asymptote a 5-point t-chart with the ldquokey pointrdquo at the center and an accurate graph with one notch on each axis labeled Show all of your work
( ) 12y 4x +minus= minus
58 Solve the following exponential equations algebraically Show all your work and give an exact answer
a 49
32 x
=⎟⎠
⎞⎜⎝
⎛minus
b 6 x 4 x 82 minus+ = c 125x 43=
59 Find the balance A for P dollars invested at rate r for t years and compounded n times
a P = $2000 r = 65 t = 10 years compounded quarterly
b P = $3000 r = 7 t = 8 years compounded continuously
60 Find the required annual interest rate to the nearest tenth of a percent for $48000 to grow to $7818694 when compounded semiannually for 5 yr Round your decimal answer to four decimal places and convert to a percent Show all of your work
61 Solve the following logarithmic equations algebraically Show all your work and give an exact answer
a 2161log x minus= b 5xlog
3= c x4log
16=
62 Graph the following logarithmic function Give the equation of the vertical asymptote the exponential form of the equation a 5-point t-chart with the ldquokeyrdquo point at the center and an accurate graph with one notch on each axis labeled Show all of your work
y = ( ) 42xlog 3 minus+
63 Use the properties of logarithms to expand or condense the following log expressions Show all of your work
a 4
3
bz
xylog b zlog32ylog
31xlog 2 aaa minusminus
64 Find the pH for lye soap which has a hydronium ion concentration of 141032 minustimes moles per liter
65 Find the [ ]+OH3 for drinking water which has a pH of 65
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
8
66 Use the change of base theorem to find an approximation for 15log21 Round your answer to four decimal places
67 Given f(x) = 3 x evaluate the following Show all of your work
a f (log 3 7) b f [log 3 (ln 3)] c f [log 3 (2 ln 3)]
68 a Find the decibel rating of a sound having an intensity of 100000 I0
b If the intensity of the sound is doubled by how much is the decibel rating increased
69 a Find the Richter scale rating for an earthquake having an amplitude of 1000000 I0
b Express the magnitude of a 67 earthquake (on the Richter scale) in terms of I0
70 Solve
a 64 13x =minus b x5x 32 =+ c 1116e 34x =minus+ d 50004
06110004t=⎟
⎠
⎞⎜⎝
⎛ +
71 a log (2x ndash 1) + log 10x = log 10 b ln x + ln (x ndash 1) = 2
c ( ) 12log3xlog4xlog 666 =minusminus d log (11x + 9) = 3 + log (x + 3)
72 Elena McDuff wants to buy an $8000 painting She has saved $6000 Find the number of years (to the nearest tenth) it will take for her savings to grow to $8000 at 57 compounded monthly
73 At what interest rate will $2500 grow to $467104 if invested for 10 years and interest is compounded quarterly
Answers
Chapter 7 Applications of Trigonometry and Vectors
1 A = 372deg a = 178 m c = 244 m
2 A = 6560deg b = 1942 cm c = 2727 cm
3 278 km
4 2934 m
5 6594 cm2
6 373 m2
7 a) 0 b) 1 c) 2
8 a) B asymp 26deg30 A asymp 112deg10 b) A1 asymp 722deg C1 asymp 596deg A2 asymp 1078deg C2 asymp 240deg c) no such triangle
9 A asymp 255deg B asymp 1022deg b asymp 739 yd
10 B1 asymp 49deg20 C1 asymp 92deg00 c1 asymp 155 km B2 asymp 130deg40 C2 asymp 10deg40 c2 asymp 288 km
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
9
11 a) A asymp 223deg B asymp 1082deg C asymp 495deg area = 2units 152231 asymp
b) A asymp 567deg C asymp 683deg b asymp 882 area = 2units 3680613547079 asymp
12 a) c asymp 283 in A asymp 449deg B asymp 1068deg area = 2units 57 7325368851 asymp
b) C asymp 107deg20 B asymp 39deg00 A asymp 33deg40 area = 2units 193040 372645000 asymp
13 52 cm 88 cm
14 264deg and 363deg
15 a) b) c)
16 a) 10 0 b) 2 8 c) 6- 4minus
17
18 17v = θ = 3319deg
19 a asymp 137 b asymp 711
20 v = 34 4
21 29 newtons
22 244 lb
23 20i + 2j
24 ndash3
25 7893deg
26 ndash24
27 orthogonal
28 2640 lb at an angle of 1672deg with the 1480-lb force
29 701deg
30 magnitude of second force 190 lb magnitude of the resultant 283 lb
31 125deg
32 305 lbs
33 226 lb
34 135 mi 504deg
35 bearing 65deg30 groundspeed 181 mph
36 groundspeed 161 mph airspeed 156 mph
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
10
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 i 212
38 i32
32x plusmnminus=
39 a) i21 b) ndash3
40 a) i+minus10 b) i162+minus c) i21
21+minus
41 a) i b) ndash i 42 i33minus
43 i23
233+minus
44 ( )oo sin225 icos225 25 +
45 a) 32020 i+ b) 308580 + 185414 i
46 iminus3
47 i21
23minusminus
48 a) i2
3272
27minus b) 128 + 128i
49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+
( )oo 350sin 350 cos23 i+
50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg
b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg
51 a) ( )o495 2minus (2 315deg) ( )2 2 minus
b) ⎟⎠⎞
⎜⎝⎛
3π 3 ⎟
⎠⎞
⎜⎝⎛minus
34π3 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
233
23
52 (1 210deg) (-1 30deg)
53 a) 2sinθ3cosθ
6rminus
= b) r = 3 or r = ndash3
54 a F b A c D d G e H f B g C h E
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
3
20 Write vector v in component form b a Show all work and or support your answer
21 Two forces act at a point in the plane One is 19 newtons the other is 32 newtons The angle between the forces is 118deg Find the magnitude of the resultant force Show all work and or support your answer
22 Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the adjacent figure Round your answer to the nearest tenth Show all work and or support your answer
23 Given u = 5 2minus and v = 3 4 find ndash2u + 4v Write your answer in the form ai + bj Show all work and or support your answer
24 Find the dot product of 3- 2 and 5 6 Show all work and or support your answer
25 Find the angle between the vectors -5i + 12j and 3i + 2j Show all work and or support your answer
26 Given u = 1 2minus v = 4 3 and w = 12 5minus evaluate ( )wvu minussdot Show all work and or support your answer
27 Determine whether the vectors 2 1 and 3 6minus are orthogonal Show all work and or support your answer
28 Two tugboats are pulling a disabled speedboat into port with forces 1240 lb and 1480 lb The angle between these forces is 282deg Find the magnitude and direction of the equilibrant Show all work and or support your answer
29 Two forces of 128 lb and 253 lb act at a point The equilibrant force is 320 lb Find the angle between the forces Show all work and or support your answer
30 A force of 176 lb makes an angle of 78deg50 with a second force The resultant of the two forces makes an angle of 41deg10 with the first force Find the magnitudes of the second force and the resultant Show all work and or support your answer
31 A 130-lb force keeps a 600-lb object from sliding down an inclined ramp Find the angle of the ramp Show all work and or support your answer
32 Find the magnitude of force necessary to keep a 3500-pound car from sliding down a ramp of 5deg Show all work and or support your answer
33 Two people are carrying a box One person exerts a force of 150 lb at an angle of 624deg with the horizontal The other person exerts a force of 114 lb at an angle of 549deg Find the weight of the box Show all work and or support your answer
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
4
34 A ship leaves port on a bearing of 340deg and travels 104 miles The ship then turns due east and travels 46 miles How far is the ship from port and what is its bearing from port Show all work and or support your answer
35 A pilot is flying at 168 mph She wants her flight path to be on a bearing of 57deg40 A wind is blowing from the south at 271 mph Find the bearing the pilot should fly and find the planersquos groundspeed Show all work and or support your answer
36 A pilot wants to fly on a bearing of 749deg By flying due east he finds a 42-mph wind blowing from the south puts him on course Find the groundspeed and airspeed Show all work and or support your answer
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 Write 288minus as the product of a real number and i Show all work and or support your answer
38 Solve 4x23x 2 minus=+ and express all nonreal complex solutions in terms of i Show all work and or support your answer
39 Perform the indicated operations and then simplify Show all work and or support your answer
a) 40
10minus
b) 8
612 minussdotminus
40 Perform the indicated operation and then simplify Write your answers in standard form Show all work and or support your answer
a) ( ) ( ) ( )iii 54324 +minus++minusminusminus b) ( )( )ii 2432 minus+minus c) ii
5132
minus+
41 Simplify each power of i Show all work and or support your answer
a) 25i b) 13minusi
42 Find the sum of i65minus and i32+minus by graphing each number in the complex plane and finding their resultant Show all work and or support your answer
43 Write the complex number 3[cos 150deg + i sin 150deg] in rectangular form Show all work and or support your answer
44 Write the complex number ndash5 ndash 5i in trigonometric form r(cos θ + i sin θ) with θ in the interval [0deg 360deg) Show all work and or support your answer
45 Find each product and write it in rectangular form On part b round to four decimal places Show all work and or support your answer
a) ( )[ ] ( )[ ]oooo 120sin 120cos5300sin 300cos8 ii +sdot+ b) (12 cis 185deg) (3 cis 125deg)
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
5
46 Find the quotient ( )( )oo
oo
150sin150cos2
120sin120cos4
i
i
+
+ amp write it in rectangular form Show all work and or support your answer
47 Convert the numerator and denominator of 31
2ii
minusminus to trigonometric form Then find the quotient and write it in
rectangular form Show all work and or support your answer
48 Find each power Write each answer in rectangular form Show all work and or support your answer
a) [ ]3100 3 ocis b) ( )522 iminusminus
49 Find all cube roots of each complex number Leave answers in trigonometric form Show all work and or support your answer
a) o300 27 cis b) iminus3
50 Find all complex solutions of each equation Leave answers in trigonometric form Show all work and or support your answer
a) 0x 4 =minus i b) 016x 4 =+
51 For each pair of polar points plot the point give two other pairs of polar coordinates (including one with a negative r) and give the rectangular coordinates for the point Show all work and or support your answer
a) ( )o135 2minus b) ⎟⎠⎞
⎜⎝⎛ minus
35 3 π
52 Given the rectangular coordinates ⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus
21
23 give two pairs of polar coordinates for the point including one with a
negative r Select θ so it is in the interval [0deg 360deg) Show all work and or support your answer
53 For each rectangular equation give its equivalent polar equation and sketch its graph Show all work and or support your answer
a) 3x ndash 2y = 6 b) 9yx 22 =+
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
6
54 Match each equation with its corresponding graph
a r = a cos θ A
b r = a sin θ B
c r = a plusmn b sin θ with 1 ltba C
d r = a plusmn b cos θ with 1 =ba D
e r = a cos 4θ E
f r = a sin 5θ F
g r = θsin b θ acos
cminus
G
h r2 = 4 sin 2θ H
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
7
55 Identify the type of polar equation and draw a complete graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 + 2 cos θ b) r = 4 cos 2θ
56 For each polar equation give its equivalent rectangular equation and sketch its graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 sin θ b) sinθ13r
minus=
Chapter 9 Exponential and Logarithmic Functions
57 Graph the following exponential function Give the equation of the horizontal asymptote a 5-point t-chart with the ldquokey pointrdquo at the center and an accurate graph with one notch on each axis labeled Show all of your work
( ) 12y 4x +minus= minus
58 Solve the following exponential equations algebraically Show all your work and give an exact answer
a 49
32 x
=⎟⎠
⎞⎜⎝
⎛minus
b 6 x 4 x 82 minus+ = c 125x 43=
59 Find the balance A for P dollars invested at rate r for t years and compounded n times
a P = $2000 r = 65 t = 10 years compounded quarterly
b P = $3000 r = 7 t = 8 years compounded continuously
60 Find the required annual interest rate to the nearest tenth of a percent for $48000 to grow to $7818694 when compounded semiannually for 5 yr Round your decimal answer to four decimal places and convert to a percent Show all of your work
61 Solve the following logarithmic equations algebraically Show all your work and give an exact answer
a 2161log x minus= b 5xlog
3= c x4log
16=
62 Graph the following logarithmic function Give the equation of the vertical asymptote the exponential form of the equation a 5-point t-chart with the ldquokeyrdquo point at the center and an accurate graph with one notch on each axis labeled Show all of your work
y = ( ) 42xlog 3 minus+
63 Use the properties of logarithms to expand or condense the following log expressions Show all of your work
a 4
3
bz
xylog b zlog32ylog
31xlog 2 aaa minusminus
64 Find the pH for lye soap which has a hydronium ion concentration of 141032 minustimes moles per liter
65 Find the [ ]+OH3 for drinking water which has a pH of 65
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
8
66 Use the change of base theorem to find an approximation for 15log21 Round your answer to four decimal places
67 Given f(x) = 3 x evaluate the following Show all of your work
a f (log 3 7) b f [log 3 (ln 3)] c f [log 3 (2 ln 3)]
68 a Find the decibel rating of a sound having an intensity of 100000 I0
b If the intensity of the sound is doubled by how much is the decibel rating increased
69 a Find the Richter scale rating for an earthquake having an amplitude of 1000000 I0
b Express the magnitude of a 67 earthquake (on the Richter scale) in terms of I0
70 Solve
a 64 13x =minus b x5x 32 =+ c 1116e 34x =minus+ d 50004
06110004t=⎟
⎠
⎞⎜⎝
⎛ +
71 a log (2x ndash 1) + log 10x = log 10 b ln x + ln (x ndash 1) = 2
c ( ) 12log3xlog4xlog 666 =minusminus d log (11x + 9) = 3 + log (x + 3)
72 Elena McDuff wants to buy an $8000 painting She has saved $6000 Find the number of years (to the nearest tenth) it will take for her savings to grow to $8000 at 57 compounded monthly
73 At what interest rate will $2500 grow to $467104 if invested for 10 years and interest is compounded quarterly
Answers
Chapter 7 Applications of Trigonometry and Vectors
1 A = 372deg a = 178 m c = 244 m
2 A = 6560deg b = 1942 cm c = 2727 cm
3 278 km
4 2934 m
5 6594 cm2
6 373 m2
7 a) 0 b) 1 c) 2
8 a) B asymp 26deg30 A asymp 112deg10 b) A1 asymp 722deg C1 asymp 596deg A2 asymp 1078deg C2 asymp 240deg c) no such triangle
9 A asymp 255deg B asymp 1022deg b asymp 739 yd
10 B1 asymp 49deg20 C1 asymp 92deg00 c1 asymp 155 km B2 asymp 130deg40 C2 asymp 10deg40 c2 asymp 288 km
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
9
11 a) A asymp 223deg B asymp 1082deg C asymp 495deg area = 2units 152231 asymp
b) A asymp 567deg C asymp 683deg b asymp 882 area = 2units 3680613547079 asymp
12 a) c asymp 283 in A asymp 449deg B asymp 1068deg area = 2units 57 7325368851 asymp
b) C asymp 107deg20 B asymp 39deg00 A asymp 33deg40 area = 2units 193040 372645000 asymp
13 52 cm 88 cm
14 264deg and 363deg
15 a) b) c)
16 a) 10 0 b) 2 8 c) 6- 4minus
17
18 17v = θ = 3319deg
19 a asymp 137 b asymp 711
20 v = 34 4
21 29 newtons
22 244 lb
23 20i + 2j
24 ndash3
25 7893deg
26 ndash24
27 orthogonal
28 2640 lb at an angle of 1672deg with the 1480-lb force
29 701deg
30 magnitude of second force 190 lb magnitude of the resultant 283 lb
31 125deg
32 305 lbs
33 226 lb
34 135 mi 504deg
35 bearing 65deg30 groundspeed 181 mph
36 groundspeed 161 mph airspeed 156 mph
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
10
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 i 212
38 i32
32x plusmnminus=
39 a) i21 b) ndash3
40 a) i+minus10 b) i162+minus c) i21
21+minus
41 a) i b) ndash i 42 i33minus
43 i23
233+minus
44 ( )oo sin225 icos225 25 +
45 a) 32020 i+ b) 308580 + 185414 i
46 iminus3
47 i21
23minusminus
48 a) i2
3272
27minus b) 128 + 128i
49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+
( )oo 350sin 350 cos23 i+
50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg
b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg
51 a) ( )o495 2minus (2 315deg) ( )2 2 minus
b) ⎟⎠⎞
⎜⎝⎛
3π 3 ⎟
⎠⎞
⎜⎝⎛minus
34π3 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
233
23
52 (1 210deg) (-1 30deg)
53 a) 2sinθ3cosθ
6rminus
= b) r = 3 or r = ndash3
54 a F b A c D d G e H f B g C h E
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
4
34 A ship leaves port on a bearing of 340deg and travels 104 miles The ship then turns due east and travels 46 miles How far is the ship from port and what is its bearing from port Show all work and or support your answer
35 A pilot is flying at 168 mph She wants her flight path to be on a bearing of 57deg40 A wind is blowing from the south at 271 mph Find the bearing the pilot should fly and find the planersquos groundspeed Show all work and or support your answer
36 A pilot wants to fly on a bearing of 749deg By flying due east he finds a 42-mph wind blowing from the south puts him on course Find the groundspeed and airspeed Show all work and or support your answer
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 Write 288minus as the product of a real number and i Show all work and or support your answer
38 Solve 4x23x 2 minus=+ and express all nonreal complex solutions in terms of i Show all work and or support your answer
39 Perform the indicated operations and then simplify Show all work and or support your answer
a) 40
10minus
b) 8
612 minussdotminus
40 Perform the indicated operation and then simplify Write your answers in standard form Show all work and or support your answer
a) ( ) ( ) ( )iii 54324 +minus++minusminusminus b) ( )( )ii 2432 minus+minus c) ii
5132
minus+
41 Simplify each power of i Show all work and or support your answer
a) 25i b) 13minusi
42 Find the sum of i65minus and i32+minus by graphing each number in the complex plane and finding their resultant Show all work and or support your answer
43 Write the complex number 3[cos 150deg + i sin 150deg] in rectangular form Show all work and or support your answer
44 Write the complex number ndash5 ndash 5i in trigonometric form r(cos θ + i sin θ) with θ in the interval [0deg 360deg) Show all work and or support your answer
45 Find each product and write it in rectangular form On part b round to four decimal places Show all work and or support your answer
a) ( )[ ] ( )[ ]oooo 120sin 120cos5300sin 300cos8 ii +sdot+ b) (12 cis 185deg) (3 cis 125deg)
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
5
46 Find the quotient ( )( )oo
oo
150sin150cos2
120sin120cos4
i
i
+
+ amp write it in rectangular form Show all work and or support your answer
47 Convert the numerator and denominator of 31
2ii
minusminus to trigonometric form Then find the quotient and write it in
rectangular form Show all work and or support your answer
48 Find each power Write each answer in rectangular form Show all work and or support your answer
a) [ ]3100 3 ocis b) ( )522 iminusminus
49 Find all cube roots of each complex number Leave answers in trigonometric form Show all work and or support your answer
a) o300 27 cis b) iminus3
50 Find all complex solutions of each equation Leave answers in trigonometric form Show all work and or support your answer
a) 0x 4 =minus i b) 016x 4 =+
51 For each pair of polar points plot the point give two other pairs of polar coordinates (including one with a negative r) and give the rectangular coordinates for the point Show all work and or support your answer
a) ( )o135 2minus b) ⎟⎠⎞
⎜⎝⎛ minus
35 3 π
52 Given the rectangular coordinates ⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus
21
23 give two pairs of polar coordinates for the point including one with a
negative r Select θ so it is in the interval [0deg 360deg) Show all work and or support your answer
53 For each rectangular equation give its equivalent polar equation and sketch its graph Show all work and or support your answer
a) 3x ndash 2y = 6 b) 9yx 22 =+
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
6
54 Match each equation with its corresponding graph
a r = a cos θ A
b r = a sin θ B
c r = a plusmn b sin θ with 1 ltba C
d r = a plusmn b cos θ with 1 =ba D
e r = a cos 4θ E
f r = a sin 5θ F
g r = θsin b θ acos
cminus
G
h r2 = 4 sin 2θ H
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
7
55 Identify the type of polar equation and draw a complete graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 + 2 cos θ b) r = 4 cos 2θ
56 For each polar equation give its equivalent rectangular equation and sketch its graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 sin θ b) sinθ13r
minus=
Chapter 9 Exponential and Logarithmic Functions
57 Graph the following exponential function Give the equation of the horizontal asymptote a 5-point t-chart with the ldquokey pointrdquo at the center and an accurate graph with one notch on each axis labeled Show all of your work
( ) 12y 4x +minus= minus
58 Solve the following exponential equations algebraically Show all your work and give an exact answer
a 49
32 x
=⎟⎠
⎞⎜⎝
⎛minus
b 6 x 4 x 82 minus+ = c 125x 43=
59 Find the balance A for P dollars invested at rate r for t years and compounded n times
a P = $2000 r = 65 t = 10 years compounded quarterly
b P = $3000 r = 7 t = 8 years compounded continuously
60 Find the required annual interest rate to the nearest tenth of a percent for $48000 to grow to $7818694 when compounded semiannually for 5 yr Round your decimal answer to four decimal places and convert to a percent Show all of your work
61 Solve the following logarithmic equations algebraically Show all your work and give an exact answer
a 2161log x minus= b 5xlog
3= c x4log
16=
62 Graph the following logarithmic function Give the equation of the vertical asymptote the exponential form of the equation a 5-point t-chart with the ldquokeyrdquo point at the center and an accurate graph with one notch on each axis labeled Show all of your work
y = ( ) 42xlog 3 minus+
63 Use the properties of logarithms to expand or condense the following log expressions Show all of your work
a 4
3
bz
xylog b zlog32ylog
31xlog 2 aaa minusminus
64 Find the pH for lye soap which has a hydronium ion concentration of 141032 minustimes moles per liter
65 Find the [ ]+OH3 for drinking water which has a pH of 65
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
8
66 Use the change of base theorem to find an approximation for 15log21 Round your answer to four decimal places
67 Given f(x) = 3 x evaluate the following Show all of your work
a f (log 3 7) b f [log 3 (ln 3)] c f [log 3 (2 ln 3)]
68 a Find the decibel rating of a sound having an intensity of 100000 I0
b If the intensity of the sound is doubled by how much is the decibel rating increased
69 a Find the Richter scale rating for an earthquake having an amplitude of 1000000 I0
b Express the magnitude of a 67 earthquake (on the Richter scale) in terms of I0
70 Solve
a 64 13x =minus b x5x 32 =+ c 1116e 34x =minus+ d 50004
06110004t=⎟
⎠
⎞⎜⎝
⎛ +
71 a log (2x ndash 1) + log 10x = log 10 b ln x + ln (x ndash 1) = 2
c ( ) 12log3xlog4xlog 666 =minusminus d log (11x + 9) = 3 + log (x + 3)
72 Elena McDuff wants to buy an $8000 painting She has saved $6000 Find the number of years (to the nearest tenth) it will take for her savings to grow to $8000 at 57 compounded monthly
73 At what interest rate will $2500 grow to $467104 if invested for 10 years and interest is compounded quarterly
Answers
Chapter 7 Applications of Trigonometry and Vectors
1 A = 372deg a = 178 m c = 244 m
2 A = 6560deg b = 1942 cm c = 2727 cm
3 278 km
4 2934 m
5 6594 cm2
6 373 m2
7 a) 0 b) 1 c) 2
8 a) B asymp 26deg30 A asymp 112deg10 b) A1 asymp 722deg C1 asymp 596deg A2 asymp 1078deg C2 asymp 240deg c) no such triangle
9 A asymp 255deg B asymp 1022deg b asymp 739 yd
10 B1 asymp 49deg20 C1 asymp 92deg00 c1 asymp 155 km B2 asymp 130deg40 C2 asymp 10deg40 c2 asymp 288 km
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
9
11 a) A asymp 223deg B asymp 1082deg C asymp 495deg area = 2units 152231 asymp
b) A asymp 567deg C asymp 683deg b asymp 882 area = 2units 3680613547079 asymp
12 a) c asymp 283 in A asymp 449deg B asymp 1068deg area = 2units 57 7325368851 asymp
b) C asymp 107deg20 B asymp 39deg00 A asymp 33deg40 area = 2units 193040 372645000 asymp
13 52 cm 88 cm
14 264deg and 363deg
15 a) b) c)
16 a) 10 0 b) 2 8 c) 6- 4minus
17
18 17v = θ = 3319deg
19 a asymp 137 b asymp 711
20 v = 34 4
21 29 newtons
22 244 lb
23 20i + 2j
24 ndash3
25 7893deg
26 ndash24
27 orthogonal
28 2640 lb at an angle of 1672deg with the 1480-lb force
29 701deg
30 magnitude of second force 190 lb magnitude of the resultant 283 lb
31 125deg
32 305 lbs
33 226 lb
34 135 mi 504deg
35 bearing 65deg30 groundspeed 181 mph
36 groundspeed 161 mph airspeed 156 mph
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
10
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 i 212
38 i32
32x plusmnminus=
39 a) i21 b) ndash3
40 a) i+minus10 b) i162+minus c) i21
21+minus
41 a) i b) ndash i 42 i33minus
43 i23
233+minus
44 ( )oo sin225 icos225 25 +
45 a) 32020 i+ b) 308580 + 185414 i
46 iminus3
47 i21
23minusminus
48 a) i2
3272
27minus b) 128 + 128i
49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+
( )oo 350sin 350 cos23 i+
50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg
b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg
51 a) ( )o495 2minus (2 315deg) ( )2 2 minus
b) ⎟⎠⎞
⎜⎝⎛
3π 3 ⎟
⎠⎞
⎜⎝⎛minus
34π3 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
233
23
52 (1 210deg) (-1 30deg)
53 a) 2sinθ3cosθ
6rminus
= b) r = 3 or r = ndash3
54 a F b A c D d G e H f B g C h E
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
5
46 Find the quotient ( )( )oo
oo
150sin150cos2
120sin120cos4
i
i
+
+ amp write it in rectangular form Show all work and or support your answer
47 Convert the numerator and denominator of 31
2ii
minusminus to trigonometric form Then find the quotient and write it in
rectangular form Show all work and or support your answer
48 Find each power Write each answer in rectangular form Show all work and or support your answer
a) [ ]3100 3 ocis b) ( )522 iminusminus
49 Find all cube roots of each complex number Leave answers in trigonometric form Show all work and or support your answer
a) o300 27 cis b) iminus3
50 Find all complex solutions of each equation Leave answers in trigonometric form Show all work and or support your answer
a) 0x 4 =minus i b) 016x 4 =+
51 For each pair of polar points plot the point give two other pairs of polar coordinates (including one with a negative r) and give the rectangular coordinates for the point Show all work and or support your answer
a) ( )o135 2minus b) ⎟⎠⎞
⎜⎝⎛ minus
35 3 π
52 Given the rectangular coordinates ⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus
21
23 give two pairs of polar coordinates for the point including one with a
negative r Select θ so it is in the interval [0deg 360deg) Show all work and or support your answer
53 For each rectangular equation give its equivalent polar equation and sketch its graph Show all work and or support your answer
a) 3x ndash 2y = 6 b) 9yx 22 =+
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
6
54 Match each equation with its corresponding graph
a r = a cos θ A
b r = a sin θ B
c r = a plusmn b sin θ with 1 ltba C
d r = a plusmn b cos θ with 1 =ba D
e r = a cos 4θ E
f r = a sin 5θ F
g r = θsin b θ acos
cminus
G
h r2 = 4 sin 2θ H
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
7
55 Identify the type of polar equation and draw a complete graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 + 2 cos θ b) r = 4 cos 2θ
56 For each polar equation give its equivalent rectangular equation and sketch its graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 sin θ b) sinθ13r
minus=
Chapter 9 Exponential and Logarithmic Functions
57 Graph the following exponential function Give the equation of the horizontal asymptote a 5-point t-chart with the ldquokey pointrdquo at the center and an accurate graph with one notch on each axis labeled Show all of your work
( ) 12y 4x +minus= minus
58 Solve the following exponential equations algebraically Show all your work and give an exact answer
a 49
32 x
=⎟⎠
⎞⎜⎝
⎛minus
b 6 x 4 x 82 minus+ = c 125x 43=
59 Find the balance A for P dollars invested at rate r for t years and compounded n times
a P = $2000 r = 65 t = 10 years compounded quarterly
b P = $3000 r = 7 t = 8 years compounded continuously
60 Find the required annual interest rate to the nearest tenth of a percent for $48000 to grow to $7818694 when compounded semiannually for 5 yr Round your decimal answer to four decimal places and convert to a percent Show all of your work
61 Solve the following logarithmic equations algebraically Show all your work and give an exact answer
a 2161log x minus= b 5xlog
3= c x4log
16=
62 Graph the following logarithmic function Give the equation of the vertical asymptote the exponential form of the equation a 5-point t-chart with the ldquokeyrdquo point at the center and an accurate graph with one notch on each axis labeled Show all of your work
y = ( ) 42xlog 3 minus+
63 Use the properties of logarithms to expand or condense the following log expressions Show all of your work
a 4
3
bz
xylog b zlog32ylog
31xlog 2 aaa minusminus
64 Find the pH for lye soap which has a hydronium ion concentration of 141032 minustimes moles per liter
65 Find the [ ]+OH3 for drinking water which has a pH of 65
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
8
66 Use the change of base theorem to find an approximation for 15log21 Round your answer to four decimal places
67 Given f(x) = 3 x evaluate the following Show all of your work
a f (log 3 7) b f [log 3 (ln 3)] c f [log 3 (2 ln 3)]
68 a Find the decibel rating of a sound having an intensity of 100000 I0
b If the intensity of the sound is doubled by how much is the decibel rating increased
69 a Find the Richter scale rating for an earthquake having an amplitude of 1000000 I0
b Express the magnitude of a 67 earthquake (on the Richter scale) in terms of I0
70 Solve
a 64 13x =minus b x5x 32 =+ c 1116e 34x =minus+ d 50004
06110004t=⎟
⎠
⎞⎜⎝
⎛ +
71 a log (2x ndash 1) + log 10x = log 10 b ln x + ln (x ndash 1) = 2
c ( ) 12log3xlog4xlog 666 =minusminus d log (11x + 9) = 3 + log (x + 3)
72 Elena McDuff wants to buy an $8000 painting She has saved $6000 Find the number of years (to the nearest tenth) it will take for her savings to grow to $8000 at 57 compounded monthly
73 At what interest rate will $2500 grow to $467104 if invested for 10 years and interest is compounded quarterly
Answers
Chapter 7 Applications of Trigonometry and Vectors
1 A = 372deg a = 178 m c = 244 m
2 A = 6560deg b = 1942 cm c = 2727 cm
3 278 km
4 2934 m
5 6594 cm2
6 373 m2
7 a) 0 b) 1 c) 2
8 a) B asymp 26deg30 A asymp 112deg10 b) A1 asymp 722deg C1 asymp 596deg A2 asymp 1078deg C2 asymp 240deg c) no such triangle
9 A asymp 255deg B asymp 1022deg b asymp 739 yd
10 B1 asymp 49deg20 C1 asymp 92deg00 c1 asymp 155 km B2 asymp 130deg40 C2 asymp 10deg40 c2 asymp 288 km
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
9
11 a) A asymp 223deg B asymp 1082deg C asymp 495deg area = 2units 152231 asymp
b) A asymp 567deg C asymp 683deg b asymp 882 area = 2units 3680613547079 asymp
12 a) c asymp 283 in A asymp 449deg B asymp 1068deg area = 2units 57 7325368851 asymp
b) C asymp 107deg20 B asymp 39deg00 A asymp 33deg40 area = 2units 193040 372645000 asymp
13 52 cm 88 cm
14 264deg and 363deg
15 a) b) c)
16 a) 10 0 b) 2 8 c) 6- 4minus
17
18 17v = θ = 3319deg
19 a asymp 137 b asymp 711
20 v = 34 4
21 29 newtons
22 244 lb
23 20i + 2j
24 ndash3
25 7893deg
26 ndash24
27 orthogonal
28 2640 lb at an angle of 1672deg with the 1480-lb force
29 701deg
30 magnitude of second force 190 lb magnitude of the resultant 283 lb
31 125deg
32 305 lbs
33 226 lb
34 135 mi 504deg
35 bearing 65deg30 groundspeed 181 mph
36 groundspeed 161 mph airspeed 156 mph
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
10
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 i 212
38 i32
32x plusmnminus=
39 a) i21 b) ndash3
40 a) i+minus10 b) i162+minus c) i21
21+minus
41 a) i b) ndash i 42 i33minus
43 i23
233+minus
44 ( )oo sin225 icos225 25 +
45 a) 32020 i+ b) 308580 + 185414 i
46 iminus3
47 i21
23minusminus
48 a) i2
3272
27minus b) 128 + 128i
49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+
( )oo 350sin 350 cos23 i+
50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg
b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg
51 a) ( )o495 2minus (2 315deg) ( )2 2 minus
b) ⎟⎠⎞
⎜⎝⎛
3π 3 ⎟
⎠⎞
⎜⎝⎛minus
34π3 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
233
23
52 (1 210deg) (-1 30deg)
53 a) 2sinθ3cosθ
6rminus
= b) r = 3 or r = ndash3
54 a F b A c D d G e H f B g C h E
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
6
54 Match each equation with its corresponding graph
a r = a cos θ A
b r = a sin θ B
c r = a plusmn b sin θ with 1 ltba C
d r = a plusmn b cos θ with 1 =ba D
e r = a cos 4θ E
f r = a sin 5θ F
g r = θsin b θ acos
cminus
G
h r2 = 4 sin 2θ H
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
7
55 Identify the type of polar equation and draw a complete graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 + 2 cos θ b) r = 4 cos 2θ
56 For each polar equation give its equivalent rectangular equation and sketch its graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 sin θ b) sinθ13r
minus=
Chapter 9 Exponential and Logarithmic Functions
57 Graph the following exponential function Give the equation of the horizontal asymptote a 5-point t-chart with the ldquokey pointrdquo at the center and an accurate graph with one notch on each axis labeled Show all of your work
( ) 12y 4x +minus= minus
58 Solve the following exponential equations algebraically Show all your work and give an exact answer
a 49
32 x
=⎟⎠
⎞⎜⎝
⎛minus
b 6 x 4 x 82 minus+ = c 125x 43=
59 Find the balance A for P dollars invested at rate r for t years and compounded n times
a P = $2000 r = 65 t = 10 years compounded quarterly
b P = $3000 r = 7 t = 8 years compounded continuously
60 Find the required annual interest rate to the nearest tenth of a percent for $48000 to grow to $7818694 when compounded semiannually for 5 yr Round your decimal answer to four decimal places and convert to a percent Show all of your work
61 Solve the following logarithmic equations algebraically Show all your work and give an exact answer
a 2161log x minus= b 5xlog
3= c x4log
16=
62 Graph the following logarithmic function Give the equation of the vertical asymptote the exponential form of the equation a 5-point t-chart with the ldquokeyrdquo point at the center and an accurate graph with one notch on each axis labeled Show all of your work
y = ( ) 42xlog 3 minus+
63 Use the properties of logarithms to expand or condense the following log expressions Show all of your work
a 4
3
bz
xylog b zlog32ylog
31xlog 2 aaa minusminus
64 Find the pH for lye soap which has a hydronium ion concentration of 141032 minustimes moles per liter
65 Find the [ ]+OH3 for drinking water which has a pH of 65
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
8
66 Use the change of base theorem to find an approximation for 15log21 Round your answer to four decimal places
67 Given f(x) = 3 x evaluate the following Show all of your work
a f (log 3 7) b f [log 3 (ln 3)] c f [log 3 (2 ln 3)]
68 a Find the decibel rating of a sound having an intensity of 100000 I0
b If the intensity of the sound is doubled by how much is the decibel rating increased
69 a Find the Richter scale rating for an earthquake having an amplitude of 1000000 I0
b Express the magnitude of a 67 earthquake (on the Richter scale) in terms of I0
70 Solve
a 64 13x =minus b x5x 32 =+ c 1116e 34x =minus+ d 50004
06110004t=⎟
⎠
⎞⎜⎝
⎛ +
71 a log (2x ndash 1) + log 10x = log 10 b ln x + ln (x ndash 1) = 2
c ( ) 12log3xlog4xlog 666 =minusminus d log (11x + 9) = 3 + log (x + 3)
72 Elena McDuff wants to buy an $8000 painting She has saved $6000 Find the number of years (to the nearest tenth) it will take for her savings to grow to $8000 at 57 compounded monthly
73 At what interest rate will $2500 grow to $467104 if invested for 10 years and interest is compounded quarterly
Answers
Chapter 7 Applications of Trigonometry and Vectors
1 A = 372deg a = 178 m c = 244 m
2 A = 6560deg b = 1942 cm c = 2727 cm
3 278 km
4 2934 m
5 6594 cm2
6 373 m2
7 a) 0 b) 1 c) 2
8 a) B asymp 26deg30 A asymp 112deg10 b) A1 asymp 722deg C1 asymp 596deg A2 asymp 1078deg C2 asymp 240deg c) no such triangle
9 A asymp 255deg B asymp 1022deg b asymp 739 yd
10 B1 asymp 49deg20 C1 asymp 92deg00 c1 asymp 155 km B2 asymp 130deg40 C2 asymp 10deg40 c2 asymp 288 km
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
9
11 a) A asymp 223deg B asymp 1082deg C asymp 495deg area = 2units 152231 asymp
b) A asymp 567deg C asymp 683deg b asymp 882 area = 2units 3680613547079 asymp
12 a) c asymp 283 in A asymp 449deg B asymp 1068deg area = 2units 57 7325368851 asymp
b) C asymp 107deg20 B asymp 39deg00 A asymp 33deg40 area = 2units 193040 372645000 asymp
13 52 cm 88 cm
14 264deg and 363deg
15 a) b) c)
16 a) 10 0 b) 2 8 c) 6- 4minus
17
18 17v = θ = 3319deg
19 a asymp 137 b asymp 711
20 v = 34 4
21 29 newtons
22 244 lb
23 20i + 2j
24 ndash3
25 7893deg
26 ndash24
27 orthogonal
28 2640 lb at an angle of 1672deg with the 1480-lb force
29 701deg
30 magnitude of second force 190 lb magnitude of the resultant 283 lb
31 125deg
32 305 lbs
33 226 lb
34 135 mi 504deg
35 bearing 65deg30 groundspeed 181 mph
36 groundspeed 161 mph airspeed 156 mph
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
10
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 i 212
38 i32
32x plusmnminus=
39 a) i21 b) ndash3
40 a) i+minus10 b) i162+minus c) i21
21+minus
41 a) i b) ndash i 42 i33minus
43 i23
233+minus
44 ( )oo sin225 icos225 25 +
45 a) 32020 i+ b) 308580 + 185414 i
46 iminus3
47 i21
23minusminus
48 a) i2
3272
27minus b) 128 + 128i
49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+
( )oo 350sin 350 cos23 i+
50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg
b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg
51 a) ( )o495 2minus (2 315deg) ( )2 2 minus
b) ⎟⎠⎞
⎜⎝⎛
3π 3 ⎟
⎠⎞
⎜⎝⎛minus
34π3 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
233
23
52 (1 210deg) (-1 30deg)
53 a) 2sinθ3cosθ
6rminus
= b) r = 3 or r = ndash3
54 a F b A c D d G e H f B g C h E
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
7
55 Identify the type of polar equation and draw a complete graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 + 2 cos θ b) r = 4 cos 2θ
56 For each polar equation give its equivalent rectangular equation and sketch its graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer
a) r = 2 sin θ b) sinθ13r
minus=
Chapter 9 Exponential and Logarithmic Functions
57 Graph the following exponential function Give the equation of the horizontal asymptote a 5-point t-chart with the ldquokey pointrdquo at the center and an accurate graph with one notch on each axis labeled Show all of your work
( ) 12y 4x +minus= minus
58 Solve the following exponential equations algebraically Show all your work and give an exact answer
a 49
32 x
=⎟⎠
⎞⎜⎝
⎛minus
b 6 x 4 x 82 minus+ = c 125x 43=
59 Find the balance A for P dollars invested at rate r for t years and compounded n times
a P = $2000 r = 65 t = 10 years compounded quarterly
b P = $3000 r = 7 t = 8 years compounded continuously
60 Find the required annual interest rate to the nearest tenth of a percent for $48000 to grow to $7818694 when compounded semiannually for 5 yr Round your decimal answer to four decimal places and convert to a percent Show all of your work
61 Solve the following logarithmic equations algebraically Show all your work and give an exact answer
a 2161log x minus= b 5xlog
3= c x4log
16=
62 Graph the following logarithmic function Give the equation of the vertical asymptote the exponential form of the equation a 5-point t-chart with the ldquokeyrdquo point at the center and an accurate graph with one notch on each axis labeled Show all of your work
y = ( ) 42xlog 3 minus+
63 Use the properties of logarithms to expand or condense the following log expressions Show all of your work
a 4
3
bz
xylog b zlog32ylog
31xlog 2 aaa minusminus
64 Find the pH for lye soap which has a hydronium ion concentration of 141032 minustimes moles per liter
65 Find the [ ]+OH3 for drinking water which has a pH of 65
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
8
66 Use the change of base theorem to find an approximation for 15log21 Round your answer to four decimal places
67 Given f(x) = 3 x evaluate the following Show all of your work
a f (log 3 7) b f [log 3 (ln 3)] c f [log 3 (2 ln 3)]
68 a Find the decibel rating of a sound having an intensity of 100000 I0
b If the intensity of the sound is doubled by how much is the decibel rating increased
69 a Find the Richter scale rating for an earthquake having an amplitude of 1000000 I0
b Express the magnitude of a 67 earthquake (on the Richter scale) in terms of I0
70 Solve
a 64 13x =minus b x5x 32 =+ c 1116e 34x =minus+ d 50004
06110004t=⎟
⎠
⎞⎜⎝
⎛ +
71 a log (2x ndash 1) + log 10x = log 10 b ln x + ln (x ndash 1) = 2
c ( ) 12log3xlog4xlog 666 =minusminus d log (11x + 9) = 3 + log (x + 3)
72 Elena McDuff wants to buy an $8000 painting She has saved $6000 Find the number of years (to the nearest tenth) it will take for her savings to grow to $8000 at 57 compounded monthly
73 At what interest rate will $2500 grow to $467104 if invested for 10 years and interest is compounded quarterly
Answers
Chapter 7 Applications of Trigonometry and Vectors
1 A = 372deg a = 178 m c = 244 m
2 A = 6560deg b = 1942 cm c = 2727 cm
3 278 km
4 2934 m
5 6594 cm2
6 373 m2
7 a) 0 b) 1 c) 2
8 a) B asymp 26deg30 A asymp 112deg10 b) A1 asymp 722deg C1 asymp 596deg A2 asymp 1078deg C2 asymp 240deg c) no such triangle
9 A asymp 255deg B asymp 1022deg b asymp 739 yd
10 B1 asymp 49deg20 C1 asymp 92deg00 c1 asymp 155 km B2 asymp 130deg40 C2 asymp 10deg40 c2 asymp 288 km
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
9
11 a) A asymp 223deg B asymp 1082deg C asymp 495deg area = 2units 152231 asymp
b) A asymp 567deg C asymp 683deg b asymp 882 area = 2units 3680613547079 asymp
12 a) c asymp 283 in A asymp 449deg B asymp 1068deg area = 2units 57 7325368851 asymp
b) C asymp 107deg20 B asymp 39deg00 A asymp 33deg40 area = 2units 193040 372645000 asymp
13 52 cm 88 cm
14 264deg and 363deg
15 a) b) c)
16 a) 10 0 b) 2 8 c) 6- 4minus
17
18 17v = θ = 3319deg
19 a asymp 137 b asymp 711
20 v = 34 4
21 29 newtons
22 244 lb
23 20i + 2j
24 ndash3
25 7893deg
26 ndash24
27 orthogonal
28 2640 lb at an angle of 1672deg with the 1480-lb force
29 701deg
30 magnitude of second force 190 lb magnitude of the resultant 283 lb
31 125deg
32 305 lbs
33 226 lb
34 135 mi 504deg
35 bearing 65deg30 groundspeed 181 mph
36 groundspeed 161 mph airspeed 156 mph
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
10
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 i 212
38 i32
32x plusmnminus=
39 a) i21 b) ndash3
40 a) i+minus10 b) i162+minus c) i21
21+minus
41 a) i b) ndash i 42 i33minus
43 i23
233+minus
44 ( )oo sin225 icos225 25 +
45 a) 32020 i+ b) 308580 + 185414 i
46 iminus3
47 i21
23minusminus
48 a) i2
3272
27minus b) 128 + 128i
49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+
( )oo 350sin 350 cos23 i+
50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg
b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg
51 a) ( )o495 2minus (2 315deg) ( )2 2 minus
b) ⎟⎠⎞
⎜⎝⎛
3π 3 ⎟
⎠⎞
⎜⎝⎛minus
34π3 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
233
23
52 (1 210deg) (-1 30deg)
53 a) 2sinθ3cosθ
6rminus
= b) r = 3 or r = ndash3
54 a F b A c D d G e H f B g C h E
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
8
66 Use the change of base theorem to find an approximation for 15log21 Round your answer to four decimal places
67 Given f(x) = 3 x evaluate the following Show all of your work
a f (log 3 7) b f [log 3 (ln 3)] c f [log 3 (2 ln 3)]
68 a Find the decibel rating of a sound having an intensity of 100000 I0
b If the intensity of the sound is doubled by how much is the decibel rating increased
69 a Find the Richter scale rating for an earthquake having an amplitude of 1000000 I0
b Express the magnitude of a 67 earthquake (on the Richter scale) in terms of I0
70 Solve
a 64 13x =minus b x5x 32 =+ c 1116e 34x =minus+ d 50004
06110004t=⎟
⎠
⎞⎜⎝
⎛ +
71 a log (2x ndash 1) + log 10x = log 10 b ln x + ln (x ndash 1) = 2
c ( ) 12log3xlog4xlog 666 =minusminus d log (11x + 9) = 3 + log (x + 3)
72 Elena McDuff wants to buy an $8000 painting She has saved $6000 Find the number of years (to the nearest tenth) it will take for her savings to grow to $8000 at 57 compounded monthly
73 At what interest rate will $2500 grow to $467104 if invested for 10 years and interest is compounded quarterly
Answers
Chapter 7 Applications of Trigonometry and Vectors
1 A = 372deg a = 178 m c = 244 m
2 A = 6560deg b = 1942 cm c = 2727 cm
3 278 km
4 2934 m
5 6594 cm2
6 373 m2
7 a) 0 b) 1 c) 2
8 a) B asymp 26deg30 A asymp 112deg10 b) A1 asymp 722deg C1 asymp 596deg A2 asymp 1078deg C2 asymp 240deg c) no such triangle
9 A asymp 255deg B asymp 1022deg b asymp 739 yd
10 B1 asymp 49deg20 C1 asymp 92deg00 c1 asymp 155 km B2 asymp 130deg40 C2 asymp 10deg40 c2 asymp 288 km
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
9
11 a) A asymp 223deg B asymp 1082deg C asymp 495deg area = 2units 152231 asymp
b) A asymp 567deg C asymp 683deg b asymp 882 area = 2units 3680613547079 asymp
12 a) c asymp 283 in A asymp 449deg B asymp 1068deg area = 2units 57 7325368851 asymp
b) C asymp 107deg20 B asymp 39deg00 A asymp 33deg40 area = 2units 193040 372645000 asymp
13 52 cm 88 cm
14 264deg and 363deg
15 a) b) c)
16 a) 10 0 b) 2 8 c) 6- 4minus
17
18 17v = θ = 3319deg
19 a asymp 137 b asymp 711
20 v = 34 4
21 29 newtons
22 244 lb
23 20i + 2j
24 ndash3
25 7893deg
26 ndash24
27 orthogonal
28 2640 lb at an angle of 1672deg with the 1480-lb force
29 701deg
30 magnitude of second force 190 lb magnitude of the resultant 283 lb
31 125deg
32 305 lbs
33 226 lb
34 135 mi 504deg
35 bearing 65deg30 groundspeed 181 mph
36 groundspeed 161 mph airspeed 156 mph
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
10
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 i 212
38 i32
32x plusmnminus=
39 a) i21 b) ndash3
40 a) i+minus10 b) i162+minus c) i21
21+minus
41 a) i b) ndash i 42 i33minus
43 i23
233+minus
44 ( )oo sin225 icos225 25 +
45 a) 32020 i+ b) 308580 + 185414 i
46 iminus3
47 i21
23minusminus
48 a) i2
3272
27minus b) 128 + 128i
49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+
( )oo 350sin 350 cos23 i+
50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg
b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg
51 a) ( )o495 2minus (2 315deg) ( )2 2 minus
b) ⎟⎠⎞
⎜⎝⎛
3π 3 ⎟
⎠⎞
⎜⎝⎛minus
34π3 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
233
23
52 (1 210deg) (-1 30deg)
53 a) 2sinθ3cosθ
6rminus
= b) r = 3 or r = ndash3
54 a F b A c D d G e H f B g C h E
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
9
11 a) A asymp 223deg B asymp 1082deg C asymp 495deg area = 2units 152231 asymp
b) A asymp 567deg C asymp 683deg b asymp 882 area = 2units 3680613547079 asymp
12 a) c asymp 283 in A asymp 449deg B asymp 1068deg area = 2units 57 7325368851 asymp
b) C asymp 107deg20 B asymp 39deg00 A asymp 33deg40 area = 2units 193040 372645000 asymp
13 52 cm 88 cm
14 264deg and 363deg
15 a) b) c)
16 a) 10 0 b) 2 8 c) 6- 4minus
17
18 17v = θ = 3319deg
19 a asymp 137 b asymp 711
20 v = 34 4
21 29 newtons
22 244 lb
23 20i + 2j
24 ndash3
25 7893deg
26 ndash24
27 orthogonal
28 2640 lb at an angle of 1672deg with the 1480-lb force
29 701deg
30 magnitude of second force 190 lb magnitude of the resultant 283 lb
31 125deg
32 305 lbs
33 226 lb
34 135 mi 504deg
35 bearing 65deg30 groundspeed 181 mph
36 groundspeed 161 mph airspeed 156 mph
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
10
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 i 212
38 i32
32x plusmnminus=
39 a) i21 b) ndash3
40 a) i+minus10 b) i162+minus c) i21
21+minus
41 a) i b) ndash i 42 i33minus
43 i23
233+minus
44 ( )oo sin225 icos225 25 +
45 a) 32020 i+ b) 308580 + 185414 i
46 iminus3
47 i21
23minusminus
48 a) i2
3272
27minus b) 128 + 128i
49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+
( )oo 350sin 350 cos23 i+
50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg
b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg
51 a) ( )o495 2minus (2 315deg) ( )2 2 minus
b) ⎟⎠⎞
⎜⎝⎛
3π 3 ⎟
⎠⎞
⎜⎝⎛minus
34π3 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
233
23
52 (1 210deg) (-1 30deg)
53 a) 2sinθ3cosθ
6rminus
= b) r = 3 or r = ndash3
54 a F b A c D d G e H f B g C h E
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
10
Chapter 8 Complex Numbers Polar Equations and Parametric Equations
37 i 212
38 i32
32x plusmnminus=
39 a) i21 b) ndash3
40 a) i+minus10 b) i162+minus c) i21
21+minus
41 a) i b) ndash i 42 i33minus
43 i23
233+minus
44 ( )oo sin225 icos225 25 +
45 a) 32020 i+ b) 308580 + 185414 i
46 iminus3
47 i21
23minusminus
48 a) i2
3272
27minus b) 128 + 128i
49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+
( )oo 350sin 350 cos23 i+
50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg
b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg
51 a) ( )o495 2minus (2 315deg) ( )2 2 minus
b) ⎟⎠⎞
⎜⎝⎛
3π 3 ⎟
⎠⎞
⎜⎝⎛minus
34π3 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
233
23
52 (1 210deg) (-1 30deg)
53 a) 2sinθ3cosθ
6rminus
= b) r = 3 or r = ndash3
54 a F b A c D d G e H f B g C h E
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
11
55a
55b
56a
56b
57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625
59 a $381112 b $525202
60 10
61 a x = 4 b x = 243 c 21x =
x y 2
43
3 21
4 0 5 -1 6 -3
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63
Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien
12
62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +
x y
917minus -6
35minus -5
-1 -4 1 -3 7 -2
63 a zlog 4xlog21ylog 3 bbb minus+ b
3 2
2
ayz
xlog
64 135
65 71032 minustimes
66 ndash39069
67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972
68 a 50 b about 3 decibels
69 a 6 b about 5000000 I0
70 a 7642 b 85476 c ndash5767 d 270246
71 a 1 b 32639 c 45 d Oslash
72 51 years
73 63