Download Math 114 - Exam 3 Review 1. For the following figure, a, b and c are

Math 114 - Exam 3 Review
1. For the following figure, a, b and c are the lengths of the legs of the triangle, u and v
are the measure of the angles.
SS
vS
S
S
b
S
Sc
S
S
S
a
uSS
(a) For a = 8, b = 15, find sec v, sec u. Also find csc u, csc v csc u = sec v =
sec u = 17
8
17
, csc v
15
=
(b) For a = 5, c = 13, find csc u, csc v. csc u = 1312, csc v = 135
√
√
(c) For a = 8, b = 16 2, find csc u, csc v csc u = 3 4 2 , csc v = 8
(d) For a = 21, b = 7, find cos u, cos v. cos u =
√
2 2
, cos v
3
=
1
3
(e) For a = 20, c = 25, find cot u, cot v. Also find tan u, tan v. tan v = cot u =
4
, tan u = cot v = 34 .
3
√
√
(f) For a = 2 5, b = 4, find cot u, cot v. cot u = 25 , cot v = √25
(g) For a = 12, b = 24, find tan u, cot u. cot u = 12 , tan u = 2
√11 , cot u =
23
√
13
6
,
cos
u
=
7
7
(h) For b = 22, c = 24, find tan u, cot u. tan u =
(i) For b = 6, c = 7, find cos u, sin u. sin u =
√
23
11
2. For the same triangle in Figure 1.
(a) Suppose sin u = 53 . What are possible values for a, b, and c? What is another set
of possible values? a = 4, b = 3, c = 5. Any multiple of those 3 numbers will
work. For example, a = 8, b = 6, c = 10.
(b) Suppose tan v = 76 . Find two sets of possible values for a, b, and c. a = 6, b =
√
7, c = 85.
√ Any multiple of those 3 numbers will work. For example, a = −6, b =
−7, c = − 85.
1
(c) Suppose cos u = 21 . Find two sets of possible values for a, b and c.
Moreover, find
√
3
1
u and v. Express u and v in radians and degrees. a = 2 , b = 2 , c = 1. Again,
any multiple will work. u = π3 radians = 60 degrees, v = π6 radians = 30 degrees.
3. (a) Suppose tan θ = 4, find cos θ, sin θ. cos θ = √117 , sin θ =
q
√
3
(b) Suppose sin θ = 43 . Find tan θ. tan θ = 13
(c) Suppose tan θ =
(d) Suppose csc θ =
√11 .
23
√
3 2
.
4
Find sin θ, cos θ. sin θ =
√
Find tan θ. tan θ = 2 2
(e) Suppose csc θ = 1312. Find cos θ. cos θ =
4. Solve for x
ln(x + 1) − ln(x − 1) = 3.
x=
−1 − e3
1 − e3
5. Solve for x
ln(x + 3) − ln(x + 2) = 5.
x=
−3 + 2e5
1 − e5
6. Solve for x
ex−1 − 5 = 5
x = ln(10) + 1
7. Solve for x
ex+2 ex−2 = 6.
x = 21 ln(6)
8. Solve for x
819−2x−2 = 27.
x=
−3
4
9. Solve for x
ln(x + 9) = 1 − ln x.
x=
√
−9± 81−4e
2
2
5
.
13
11
, cos θ
12
√4 .
17
=
√
23
.
12
10. For x in [0, π2 ], solve for x
e3 sec
x=
2
x−1
= 1.
π
6
11. For x in [0, π2 ], solve for x
ln(tan x) = ln(4 tan x − 9).
x=
π
3
12. For x in [0, π2 ], solve for x
−4 log3 (9 tan2 x) = −4.
x=
π
6
13. For x in [0, π], solve for x
−8 + log9 (tan x + 1) = −8.
x = 0, π
14. Find the points of intersection for the line y = 2x + 3 with the circle of radius 2,
3 18
centered at (1, 3). ( 10
, 5 ), ( −13
,2
10 5
15. Find the points of intersection
for the √
line y =√ 1/2x + 1 with the circle of radius 3,
√
√
6+4 11 8+2 11
6−4 11 8−2 11
centered at (1, 2). ( 5 , 5 ), ( 5 , 5 ).
√
16. Find √the points
of
intersection
for
the
line
y
=
−
3x + 1 with the unit circle.
√
3
3
1
1
(− 2 , 2 ), ( 2 , − 2 )
17. Find the midpoint between (8, 2) and (9, 8). ( 17
, 5)
2
18. Find the midpoint between (6, 1) and (−2, 5) (2, 3).
19. Find the coordinates of a point on the circle of
radius
√
3
1
π
angle to the positive horizontal axis is 3 . ( 4 , 4 ).
1
2
centered at the origin, whose
20. Find the coordinates of a point on the circle
√ of radius 2 centered at the origin, whose
π
angle to the positive horizontal axis is 6 . ( 3, 1).
21. Find the shortest distance between the line y = x + 2 and the point (0,0).√Find the
coordinates of the point which give the shortest distance. shortest distance: 2. point:
(-1,1)
3
√
22. Find the shortest distance between the line y = 2x + 3 and the point (0,0).√ Find the
coordinates
of the point which give the shortest distance. shortest distance: 3, point :
p
(− (2), 1)
and the unit circle. Give
23. Find the shortest distance between the line y = 34 x + 25
4
the coordinates of the point on the line which gives the shortest distance. shortest
distance: 4, point: (−3, 4)
24. Find the shortest distance between the line 12y + 5x = 169 and the circle of radius 3
centered at the origin. Give the coordinates of the point on the line which give the
shortest distance. shortest distance: 10, point:(5, 12)
25. Find the shortest distance between the line y = 2x + 4 and the unit circle. Give the
coordinates of the point on the line which give the shortest distance. shortest distance:
√3 − 1, point: ( −6 , 8
5 5
5
26. Sketch the graphs of the following equations.
(a) 3y + 2x = 3 Line with slope
−2
,
3
and y-intercept= 1
(b) 4x2 + y 2 − 48x + 140 = 0 Ellipse with center (6, 0), stretched vertically by 2.
(c) x2 + y 2 + 6x − 2y + 9 = 0 Circle with center (-3,1), and radius 1
(d) x2 + 2x + y − 1 Parabola with vertex (-1,2), pointed downward.
(e) 25x2 + y 2 + 50x = 26. Ellipse with center (1,0), stretched vertically by 5
27. On a blank sheet of paper, draw the unit circle with coordinates for points that make
an angle of 0, π6 , π2 , π3 , π2 , 5π
, 3π
, 2π
, π, etc.
6
4
3
28. (a) Find cos(sin−1 ( 23 )).
√
5
3
(b) Find sin(cos−1 (1)) 0
√1
10
12
)) 17
sec(cos−1 ( 17
12
√
−1 12
145
tan(sin ( 17 )) 12
tan(sin−1 ( 76 )) √613
7
cos(tan−1 ( 24
)) 24
25
cot(cos−1 ( √15 )) 12
tan(cos−1 ( 11
)) √1123
12
√
cot(cos−1 ( 23 )) 25
(c) Find cos(tan−1 (3))
(d) Find
(e) Find
(f) Find
(g) Find
(h) Find
(i) Find
(j) Find
29. In a given year, the minimum wage was only $1.60 per hour. Use the exponential
growth formula to predict when that minimum wage in the United States will reach
1
$8.50 per hour if the rate of growth in the minimum wage is 3.9%. t = 3.9
ln( 85
)
16
4
30. A biologist is researching a newly-discovered species of bacteria. At time t = 0 hours,
he puts one hundred bacteria into what he has determined to be a favorable growth
medium. Six hours later, he measures 450 bacteria. Assuming exponential growth,
what is the growth constant “k ′′ , i.e. the constant which determines the rate of growth
for the bacteria? ln(4.5)
6
31. Growth of bacteria in food products causes a need to time-date some products (like
milk) so that shoppers will buy the product and consume it before the number of
bacteria grows too large and the product goes bad. Bacteria in a controlled environment
grows exponentially.
(a) An initial population of 500 bacteria will grow to 552 bacteria at the end of one
day. Find the rate of growth constant at which the bacteria is growing. k = ln 552
500
(b) If the product cannot be eaten after the bacteria count reaches 4,000,000, how
long will it take? t = ln(8000)
ln 552
500
32. A certain type of bacteria, given a favorable growth medium, doubles in population
every 6.5 hours. Given that there were approximately 100 bacteria to start with, how
ln 2
many bacteria will there be in a day and a half? A = 100e36( 6.5 )
33. Carbon-14 dating
(a) Carbon-14 is a radioactive material used in dating artifacts. If Carbon-14 has a
half life of 5730 years. What is the decay constant, i.e. the constant for the rate
of decay?
Remark. I was recently at the Fields museum in Chicago in which some of the
exhibits mentions using Carbon 14 to determine whether an object is authentic.
Here’s how it works. Carbon has a stable form, Carbon 12 and Carbon 13 and a
non-stable form Carbon 14, which is subject to decay. The ratio of stable carbon
to nonstable carbon in the atmosphere is constant. That is, knowing how much
stable carbon there is tells us how much nonstable carbon there should be. Living
things exchange carbon with the atmosphere in the form of carbon dioxide. This
exchange of carbon with the atmosphere stops when the living thing dies and
carbon 14 begins to break down. Thus, with a fossil, one can measure the amount
of stable carbon present to determine how much carbon 14 is in the fossil at the
time of the animal’s death. Then we can determine how many years ago it was.
(b) Suppose you found some bones in your backyard. Using fancy equipment, you
determine that the amount of Carbon-14 that should’ve been in the animal at
the time it died is 1.3 × 1012 grams. According to your equipment, there is
currently only 1.0 × 10−12 grams of Carbon-14. How long ago did this animal die?
5730 ln( 1 )
t = ln(0.5)1.3
5
34. A person 100 meters from the base of a tree, observes that the angle between the ground
and the top of the tree is 18 degrees. What is the height of the tree? 100 tan 18π
180
35. If the distance from a helicopter to a tower is 300 feet and the angle of depression is
30◦ , find the
√ distance on the ground from a point directly below the helicopter to the
tower. 150 3
36. The angle of elevation of a hot air balloon, climbing vertically, changes from 45 degrees
at 10:00 am to 60 degrees at 10:02 am. The point of observation of the angle of
elevation is situated 300 meters away from the
take off point. What is the upward
√
300( 3−1)
speed, assumed constant, of the balloon?
meters per second.
120
37. When the top of a mountain is viewed from point A, 2000 m from ground, the angle of
depression a is equal to 45 degrees and when it is viewed from point B on the ground
the angle of elevation b is equal to 30 degrees. If points A and B are on the same
vertical line, find the height of the mountain. √2000
3+1
38. Show that
cot x − tan x
= csc2 x − sec2 x.
sin x cos x
cos x
− sin x
cot x − tan x
= sin x cos x
sin x cos x
sin x cos x cos x sin x
1
−
=
sin x cos x sin x cos x
1
1
−
=
2
sin x cos2 x
= csc2 x − sec2 x
39. Show that
tan2 x − sin2 x = tan2 x sin2 x.
40. Show that
csc x
csc x
−
= 2 sec2 x.
1 + csc x 1 + csc x
41. Show that
sin x +
cos x
= csc x.
cot x
6
42. Show that
(sin x + cos x)2 + (sin x − cos x)2 = 2.
43. Show that
(cot2 x + 1) =
− cot2 x
.
sin2 x − 1
44. Show that
1
1
+
= 2 csc2 x.
1 − cos x 1 + cos x
45. Show that
sin x cos y + cos x sin y
tan x + tan y
=
.
cos x cos y − sin x sin y
1 − tan x tan y
46. Solve for x in the interval [0, 2π),
−5 cos2 x + 9 sin x = −3.
x = sin−1 (0.2), π − sin−1 (0.2)
47. A backpacker notes that from a certain point on level ground, the angle of elevation
to a point at the top of a tree is 34◦ . After walking 50 feet closer to the tree, the
backpacker notes
that the angle of elevation is 54 degrees. Find the height of the tree.
50 tan(54◦ ) tan(34◦ )
height= tan(54◦ )−tan(34◦ )
Note (Things not on this review that you should look over.). You should look over the
sections on finding area of your paralleograms, trapezoids, circles, ellipses, etc. Remember
how to find arclengths and areas of circles.
7