Mathematical Functions - Fall 2012
Quiz 9 November 25, 2012
This part consists of 8 multiple choice problems. Nothing more than the answer is required; consequently no partial credit will be awarded.
1. Express 80◦ in radians.
A
π
9
B
2π
9
C
4π
9
D
5π
9
E
7π
9
Solution: We have
80◦ = 80
2. Express
A
π 8π
4π
80π
rad =
rad =
rad
rad =
180
180
18
9
7π
rad in degrees.
6
210◦
B
220◦
C
230◦
D
240◦
E
250◦
Solution: We have
7π
rad =
6
7π
6
180
π
=
1
7π · 180
= 7 · 30 = 210◦
6·π
3. Find an angle that is coterminal with the angle θ = 20◦ in standard position.
A
200◦
B
160◦
C
−160◦
D
−200◦
E
−340◦
Solution: Note that
20◦ − 360◦ = −340◦
therefore −340◦ is coterminal with θ = 20◦ .
4. Find an angle that is coterminal with the angle θ =
A
2π
5
B
4π
5
C
6π
5
D
E
π
in standard position.
5
11π
5
12π
5
Solution: Note that
π
π 10π
π + 10π
11π
+ 2π = +
=
=
5
5
5
5
5
π
11π
is coterminal with θ = .
therefore
5
5
2
5. Find an angle with measure between 0◦ and 360◦ that is coterminal with the angle of measure
1567◦ in standard position.
A
107◦
B
117◦
C
127◦
D
137◦
E
None of the above
Solution: Note that
1567◦ − 4 · 360◦ = 127◦
therefore 127◦ is coterminal with θ = 1567◦ .
6. Find the length of an arc of a circle with radius 27 m that subtends a central angle of 60◦ .
A
9π
B
5π
2
C
9π
2
D
9π
4
E
27π
4
Solution: We first note that 60◦ =
π
rad. So the length of the arc is
3
s = rθ = (27)
3
π
= 9π m
3
7. A central angle θ in a circle of radius 12 m is subtended by an arc of length 22 m. Find the
measure of θ in radians.
A
6
11
B
11
6
C
6π
11
D
11π
6
E
None of the above
Solution: By the formula θ = s/r, we have
θ=
22
11
s
=
=
rad
r
12
6
8. Find the area of a sector of a circle with central angle 60◦ if the radius of the circle is
A
B
C
√
3 m.
π
2
3π
2
π
3
D
2π
3
E
π
Solution: To use the formula for the area of a circular sector, we must find the central angle of
the sector in radians:
π
60◦ = rad
3
Thus, the area of the sector is
π π
1 √ π 1
1
= (3)
= m2
A = r2 θ = ( 3)2
2
2
3
2
3
2
4
In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit.
1. A satellite in a circular orbit 1250 kilometers above Earth makes one complete revolution
every 110 minutes. What is its angular and linear speed? Assume that Earth is a sphere of
radius 6400 kilometers.
Solution: We have
Angular speed =
2π
π
θ
=
=
t
110
55
and
Linear speed = Angular speed × r =
π
π
7650π
· (6400 + 1250) =
· 7650 =
55
55
55
1530π
=
11
≈ 436.967 km/min
2. A biologist wants to know the width w of a river (see figure) in order to properly set instruments for studying the pollutants in the water. From point A, the biologist walks downstream
100 feet and sights to point C. From this sighting, it is determined that θ = 58◦ . How wide is
the river?
Solution: We have
tan 58◦ =
w
100
=⇒
w = 100 tan 58◦ ≈ 160 feet
5