Precalculus
HS Mathematics
Unit: 03 Lesson: 02
Conversions with Dimensional Analysis (pp. 1 of 2)
Sample: 40 mph is how many feet per second?
40 mi
5280 ft
1 hr
1 min
40(5280) ft
= 58 32 ft/sec
=
1 hr
1 mi
60 min
60 sec
60(60) sec
Start with the given information as a ratio. Then cancel unwanted units by multiplying by various
forms of “1” (comparing unit equivalents, such as 1 mi = 5280 ft).
1)
A car went 44 feet through an intersection in only 0.5 seconds. How fast was the car going in
miles per hour?
44 ft
0.5 sec
=
=
2)
In football, Drew runs a 40-yard dash in 5.1 seconds. Describe his rate in miles per hour.
3)
A leaky faucet in our kitchen wastes a cup of water every 10 minutes.
How fast is this water leaking in gallons per day?
4)
With my internet connection, my computer estimated that it would take 18
seconds to download an 890-KB file. What is the transfer rate in gigabytes
(GB) per day?
5)
An old-fashioned record player spins an album at a rate of 33 31 rpm
(revolutions per minute). Convert this rate into degrees per second.
©2010, TESCCC
10/20/10
1 gal = 4 qt
1 qt = 4 cups
1 MB = 1024 KB
1 GB = 1024 MB
1 rev = 360
page 9 of 36
Precalculus
HS Mathematics
Unit: 03 Lesson: 02
Conversions with Dimensional Analysis (pp. 2 of 2)
6)
The world record for the fastest spinning on ice skates is 1848 of rotation
per second. Describe this rate in revolutions per minute.
7)
On average, the earth is 93 million miles away from the center of
the sun.
A) Assuming that the earth’s orbit is circular, estimate the
number of miles our planet travels in one complete revolution
around the sun.
1 rev = 360
Earth
Sun
93 million
miles
B)
8)
The complete revolution around the sun takes 1 year. How
fast is the earth moving in this orbit in feet per second?
Suppose a car wheel is 26 inches in diameter.
A) If the tire completes one rotation, how far would the car travel?
26 in
B)
If the car is traveling at 60 miles per hour, how fast is the car
wheel spinning in revolutions per second?
©2010, TESCCC
10/20/10
page 10 of 36
Precalculus
HS Mathematics
Unit: 03 Lesson: 02
Radian Measure of Angles (pp. 1 of 2)
3 radii
(radians)
2 radii
(radians)
radians
degrees
1
57.3
2
Besides degrees, angles can also
be measured in radians. Radian
measure is defined by the following
ratio:
Length of intercepted arc
Length of the radius
3
1 radius
180
1 radian
360
radius
Another way to describe radian
measure is to ask, “How many radii
can fit along the arc included in the
angle?”
A little more than 3 radians in 180
While you can write “radians” (or “rad” for short) after a measure to indicate radians, in most cases
you don’t have to write any units at all. Since radians are defined as a ratio of two lengths (such as
inches to inches, or cm to cm), these units “cancel” each other out. In other words, radians are “unitless” measures.
To change from degrees to radians, multiply by the following conversion factor:
Convert these degree measures into radians.
1)
2)
120
90
r
r
3)
300
4)
420
r
©2010, TESCCC
10/20/10
page 17 of 36
Precalculus
HS Mathematics
Unit: 03 Lesson: 02
Radian Measure of Angles (pp. 2 of 2)
To change from radians to degrees, multiply by the following conversion factor:
Convert these radian measures into degrees.
5)
6)
5
4
4
r
r
7)
8)
5
6
7.854
r
Formulas Using Radians
A circle has the following measures:
Arc
r=
=
Sector
r
L=
S=
length of the radius
measure of central angle
(in radians)
length of the intercepted arc
area of the sector formed
The following formulas can be
used to find arc length and
sector area:
Arc length: L r
Sector area: S
1
2
r2
For each figure: A) convert the degree measure to radians, B) find the length of the intercepted arc,
and C) find the area (shaded) of the sector formed.
9)
10)
B
75
6 cm
A
240
8 in
A
B
©2010, TESCCC
10/20/10
page 18 of 36
Precalculus
HS Mathematics
Unit: 03 Lesson: 02
Circular Reasoning (pp. 1 of 2)
Convert each degree measure to radians.
1) 105
2) 270
3) 1080
Convert each radian measure to degrees.
11
7
4)
5)
15
6
Arc length: L r
Sector area: S
1
2
6) 3.4
Here, r is the circle’s radius, and
is the measure of a central angle (in radians)
r2
In items #7 and #8: A) convert the degree measure to radians, B) find the length of the intercepted
arc, and C) find the area (shaded) of the sector formed.
7)
8)
B
135
20 in
300
A
18 cm
A
B
9)
45
0
fe
et
A particular softball field is a 90 sector with a radius of 450 feet.
A) A fence surrounds the entire field. What is the total length
of the fence?
B)
What area is enclosed by the fence?
90
©2010, TESCCC
10/20/10
page 21 of 36
Precalculus
HS Mathematics
Unit: 03 Lesson: 02
Circular Reasoning (pp. 2 of 2)
10)
Dave
Gary
50
30
Three friends order a 16-inch (diameter) pizza to share.
A) Sarah eats only a 40 slice of pizza. How many square inches is
this?
60
40
B)
Sarah
Dave eats both a 30 slice and a 50 slice. How many square
inches of pizza did he eat?
C) Gary eats all of a 60 slice of pizza—except the crust. What is the length of this crust?
11)
North
Pole
ole
Wichita
(90 )
37.69 N
30.27 N
Austin
Austin, Texas, and Wichita, Kansas, fall approximately on the
same line of longitude. However, Austin lies at 30.27 N latitude,
and Wichita’s latitude is 37.69 N.
Degrees latitude is measured as an angle central to the earth
starting at the equator, where the earth has an approximate
radius of 4000 miles.
(0 )
Eq
Equator
r 4000 mi
A)
What is the measure of the arc on the earth’s surface
between these two cities in degrees? …in radians?
B) What is the approximate distance between the two cities (along the earth’s surface)?
12)
On a circle with a radius of 12 cm is an arc
of length 20 cm. What is the degree
measure of the central angle used to make
this arc?
©2010, TESCCC
13)
10/20/10
In a circle with a radius of 50 units is a
sector with an area of 3,200 un2. What is
the degree measure of the central angle
used to make this sector?
page 22 of 36
Precalculus
HS Mathematics
Unit: 03 Lesson: 02
Angles, Circles, Velocity (pp. 1 of 3)
1 mile = 5,280 feet
1 km = 1,000 m
1 rev = 360
(radians) = 180
Arc length: L
Sector area: S
vL
r
1
2
2
r
( measured in radians)
r v
Linear Velocity = r (Angular Velocity)
( v measured in radians per time)
1)
30 miles per hour is how many feet per second?
2)
2.4 meters per second is how many kilometers per hour?
3)
The second hand on a clock completes one revolution every minute.
Convert this rate into degrees per day.
4)
On its axis, the earth spins once every day. Convert this rate into degrees per minute.
Convert each degree measure to radians.
5) 120
6) 315
7) 540
Convert each radian measure to degrees.
13
5
8)
9)
12
6
10) 3.578
For each figure below: A) convert the measure of the central angle to radians, B) find the length of the
intercepted arc, and C) find the area (shaded) of the sector formed.
11)
12)
B
150
20 in
285
4 cm
A
A
B
©2010, TESCCC
10/20/10
page 34 of 36
Precalculus
HS Mathematics
Unit: 03 Lesson: 02
Angles, Circles, Velocity (pp. 2 of 3)
Answer questions involving radian measure, arc length, and linear and angular velocity.
13)
The radius of the earth is approximately 4,000 miles. A satellite orbits 5 miles above its surface.
The satellite completes 220 of a revolution around the earth in a day.
A) How many miles did the satellite travel in orbit during the day?
B)
14)
What was the average speed of the satellite in miles per hour?
A 50-cm pendulum in a grandfather clock swings on a 12-cm arc every
second.
In this second, how many degrees does the pendulum move?
?
50 cm
12 cm
15)
When serving, a tennis player swings her racket with a rotational speed equivalent to 180 in
0.1 second.
A) Find the angular velocity of the racket in radians per second.
B)
The player’s racket strikes a tennis ball at a distance of 38 inches from the center of
rotation. How fast would the ball leave the racket in miles per hour?
©2010, TESCCC
10/20/10
page 35 of 36
Precalculus
HS Mathematics
Unit: 03 Lesson: 02
Angles, Circles, Velocity (pp. 3 of 3)
16)
A piece of exercise equipment uses a two-pulley system that allows users to lift weights by
pulling up on a handle attached to a cord. The upper pulley has a radius of 6 cm, and the lower
pulley’s radius is 4 cm. An athlete lifts the weights so that they rise at the rate of 10 cm/sec.
A) Find the angular velocity of the upper pulley in degrees per
r = 6 cm
second.
B)
Find the angular velocity of the lower pulley in degrees per
second
r = 4 cm
17)
The motor in a clock is powered to turn a big gear (B, of radius 5
units) at a rate of 144 every minute. The second hand on the
clock, however, needs to turn at a rate of 1 revolution per minute.
r = 5 un
So, the big gear (B) is coupled with a smaller gear (S) which will
turn at a faster angular velocity.
B
r=?
S
A)
Find the angular velocity of gear B in radians per minute.
B)
Find the linear velocity of the teeth on the rim of gear B (in units per minute).
C)
Determine the radius that could be used for gear S so that it has the same linear velocity
as gear B, but rotates at 1 revolution per minute.
©2010, TESCCC
10/20/10
page 36 of 36