Download Ch 5 - Bremen High School District 228

Semester 2 Review
Inverse functions & Restrictions:
(check INNER domain on composites)
  
sin
sin-1  1,1
  2 , 2 
cos
cos-1  1,1
0,  
tan
  
 , 
 2 2
 ,  
tan-1
Simple Harmonic Motion Equations:
At rest:
d  a sin t
At max displacement:
d  a cos t
Max displacement: a
Period:
2

frequency:
Graphs
y = Asin(Bx – C) + D
Amplitude = A
period =
2
B
phase shift =

1
or
2
period
y = Acsc(Bx – C) + D
first graph the associated sine equation
Zeros become asymptotes.
C
B
Key points every
period
4
y = Atan(Bx – C)


Asymptotes:   Bx  C 
2
2
Zeros: midway between asymptotes
points:
(midway between asy. and zero,  A )
y = Acot(Bx – C)
(midway between zero and asy., A )
(midway between zero and asy., - A )
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Asymptotes: 0  Bx  C  
Zeros: midway between asymptotes
points:
(midway between asy. and zero, A )
Page 1 of 26
Semester 2______
Review
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
s
r
s  r
Revolution x 360 = Degrees
opposite
hypotenuse
adjacent
cos  
hypotenuse
opposite
tan  
adjacent
sin  
hypotenuse
opposite
hypotenuse
sec =
adjacent
adjacent
cot =
opposite
1
csc 
1
cos  
sec 
1
tan  
cot 
1
sin 
1
sec =
cos 
1
cot =
tan 
csc =
Quotient Identities (2)
sin 
tan  
cos 
______
______
______
______
______
Deg 

180 
cos 
cot  
sin 
  r
Revolution x 2π = radians
csc =
Reciprocal Identities (6)
sin  
______
______
______
0
30
45
sin Θ
cos Θ
tan Θ
Cofunction Identities (6)
sin   cos  90    
cos   sin  90    
sec  csc  90    
csc  sec  90    
tan   cot  90    
cot  tan  90    
for radians, replace 90with  / 2.
Pythagorean Identities (3)
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   cot 2 
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60
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90
Semester 2 Review
Ch 5
5.1 Find the radian measure of Θ.
1. A central angle, Θ, in a circle of radius 12 in intercepts an arc of length 42 in.
2. A central angle, Θ, in a circle of radius 6 cm intercepts an arc of length 15 cm.
3. A central angle, Θ, in a circle of radius 10 km intercepts an arc of length 45 km.
Convert each angle in degrees to radians:
4.
60
5.
270
Convert each angle in radians to degrees:

4

7.
8.
4
3
Draw and label each angle in standard position:

3
 

10.
11.
12.
4
4
6.
-300
9.
6
 
7
4
Find a positive angle less than 360 that is coterminal with each of the following:
13.
400
14.
-135
15.
750
Find a positive angle less than 2π that is coterminal with each of the following:
17
13
17

16.
17.
18.
6
5
3
Find the length of the arc intercepted by the central angle in terms of π. Then round to
two decimal places.
19.
A circle has a radius of 6 inches. A central angle is 45.
20.
A circle has a radius of 10 inches. A central angle is 120.
21.
A circle has a radius of 8 feet. A central angle is 270.
22. A 45-rpm record has an angular speed of 45 revolutions per minute. Find the linear
speed, in inches per minute, at the point where the needle is 1.5 inches from the record’s
center.
23. A wind machine generates electricity using blades that are 10 feet long. These
propellers rotate at 4 revolutions per minute. Find the linear speed, in feet per second,
of the tips of the blades.
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Semester 2 Review
5.2 Find the value of each of the 6 trigonometric functions of Θ in the figure.
1.
2.
c=5
Θ
a=1
Θ
b=4
a=3
Find each of the following. If a radical appears in the denominator, rationalize.
3.
tan 60
4.
sin 30
5.
cos 45
6.
csc 45
7.
cot 45
8.
sec 60
9. Given sin  
5
2
and cos  
, find the value of the four remaining trig functions.
3
3
10. Given sin  
21
2
and cos  
, find the value of the four remaining functions.
5
5
1
and Θ is an acute angle, find the value of cosΘ using a
2
trigonometric identity.
11. Given that sin  
3
and Θ is an acute angle, find the value of cosΘ using a
5
trigonometric identity.
12. Given that sin  
Find a cofunction with the same value as the given expression:


cot
csc
13.
sin 46
14.
15.
16.
12
3
Use a calculator to find the value to four decimal places.
17.
sin 72.8
18.
csc 1.5
19.
cos 48.2
20.
Find the value of the variable.
21.
a
24
a 750 yd
22.
c
10
a
sin 72
cot 1.2
500 ft
23.
A flagpole 14 meters tall has a shadow 10 meters long. Find the angle of elevation
of the sun to the nearest degree.
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Semester 2 Review
5.3
1. Let P(1, -3) be a point on the terminal side of Θ. Find each of the six trig functions of
Θ.
2. Let P(-3, -5) be a point on the terminal side of Θ. Find each of the six trig functions
of Θ.
Evaluate, if possible, the cosine and cosecant functions at the four following quadrantal
angles:

3
3.   0  0
4.   90 
5.   180  
6.   270 
2
2
7.
8.
If sin   0 and cos  0 , name the quadrant in which Θ lies.
If tan   0 and cos  0 , name the quadrant in which Θ lies.
9.
Given tan   
10.
1
and cos  0 , find sinΘ and secΘ.
3
2
Given tan    and cos  0 , find cosΘ and cscΘ.
3
Find the reference angle, Θ’, for each of the following angles:
7
11. Θ = 210
12.  
13. Θ = -240
14. Θ = 3.6
4
15. Θ = 665
16.  
15
4
17.   
11
3
18. Θ = 580
Use reference angles to find the exact value of the following trig functions:
5
17
 
19. sin 300
20. tan
21. sec   
22. cos
4
6
 6
 22 
23. sin  

 3 
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24. tan
14
3
 17 
25. sec  

 4 
26. sin 135
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Semester 2 Review
5.4 Find the values of the trig functions at t on the unit circle.
1.  1 3 
2.
 0,1
  ,

 2 2 
•
•
t
t
Find the exact value of each trig function:
 
tan   
3.
4.
cos  60
 3
5.
 
tan   
 6
6.
cos  45
7.
cos 420
8.
sin
9
4
9.
cos 405
10.
sin
7
3
11.
csc
7
6
12.
 5 
sin  

 6 
13.
tan17
14,
sin
47
4
5.5 Determine the amplitude, period, and phase shift, then graph.
1
y  sin x
y  3sin x
y  2sin x
1.
2.
3.
2
4.
y  3sin 2 x
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5.
y  2 sin
1
x
2
6.
y  3cos

2
x
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7.
y  4 cos  x
10.
y
1
cos  4 x   
2
Semester 2 Review
2 

8.
y  4sin  2 x 

3 

11.
y
3
cos  2 x   
2
9.


y  3sin  2 x  
3

12.
y
1
cos x  1
2
Write an equation of the form y  A sin( Bx  C ) .
13.
14.
Window
Xmin=-π/4
Xmax=π/2
Xscl=π/8
Ymin=-4
Ymax=4
Yscl=1
15. The depth of water at a boat dock varies with the tides. The depth is 5 feet at low
tide and 13 feet at high tide. On a certain day, low tide occurs at 4a.m. and high tide at
10 a.m. If y represents the depth of water, in feet, x hours after midnight, use a sine
function of the form y  A sin( Bx  C )  D to model the water’s depth.
16.
A region that is 30 north of the Equator averages a minimum of 10 hours of
daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x
represent the month of the year, with 1 for January, and 12 for December. If y
represents the number of hours of daylight in month x, use a sine function of the form
y  A sin( Bx  C )  D to model the hours of daylight.
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Page 7 of 26
Semester 2 Review
5.6 Graph.
17. y  2 tan
x
for -   x  3
2
18.
y  3 tan 2 x for 

4
x
3
4
19.


y  tan  x  
4

20.


y  tan  x  
2

21.
y  3cot 2 x
22.
y
1

cot x
2
2
23.
y  2 csc 2 x
24.


y  csc  x  
4

25.
y  3sec

2
x for –π<x<5π
5.7 Find the exact values.
2
sin 1
27.
28.
2
30.

2
sin 1  

 2 
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31.
26.
sin 1
3
2
 3
cos 1 

 2 
y  2sec 2 x
for 
3
3
x
4
4
29.
 1
sin 1   
 2
32.
 1
cos 1   
 2
Page 8 of 26
33.
tan
36.
 3 
sin 1  sin

2 

39.
cos cos 1  1.2 
42.
1
Semester 2 Review
34.
tan 1  1
3
cos cos 1 0.6
37.
cos cos 1 1.5

38.
sin 1  sin  

40.
5

cos  tan 1 
12 

41.
3

sin  tan 1 
4


 1 
cot  sin 1    
 3 

43.

 1 
cos  sin 1    
 2 




35.
Write as an algebraic expression in x.
44.
If 0  x  1 , cos  sin 1 x 
45.
Find the value to four decimal places.
1
sin 1
46.
47.
tan 1  9.65
4

If x>0, sec  tan 1 x 
48.
cos 1
1
3
5.8 Solve the triangle. Round to two decimal places.
49.
A = 34.5 , and b = 10.5
50.
A = 62.7 , and a = 8.4.
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Semester 2 Review
51.
From a point on level ground 125 feet from the base of a tower, the angle of
elevation is 57.2 . Approximate the height of the tower to the nearest foot.
52.
From a point on level ground 80 feet from the base of the Eiffel Tower, the angle
of elevation is 85.4 . Approximate the height of the Eiffel Tower to the nearest foot.
53.
A kite flies at a height of 30 feet when 65 feet of string is out. If the string is in
a straight line, find the angle it makes with the ground to the nearest tenth of a degree.
54.
A guy wire is 13.8 yards long and is attached from the ground to a pole 6.7 yards
above the ground. Find the angle, to the nearest tenth of a degree, that the wire makes
with the ground.
55.
You are taking your first hot-air balloon ride. Your friend is standing on the
ground 100 feet away from your point of launch. At one instant, the angle of elevation to
you is 31.7. One minute later, the angle of elevation is76.2. How far did you travel, to
the nearest tenth of a foot, during that minute?
56.
You are standing on level ground 800 feet from Mt. Rushmore, looking at the
sculpture of Abraham Lincoln’s face. The angle of elevation to the bottom of the
sculpture is 32 and the angle of elevation to the top is 35. Find the height of the
sculpture of Lincoln’s face to the nearest tenth of a foot.
N
57.
a.
b.
c.
d.
Find each of the following:
The bearing from O to B.
The bearing from O to A.
The bearing from O to D.
The bearing from O to C.
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B
A
40
W
75
C
20
O
25
S
E
D
Page 10 of 26
Semester 2 Review
58.
A boat leaves the entrance to a harbor and travels 25 miles on a bearing of N 42
E. The captain then turns the boat 90 clockwise and travels 18 miles on a bearing of S
48 E. At this time:
a.
How far is the boat, to the nearest tenth of a mile, from the harbor?
b.
What is the bearing, to the nearest tenth of a degree, of the boat from the
harbor?
59.
You leave the entrance to a system of hiking trails and hike 2.3 miles on a bearing
of S31 W. Then the trail turns 90 clockwise and you hike 3.5 miles on a bearing of N
59 W. At that time:
a.
How far are you, to the nearest tenth of a mile, from the entrance of the trail
system?
b.
What is your bearing, to the nearest tenth of a degree, from the entrance of the
trail system?
60.
A ball on a spring is pulled 4 inches below its rest position and then released. The
period of motion is 6 seconds. Write the equation for the ball’s simple harmonic motion.
61.
A ball on a spring is pulled 6 inches below its rest position and then released. The
period of motion is 4 seconds. Write the equation for the ball’s simple harmonic motion.
62.
If a mass moves in simple harmonic motion described by d  10 cos

6
t , with t
measured in seconds and d in centimeters. Find each of the following:
a.
the maximum displacement
b.
the frequency
c.
the time required for one cycle.
63.
A mass moves in simple harmonic motion described by d  12 cos

4
measured in seconds and d in centimeters. Find each of the following:
a.
the maximum displacement
b.
the frequency
c.
the time required for one cycle.
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t , with t
Page 11 of 26
6.1 Verify the identity.
sec x cot x  csc x
1.
Semester 2 Review
2.
csc x tan x  sec x
3.
sin x tan x  cos x  sec x
4.
cos x cot x  sin x  csc x
5.
cos x  cos x sin 2 x  cos3 x
6.
sin x  sin x cos 2 x  sin 3 x
7.
1  sin x
 sec x  tan x
cos x
8.
1  cos x
 csc x  cot x
sin x
9.
cos x
1  sin x

 2sec x
1  sin x
cos x
10.
sin x
1  cos x

 2 csc x
1  cos x
sin x
11.
sin x
1  cos x

1  cos x
sin x
12.
cos x
1  sin x

1  sin x
cos x
13.
15.
tan x  sin   x 
1  cos x
 tan x
1
1

 2  2 cot 2 x (Both)
1  cos x 1  cos x
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14.
16.
sec x  csc   x 
sec x csc x
 sin x  cos x
1
1

 2  2 tan 2 x
1  sin x 1  sin x
Page 12 of 26
Semester 2 Review
6.2 Find the exact values.
17.
cos 15 (Use 60-45=15)
18.
cos 30 (Use 90-60=30)
19.
cos80 cos20 + sin80 sin20
20.
cos70 cos40 +sin70 sin40
21.
sin
22.
sin
7
,
12
if
7  
 
12 3 4
Verify the identity.
cos    
23.
 cot   tan 
sin  cos 
25.
  tan x  1

tan  x   
4  tan x  1

24.
26.
5
,
12
if
cos    
cos  cos 
5  
 
12 6 4
 1  tan  tan 
tan  x     tan x
12
3
for α in quadrant II and sin   for β in quadrant I.
13
5
cosβ
27.
cos(α+β)
28.
sin(α+β)
Find the exact values if sin  
25.
cosα
26.
4
1
for α in quadrant II and sin   for β in quadrant I.
5
2
cosβ
31.
cos(α+β)
32.
sin(α+β)
Find the exact values if sin  
29.
cosα
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30.
Page 13 of 26
Semester 2 Review
5
and x lies in quadrant II, find the exact values:
13
sin 2x
34.
cos 2x
35.
tan 2x
6.3 If sin x 
33.
4
and x lies in quadrant II, find the exact values:
5
sin 2x
37.
cos 2x
38.
tan 2x
If sin x 
36.
Find the exact values:
2 tan15
39.
1  tan 2 15
40.
cos 2 15  sin 2 15
42.
tan x 
sin 2 x
1  cos 2 x
6.5 Solve the equation.
1.
3sin x  2  5sin x 1
2.
5sin x  3sin x  3
3.
tan 3x  1,
4.
tan 2 x  3,
5.
sin
6.
sin
Verify the identity.
1  cos 2 x
tan x 
41.
sin 2 x
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x
3

,
2
2
0  x  2
0  x  2
x 1
 ,
3 2
0  x  2
0  x  2
Page 14 of 26
Semester 2 Review
7.
2cos2 x  cos x  1  0, 0  x  2
8.
2sin 2 x  3sin x  1  0, 0  x  2
9.
4sin 2 x  1  0, 0  x  2
10.
4cos2 x  3  0,
11.
tan x sin 2 x  3tan x,
0  x  2
12.
sin x tan x  sin x,
13.
2cos2 x  3sin x  0,
0  x  2
14.
2sin 2 x  3cos x  0,
15.
cos 2 x  3sin x  2  0,
16.
cos 2 x  sin x  0,
0  x  2
17.
1
sin x cos x  ,
2
0  x  2
18.
1
sin x cos x   ,
2
0  x  2
19.
sin x  cos x  1,
0  x  2
20.
cos x  sin x  1,
0  x  2
0  x  2
0  x  2
0  x  2
0  x  2
Solve each equation, correct to four decimal places, for 0 ≤ x < 2π.
21.
tan x = 12.8044
22.
cos x = -0.4317
23.
tan x = 3.1044
24.
sin x = -0.2315
25.
sin 2 x  sin x  1  0
26.
cos 2 x  5cos x  3  0
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Semester 2 Review
7.1 Solve triangle ABC. Round to the nearest tenth.
27.
A = 46 , C = 63 , and c = 56 in
28.
A = 64 , C = 82 , and c = 14 cm
29.
A = 50 , C = 33.5 , and b = 76
30.
A = 40 , C = 22.5 , and b = 12
31.
A = 43 , a = 81, and b = 62
32.
A = 57 , a = 33, and b = 26
33.
A = 75 , a = 51, and b = 71
34.
A = 50 , a = 10, and b = 20
35.
A = 40 , a = 54, and b = 62
36.
A = 35 , a = 12, and b = 16
Find the area of a triangle rounded to the nearest square meter.
37.
side lengths 24m and 10m and an included angle of 62.
38.
side lengths 8m and 12m and an included angle of 135.
39.
Two fire-lookout stations are 20 miles apart, with station B directly east of station
A. Both stations spot a fire on a mountain to the north. The bearing from station A to
the fire is N50E. The bearing from station B to the fire is N36W. How far, to the
nearest tenth of a mile, is the fire from station A?
40.
Two fire-lookout stations are 13 miles apart, with station B directly east of station
A. Both stations spot a fire. The bearing from station A is N35E and the bearing of the
fire from station B to the fire is N49W. How far, to the nearest tenth of a mile, is the
fire from station B?
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Semester 2 Review
7.2
Solve the triangle. Round lengths to the nearest tenth and angles to the nearest degree.
41.
A = 60, b = 20, and c = 30.
42.
A = 120, b = 7, and c = 8
43.
a = 6, b = 9, and c = 4
44.
a =8, b = 10, and c = 5
45.
Two airplanes leave an airport at the same time on different runways. One flies on
a bearing of N66W at 325 miles per hour. The other plane flies on a bearing of S26W
at 300 miles per hour. How far apart will the planes be after 2 hours?
46.
Two airplanes leave an airport at the same time on different runways. One flies
directly north at 400 miles per hour. The other plane flies on a bearing of N75E at 350
miles per hour. How far apart will the planes be after 2 hours?
Find the area of the triangle. Round to the nearest whole unit.
47.
a= 12 yd, b = 16 yd, c = 24 yd
48.
a = 6m, b = 16m, and c = 18m
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Semester 2 Review
4.1
The exponential function f(x) = 13.49(0.967)x - 1 describes the number of O-rings
expected to fail, f(x), when the temperature is xF. Find the number of O-rings
expected to fail at the given temperature.
1.
31F
2.
60F
3.
-10F
Graph.
4.
f  x  2
7.
1
f  x   
3
x
x
5.
g  x  3
6.
1
h  x   
2
8.
g  x   3x1
9.
h  x   2x  3
x
x
10.
The function f(x) = 3.6e0.02x describes world population, f(x), in billions, x years
after 1969. Find the world population in 2020.
11.
The function f(x) = 6.4e0.0123x describes world population, f(x), in billions, x years
after 2004. Find the world population in 2050.
12.
You decide to invest $8000 for 6 years. How much will have if you invest at 7%
per year, compounded monthly? at 6.85% per year, compounded continuously?
13.
You decide to invest $10000 for 5 years at an annual rate of 8%. How much will
have if it is compounded quarterly? compounded continuously?
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Semester 2 Review
4.2
Write each equation in its equivalent exponential form:
14.
2 = log5x
15.
3 = logb64
16.
log37 = y
Write each equation in its equivalent logarithmic form:
17.
122 = x
18.
b3 = 8
19.
ey = 9
Evaluate.
20.
log216
21.
log39
22.
log255
23.
log77
24.
log51
25.
log778
26.
6log69
27.
log445
28.
log81
33.
f(x) = ln (3 - x)
Graph.
29.
f(x) = log2x
Find the domain.
31.
f(x) = log4 (x + 3)
34.
30.
32.
g(x) = log3x
g(x) = log4 (x - 5)
g(x) = ln (x - 3)2
35.
The percentage of adult height attained by a boy who is x years old can be modeled
by f(x) = 29 + 48.8 log (x + 1), where x represents the boy’s age and f(x) represents the
percentage of his adult height. An 8 year old boy has attained approximately what
percentage of his adult height?
36.
The function f(x) = 13.4 ln x - 11.6 models the temperature increase, f(x), in
degrees Fahrenheit, after x minutes in an enclosed vehicle. Find the temperature
increase after 50 minutes.
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Page 19 of 26
4.3
Expand each expression.
37.
log4(7• 5)
38.
Semester 2 Review
log (10x)
39.
log574
42.
ln x
45.
 3x 
log 6 
4 

 36 y 
40.
 19 
log 7  
 x 
41.
 e5 
ln  
 11 
43.
log (4x)5
44.
log b x 2 y


Write as a single logarithm.
1
log x  4 log  x  1
2
46.
log42 + log432
47.
log (4x - 3) - logx 48.
49.
3 ln (x + 7) - ln x
50.
1
4 log b x  2 log b 6  log b y
2
Evaluate.
51.
log5140
52.
log72506
4.4 Solve.
53.
23x-8 = 16
54.
27x+3 = 9x-1
55.
4x = 15
56.
40e0.6x - 3 = 237
57.
5x-2 = 42x+3
58.
e2x - 4ex + 3 = 0
59.
log4 (x + 3) = 2
60.
3 ln (2x) = 12
61.
log2 x + log2 (x - 7) = 3
62.
1
ln  x  2   ln  4 x  3  ln  
x
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Semester 2 Review
63.
The risk of having a car accident while under the influence of alcohol can be
modeled by R = 6e12.77x, where x is the blood alcohol concentration and R, given as a
percent, is the risk of a car accident. What blood alcohol level corresponds to a 20% risk
of a car accident?
64.
How long will it take $25,000 to grow to $500,000 at 9% annual interest
compounded monthly?
 r
A  P 1  
 n
nt
65.
The function f(x) = 34.1 ln x + 117.7 models the number of U.S. Internet users,
f(x), in millions, x years after 1999. By what year will there be 200 million Internet
users in the U.S.?
4.5
66.
In 1970, the U.S. population was 203.3 million. By 2003, it had grown to 294
million. Find the exponential growth function that models this data (A = A0ekt). Then find
the year when the population will reach 315 million.
67.
Use the fact that after 5715 years a given amount of carbon-14 will have decayed
to half the original amount to find the exponential decay model for carbon-14. Then
estimate the age of scrolls found in 1947 that contained 76% of their original carbon-14.
30000
describes the number of people, f(t), who have
1  20e 1.5t
become ill with influenza t weeks after its initial outbreak in a town with 30,000
inhabitants. How many people became ill when the epidemic began? How many people
were ill by the end of the fourth week? What is the limiting size of f(t), the population
that becomes ill?
68.
The function f (t ) 
69.
Rewrite y = 2.557 (1.017)x in terms of base e.
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Semester 2 Review
Ch 8
1.
Bottled water and medical supplies are to be shipped to victims of an earthquake
by plane. Each container of bottled water will serve 10 people and each medical kit will
aid 6 people. Let x represent the number of bottles of water to be shipped and y
represent the number of medical kits.
a. Write the objective function that describes the number of people who can be
helped.
b. Each plane can carry no more than 80,000 pounds. Bottled water weighs 20
pounds per container and each medical kit weighs 10 pounds. Write an inequality
that describes the constraint.
c. Each plane can carry a total volume of supplies that does not exceed 6000 cubic
feet. Each water bottle is 1 cubic foot and each medical kit also has a volume of 1
cubic foot. Write an inequality describing this second constraint.
d. Determine how many bottles of water and how many medical kits should be
sent on each plane to maximize the number of earthquake victims who can be
helped.
2.
A company manufactures bookshelves and desks for computers. Let x represent
the number of bookshelves manufactured daily and y the number of desks manufactured
daily. The company’s profits are $25 per bookshelf and $55 per desk.
a. Write the objective function that describes the company’s total profit, z, from
x bookshelves and y desks.
b. To maintain high quality, the company should not manufacture more that a total
of 80 bookshelves and desks per day. Write an inequality that describes this
constraint.
c. To meet customer demand, the company must manufacture between 30 and
80 bookshelves per day, inclusive. Furthermore, the company must manufacture
at least 10 and no more than 30 desks per day. Write an inequality that describes
each of these sentences.
d. How many bookshelves and how many desks should be manufactured per day to
obtain maximum profit? What is the maximum daily profit?
3.
Find the maximum value of the objective function z = 3x + 5y subject to the
constraints x ≥ 0, y ≥ 0, x + y ≥ 1, x + y ≤ 6.
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Semester 2 Review
Ch 9
Write the solution set for a system of equations represented by the matrix.
8 
1 2 5 19
1 1 1




4.
5.
9 
0 1 3
0 1 12 15
0 0 1
0 0
4 
1
1 
Use matrices to solve the system.
Gauss (REF)
3 x  y  2 z  31
x  y  2 z  19
6.
7.
x  3 y  2 z  25
2w  x  3 y  z  6
w  x  2 y  2 z  1
w  x  y  z  4
8.
9.
 w  2 x  2 y  z  7
10.
x  3 y  2 z  25
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x  2y  z  6
w  3 x  2 y  z  3
2w  7 x  y  2 z  1
3w  7 x  3 y  3z  5
5w  x  4 y  2 z  18
Gauss-Jordan (RREF)
3 x  y  2 z  31
x  y  2 z  19
2 x  y  2 z  18
x  y  2z  9
w x y  z  4
11.
2w  x  2 y  z  0
w  2 x  y  2 z  2
3w  2 x  y  3z  4
Page 23 of 26
Semester 2 Review
11.1
Write the first four terms of the sequence whose general term is given.
1.
3.
an = 3n + 4
an = 3an-1 + 2, for a1 = 5 and n≥ 2
Evaluate.
10!
5.
2!8!
8.
6.
7
  2  k  5 



k 4
 n  1!
n!
 1

n
2.
an
4.
2n
an 
 n  1!
7.
 i
6
i 1
9.
3n  1
2

1
5
3
i1
Express in summation notation.
10. 13 + 23 + 33 + … +73
1 1 1

11. 1   
3 9 27

1
3n1
11.2
12.
Find the first five terms of the arithmetic sequence in which a1 = 77.4 and
an = an-1 - 0.67.
13.
Find the eighth term of the arithmetic sequence whose first term is 4 and whose
common difference is -7.
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Semester 2 Review
14.
According to the U.S. Census Bureau, new one-family houses sold for an average of
$159,000 in 1995. This average sales price has increased by approximately $9700 per
year. Write a formula for the nth term of the arithmetic sequence that describes the
average cost for new one-family houses n years after 1994. How much will new onefamily houses cost, on average, by the year 2010?
15.
Find the sum of the first 100 terms of the arithmetic sequence: 1, 3, 5, 7, …
16.
Find the following sum:
25
  5i  9 
i 1
17.
Your grandmother has assets of $400,000. One option that she is considering
involves an adult residential community for a six-year period beginning in 2006. The
model an = 1800n + 58,730
describes yearly adult residential community costs n
years after 2005. Does your grandmother have enough to pay for the facility?
11.3
18.
Write the first six terms of the geometric sequence with first term 6 and common
ratio 1/3.
19.
Find the eighth term of the geometric sequence whose first term is -4 and whose
common ratio is -2.
20.
The population in Florida from 1990 to 1997 is shown in the following table:
Year
1990 1991 1992 1993 1994 1995 1996 1997
Population in millions 12.94 13.20 13.46 13.73 14.00 14.28 14.57 14.86
a.
b.
c.
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Show that the population is increasing geometrically.
Write the general term for the sequence describing population growth for
Florida n years after 1989.
Estimate Florida’s population, in millions, for the year 2000.
Page 25 of 26
21.
Semester 2 Review
Find the sum of the first 18 terms of the geometric sequence:
2, -8, 32, -128, …
22.
Find the following sum:
10
6 2
i
i1
23.
A union contract specifies that each worker will receive a 5% pay increase each
year for the next 30 years. One worker is paid $20,000 the first year. What is this
person’s total lifetime salary over a 30-year period?
24.
To save for retirement, you decide to deposit $1000 into an IRA at the end of
each year for the next 30 years. If the interest rate is 10% per year compounded
annually, find the value of the IRA after 30 years.
25.
Find the sum of the infinite geometric series:
26.
Express 0.7 as a fraction in lowest terms.
3 3 3
3
   
8 16 32 64
.
27.
Suppose the government reduces taxes so that each consumer has $2000 more
income. The government assumes that each person will spend 70% of this ($1400). The
individuals and businesses receiving this $1400 in turn spend 70% of it (980), creating
extra income for other people to spend, and so on. Determine the total amount spent on
consumer goods from the initial $2000 tax rebate.
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