Precalculus

Michael Sullivan

Chapter 6

Trigonometric Functions - all with Video Answers

Educators

Section 1

Angles and Their Measure

Problem 1

What is the formula for the circumference $C$ of a circle of radius $r ?$ What is the formula for the area $A$ of a circle of radius $r ?$

Julie Silva

Julie Silva

Numerade Educator

Problem 2

If a particle has a speed of $r$ feet per second and travels a distance $d$ (in feet) in time $t$ (in seconds), then $d=$ ___________.

Julie Silva

Numerade Educator

Problem 3

An angle $\theta$ is in ____________ ______________ if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive $x$ -axis.

Katelyn Chen

Numerade Educator

Problem 4

A ___________ _____________ is a positive angle whose vertex is at the center of a circle.

Katelyn Chen

Numerade Educator

Problem 5

If the radius of a circle is $r$ and the length of the arc subtended by a central angle is also $r,$ then the measure of the angle is 1 ___________.
(a) degree
(b) minute
(c) second
(d) radian

Katelyn Chen

Numerade Educator

Problem 6

On a circle of radius $r,$ a central angle of $\theta$ radians subtends an arc of length $s=$ ___________; the area of the sector formed by this angle $\theta$ is $A=$ ___________.

Katelyn Chen

Numerade Educator

Problem 7

$180^{\circ}=$ __________ radians
(a) $\frac{\pi}{2}$
(b) $\pi$
(c) $\frac{3 \pi}{2}$
(d) $2 \pi$

Katelyn Chen

Numerade Educator

Problem 8

An object travels around a circle of radius $r$ with constant speed. If $s$ is the distance traveled in time $t$ around the circle and $\theta$ is the central angle (in radians) swept out in time $t,$ then the linear speed of the object is $v=$ ________ and the angular speed of the object is $\omega=$ _____.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 9

True or False
The angular speed $\omega$ of an object traveling around a circle of radius $r$ is the angle $\theta$ (measured in radians) swept out, divided by the elapsed time $t$

Julie Silva

Numerade Educator

Problem 10

True or False
For circular motion on a circle of radius $r$, linear speed equals angular speed divided by $r$.

Katelyn Chen

Numerade Educator

Problem 11

draw each angle.
$$
30^{\circ}
$$

Julie Silva

Numerade Educator

Problem 12

Draw each angle.
$$
60^{\circ}
$$

Julie Silva

Numerade Educator

Problem 13

Draw each angle.
$$
135^{\circ}
$$

Julie Silva

Numerade Educator

Problem 14

Draw each angle.
$$
-120^{\circ}
$$

Julie Silva

Numerade Educator

Problem 15

Draw each angle.
$$
450^{\circ}
$$

Julie Silva

Numerade Educator

Problem 16

Draw each angle.
$$
540^{\circ}
$$

Julie Silva

Numerade Educator

Problem 17

Draw each angle.
$$
\frac{3 \pi}{4}
$$

Julie Silva

Numerade Educator

Problem 18

Draw each angle.
$$
\frac{4 \pi}{3}
$$

Julie Silva

Numerade Educator

Problem 19

Draw each angle.
$$
-\frac{\pi}{6}
$$

Julie Silva

Numerade Educator

Problem 20

Draw each angle.
$$
-\frac{2 \pi}{3}
$$

Julie Silva

Numerade Educator

Problem 21

Draw each angle.
$$
\frac{16 \pi}{3}
$$

Julie Silva

Numerade Educator

Problem 22

Draw each angle.
$$
\frac{21 \pi}{4}
$$

Julie Silva

Numerade Educator

Problem 23

Convert each angle to a decimal in degrees. Round your answer to two decimal places.
$$
40^{\circ} 10^{\prime} 25^{\prime \prime}
$$

Julie Silva

Numerade Educator

Problem 24

Convert each angle to a decimal in degrees. Round your answer to two decimal places.
$$
61^{\circ} 42^{\prime} 21^{\prime \prime}
$$

Julie Silva

Numerade Educator

Problem 25

Convert each angle to a decimal in degrees. Round your answer to two decimal places.
$$
50^{\circ} 14^{\prime} 20^{\prime \prime}
$$

Abhijith V

Numerade Educator

Problem 26

Convert each angle to a decimal in degrees. Round your answer to two decimal places.
$$
73^{\circ} 40^{\prime} 40^{\prime \prime}
$$

Julie Silva

Numerade Educator

Problem 27

Convert each angle to a decimal in degrees. Round your answer to two decimal places.
$$
9^{\circ} 9^{\prime} 9^{\prime \prime}
$$

Julie Silva

Numerade Educator

Problem 28

Convert each angle to a decimal in degrees. Round your answer to two decimal places.
$$
98^{\circ} 22^{\prime} 45^{\prime \prime}
$$

Julie Silva

Numerade Educator

Problem 29

Convert each angle to $D^{\circ} M^{\prime} S^{\prime \prime}$ form. Round your answer to the nearest second.
$$
40.32^{\circ}
$$

Julie Silva

Numerade Educator

Problem 30

Convert each angle to $D^{\circ} M^{\prime} S^{\prime \prime}$ form. Round your answer to the nearest second.
$$
61.24^{\circ}
$$

Julie Silva

Numerade Educator

Problem 31

Convert each angle to $D^{\circ} M^{\prime} S^{\prime \prime}$ form. Round your answer to the nearest second.
$$
18.255^{\circ}
$$

Julie Silva

Numerade Educator

Problem 32

Convert each angle to $D^{\circ} M^{\prime} S^{\prime \prime}$ form. Round your answer to the nearest second.
$$
29.411^{\circ}
$$

Julie Silva

Numerade Educator

Problem 33

Convert each angle to $D^{\circ} M^{\prime} S^{\prime \prime}$ form. Round your answer to the nearest second.
$$
19.99^{\circ}
$$

Julie Silva

Numerade Educator

Problem 34

Convert each angle to $D^{\circ} M^{\prime} S^{\prime \prime}$ form. Round your answer to the nearest second.
$$
44.01^{\circ}
$$

Julie Silva

Numerade Educator

Problem 35

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
30^{\circ}
$$

Julie Silva

Numerade Educator

Problem 36

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
120^{\circ}
$$

Julie Silva

Numerade Educator

Problem 37

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
240^{\circ}
$$

Julie Silva

Numerade Educator

Problem 38

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
330^{\circ}
$$

Julie Silva

Numerade Educator

Problem 39

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
-60^{\circ}
$$

Julie Silva

Numerade Educator

Problem 40

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
-30^{\circ}
$$

Julie Silva

Numerade Educator

Problem 41

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
180^{\circ}
$$

Julie Silva

Numerade Educator

Problem 42

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
270^{\circ}
$$

Julie Silva

Numerade Educator

Problem 43

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
-135^{\circ}
$$

Julie Silva

Numerade Educator

Problem 44

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
-225^{\circ}
$$

Julie Silva

Numerade Educator

Problem 45

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
-90^{\circ}
$$

Julie Silva

Numerade Educator

Problem 46

Convert each angle in degrees to radians. Express your answer as a multiple of $\pi$.
$$
-180^{\circ}
$$

Julie Silva

Numerade Educator

Problem 47

Convert each angle in radians to degrees.
$$
\frac{\pi}{3}
$$

Abhijith V

Numerade Educator

Problem 48

Convert each angle in radians to degrees.
$$
\frac{5 \pi}{6}
$$

Julie Silva

Numerade Educator

Problem 49

Convert each angle in radians to degrees.
$$
-\frac{5 \pi}{4}
$$

Arin Asawa

Numerade Educator

Problem 50

Convert each angle in radians to degrees.
$$
-\frac{2 \pi}{3}
$$

Julie Silva

Numerade Educator

Problem 51

Convert each angle in radians to degrees.
$$
\frac{\pi}{2}
$$

Julie Silva

Numerade Educator

Problem 52

Convert each angle in radians to degrees.
$$
4 \pi
$$

Julie Silva

Numerade Educator

Problem 53

Convert each angle in radians to degrees.
$$
\frac{\pi}{12} 15^{\circ}
$$

ZS

Zaid Simjee

Numerade Educator

Problem 54

Convert each angle in radians to degrees.
$$
\frac{5 \pi}{12}
$$

Julie Silva

Numerade Educator

Problem 55

Convert each angle in radians to degrees.
$$
-\frac{\pi}{2}
$$

Julie Silva

Numerade Educator

Problem 56

Convert each angle in radians to degrees.
$$
-\pi
$$

Julie Silva

Numerade Educator

Problem 57

Convert each angle in radians to degrees.
$$
-\frac{\pi}{6}
$$

Julie Silva

Numerade Educator

Problem 58

Convert each angle in radians to degrees.
$$
-\frac{3 \pi}{4}
$$

Julie Silva

Numerade Educator

Problem 59

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places.
$$
17^{\circ}
$$

Julie Silva

Numerade Educator

Problem 60

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places.
$$
73^{\circ}
$$

Julie Silva

Numerade Educator

Problem 61

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places.
$$
-40^{\circ}
$$

Julie Silva

Numerade Educator

Problem 62

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places.
$$
-51^{\circ}
$$

Julie Silva

Numerade Educator

Problem 63

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places.
$$
125^{\circ}
$$

Julie Silva

Numerade Educator

Problem 64

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places.
$$
350^{\circ}
$$

Julie Silva

Numerade Educator

Problem 65

Convert each angle in radians to degrees. Express your answer in decimal form, rounded to two decimal places.
$$
3.14
$$

Abhijith V

Numerade Educator

Problem 66

Convert each angle in radians to degrees. Express your answer in decimal form, rounded to two decimal places.
$$
0.75
$$

Abhijith V

Numerade Educator

Problem 67

Convert each angle in radians to degrees. Express your answer in decimal form, rounded to two decimal places.
$$
2
$$

Julie Silva

Numerade Educator

Problem 68

Convert each angle in radians to degrees. Express your answer in decimal form, rounded to two decimal places.
$$
3
$$

Julie Silva

Numerade Educator

Problem 69

Convert each angle in radians to degrees. Express your answer in decimal form, rounded to two decimal places.
$$
6.32
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 70

Convert each angle in radians to degrees. Express your answer in decimal form, rounded to two decimal places.
$$
\sqrt{2}
$$

Julie Silva

Numerade Educator

Problem 71

Denotes the length of the arc of a circle of radius r subtended by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=10$ meters, $\theta=\frac{1}{2}$ radian, $s=?$

Julie Silva

Numerade Educator

Problem 72

Denotes the length of the arc of a circle of radius r subtended by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=6$ feet, $\quad \theta=2$ radians, $\quad s=?$

Abhijith V

Numerade Educator

Problem 73

Denotes the length of the arc of a circle of radius r subtended by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$\theta=\frac{1}{3}$ radian, $s=2$ feet, $\quad r=?$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 74

Denotes the length of the arc of a circle of radius r subtended by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$\theta=\frac{1}{4}$ radian, $s=6$ centimeters, $r=?$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 75

Denotes the length of the arc of a circle of radius r subtended by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=5$ miles, $s=3$ miles, $\theta=?$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 76

Denotes the length of the arc of a circle of radius r subtended by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=6$ meters, $s=8$ meters, $\theta=?$

Abhijith V

Numerade Educator

Problem 77

Denotes the length of the arc of a circle of radius r subtended by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=2$ inches, $\quad \theta=30^{\circ}, \quad s=?$

Abhijith V

Numerade Educator

Problem 78

Denotes the length of the arc of a circle of radius r subtended by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=3$ meters, $\quad \theta=120^{\circ}, \quad s=?$

Abhijith V

Numerade Educator

Problem 79

A denotes the area of the sector of a circle of radius $r$ formed by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=10$ meters, $\theta=\frac{1}{2}$ radian, $A=?$

Abhijith V

Numerade Educator

Problem 80

A denotes the area of the sector of a circle of radius $r$ formed by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=6$ feet, $\theta=2$ radians, $A=?$

Abhijith V

Numerade Educator

Problem 81

A denotes the area of the sector of a circle of radius $r$ formed by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$\theta=\frac{1}{3}$ radian, $A=2$ square feet, $r=?$

Abhijith V

Numerade Educator

Problem 82

A denotes the area of the sector of a circle of radius $r$ formed by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$\theta=\frac{1}{4}$ radian, $\quad A=6$ square centimeters, $r=?$

Abhijith V

Numerade Educator

Problem 83

A denotes the area of the sector of a circle of radius $r$ formed by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=5$ miles, $A=3$ square miles, $\theta=?$

Abhijith V

Numerade Educator

Problem 84

A denotes the area of the sector of a circle of radius $r$ formed by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=6$ meters, $A=8$ square meters, $\theta=?$

Abhijith V

Numerade Educator

Problem 85

A denotes the area of the sector of a circle of radius $r$ formed by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=2$ inches, $\quad \theta=30^{\circ}, \quad A=?$

Abhijith V

Numerade Educator

Problem 86

A denotes the area of the sector of a circle of radius $r$ formed by the central angle $\theta .$ Find the missing quantity. Round answers to three decimal places.
$r=3$ meters, $\quad \theta=120^{\circ}, \quad A=?$

Abhijith V

Numerade Educator

Problem 87

Find the length s and area $A .$ Round answers to three decimal places.

Katelyn Chen

Numerade Educator

Problem 88

Find the length s and area $A .$ Round answers to three decimal places.

Katelyn Chen

Numerade Educator

Problem 89

Find the length s and area $A .$ Round answers to three decimal places.

Katelyn Chen

Numerade Educator

Problem 90

Find the length s and area $A .$ Round answers to three decimal places.

Katelyn Chen

Numerade Educator

Problem 91

The minute hand of a clock is 6 inches long. How far does the tip of the minute hand move in 15 minutes? How far does it move in 25 minutes? Round answers to two decimal places.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 92

A pendulum swings through an angle of $20^{\circ}$ each second. If the pendulum is 40 inches long, how far does its tip move each second? Round answers to two decimal places.

Julie Silva

Numerade Educator

Problem 93

Find the area of the sector of a circle of radius 4 meters formed by an angle of $45^{\circ} .$ Round the answer to two decimal places.

Katelyn Chen

Numerade Educator

Problem 94

Find the area of the sector of a circle of radius 3 centimeters formed by an angle of $60^{\circ} .$ Round the answer to two decimal places.

Katelyn Chen

Numerade Educator

Problem 95

A water sprinkler sprays water over a distance of 30 feet while rotating through an angle of $135^{\circ}$. What area of lawn receives water?

Julie Silva

Numerade Educator

Problem 96

An engineer is asked to design a water sprinkler that will cover a field of 100 square yards that is in the shape of a sector of a circle of radius 15 yards. Through what angle should the sprinkler rotate?

Katelyn Chen

Numerade Educator

Problem 97

The arm and blade of a windshield wiper have a total length of 34 inches. If the blade is 25 inches long and the wiper sweeps out an angle of $120^{\circ},$ how much window area can the blade clean?

Katelyn Chen

Numerade Educator

Problem 98

The arm and blade of a windshield wiper have a total length of 30 inches. If the blade is 24 inches long and the wiper sweeps out an angle of $125^{\circ},$ how much window area can the blade clean?

Katelyn Chen

Numerade Educator

Problem 99

An object is traveling around a circle with a radius of 5 centimeters. If in 20 seconds a central angle of $\frac{1}{3}$ radian is swept out, what is the angular speed of the object? What is its linear speed?

Julie Silva

Numerade Educator

Problem 100

An object is traveling around a circle with a radius of 2 meters. If in 20 seconds the object travels 5 meters, what is its angular speed? What is its linear speed?

Julie Silva

Numerade Educator

Problem 101

A gondola on an amusement park ride, similar to the Spin Cycle at Silverwood Theme Park, spins at a speed of 13 revolutions per minute. If the gondola is 25 feet from the ride's center, what is the linear speed of the gondola in miles per hour?

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 102

A centrifugal force ride, similar to the Gravitron, spins at a speed of 22 revolutions per minute. If the diameter of the ride is 13 meters, what is the linear speed of the passengers in kilometers per hour?

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 103

A Blu-ray drive has a maximum speed of 10,000 revolutions per minute. If a Blu-ray disc has a diameter of $12 \mathrm{~cm},$ what is the linear speed, in $\mathrm{km} / \mathrm{h},$ of a point $4 \mathrm{~cm}$ from the center if the disc is spinning at a rate of 8000 revolutions per minute?

Narayan Hari

Numerade Educator

Problem 104

A DVD drive has a maximum speed of 7200 revolutions per minute. If a DVD has a diameter of $12 \mathrm{~cm}$, what is the linear speed, in $\mathrm{km} / \mathrm{h},$ of a point $5 \mathrm{~cm}$ from the disc's center if it is spinning at a rate of 5400 revolutions per minute?

Katelyn Chen

Numerade Educator

Problem 105

The diameter of each wheel of a bicycle is 26 inches. If you are traveling at a speed of 35 miles per hour on this bicycle, through how many revolutions per minute are the wheels turning?

PR

Paul Ridder

Numerade Educator

Problem 106

The radius of each wheel of a car is 15 inches. If the wheels are turning at the rate of 3 revolutions per second, how fast is the car moving? Express your answer in inches per second and in miles per hour.

Katelyn Chen

Numerade Educator

Problem 107

The latitude of a location $L$ is the angle formed by a ray drawn from the center of Earth to the Equator and a ray drawn from the center of Earth to $L$. See the figure.
Memphis, Tennessee, is due north of New Orleans, Louisiana. Find the distance between Memphis $\left(35^{\circ} 9^{\prime}\right.$ north latitude) and New Orleans $\left(29^{\circ} 57^{\prime}\right.$ north latitude). Assume that the radius of Earth is 3960 miles.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 108

The latitude of a location $L$ is the angle formed by a ray drawn from the center of Earth to the Equator and a ray drawn from the center of Earth to $L$. See the figure.
Charleston, West Virginia, is due north of Jacksonville, Florida. Find the distance between Charleston $\left(38^{\circ} 21^{\prime}\right.$ north latitude) and Jacksonville $\left(30^{\circ} 20^{\prime}\right.$ north latitude $) .$ Assume that the radius of Earth is 3960 miles.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 109

The latitude of a location $L$ is the angle formed by a ray drawn from the center of Earth to the Equator and a ray drawn from the center of Earth to $L$. See the figure.
Earth rotates on an axis through its poles. The distance from the axis to a location on Earth at $30^{\circ}$ north latitude is about 3429.5 miles. Therefore, a location on Earth at $30^{\circ}$ north latitude is spinning on a circle of radius 3429.5 miles. Compute the linear speed on the surface of Earth at $30^{\circ}$ north latitude.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 110

The latitude of a location $L$ is the angle formed by a ray drawn from the center of Earth to the Equator and a ray drawn from the center of Earth to $L$. See the figure.
Earth rotates on an axis through its poles The distance from the axis to a location on Earth at $40^{\circ}$ north latitude is about 3033.5 miles. Therefore, a location on Earth at $40^{\circ}$ north latitude is spinning on a circle of radius 3033.5 miles. Compute the linear speed on the surface of Earth at $40^{\circ}$ north latitude.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 111

The mean distance of the moon from Earth is $2.39 \times 10^{5}$ miles. Assuming that the orbit of the moon around Earth is circular and that 1 revolution takes 27.3 days, find the linear speed of the moon. Express your answer in miles per hour.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 112

The mean distance of Earth from the Sun is $9.29 \times 10^{7}$ miles. Assuming that the orbit of Earth around the Sun is circular and that 1 revolution takes 365 days, find the linear speed of Earth. Express your answer in miles per hour.

Katelyn Chen

Numerade Educator

Problem 113

Two pulleys, one with radius 2 inches and the other with radius 8 inches, are connected by a belt. (See the figure.) If the 2 -inch pulley is caused to rotate at 3 revolutions per minute, determine the revolutions per minute of the 8-inch pulley.

Katelyn Chen

Numerade Educator

Problem 114

A neighborhood carnival has a Ferris wheel whose radius is 30 feet. You measure the time it takes for one revolution to be 70 seconds. What is the linear speed (in feet per second) of this Ferris wheel? What is the angular speed in radians per second?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 115

To approximate the speed of the current of a river, a circular paddle wheel with radius 4 feet is lowered into the water. If the current causes the wheel to rotate at a speed of 10 revolutions per minute, what is the speed of the current? Express your answer in miles per hour.

Katelyn Chen

Numerade Educator

Problem 116

A spin balancer rotates the wheel of a car at 480 revolutions per minute. If the diameter of the wheel is 26 inches, what road speed is being tested? Express your answer in miles per hour. At how many revolutions per minute should the balancer be set to test a road speed of 80 miles per hour?

Katelyn Chen

Numerade Educator

Problem 117

At the Cable Car Museum you can see the four cable lines that are used to pull cable cars up and down the hills of San Francisco. Each cable travels at a speed of 9.55 miles per hour, caused by a rotating wheel whose diameter is 8.5 feet. How fast is the wheel rotating? Express your answer in revolutions per minute.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 118

Naples, Florida, is about 90 miles due west of Ft. Lauderdale. How much sooner would a person in Ft. Lauderdale first see the rising Sun than a person in Naples? See the hint. [Hint: Consult the figure. When a person at $Q$ sees the first rays of the Sun, a person at $P$ is still in the dark. The person at $P$ sees the first rays after Earth has rotated so that $P$ is at the location $Q .$ Now use the fact that at the latitude of Ft. Lauderdale in 24 hours an arc of length $2 \pi(3559)$ miles is subtended. $]$

Katelyn Chen

Numerade Educator

Problem 119

A dog is attached to a 9-foot rope fastened to the outside corner of a fenced-in garden that measures 6 feet by 10 feet. Assuming that the dog cannot enter the garden, compute the exact area that the dog can wander. Write the exact area in square feet.

Katelyn Chen

Numerade Educator

Problem 120

The measure of arc $\widehat{B E}$ is $2 \pi .$ Find the exact area of the portion of the rectangle $A B C D$ that falls outside of the circle whose center is at $A .^{*}$

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 121

How fast would you have to travel on the surface of Earth at the equator to keep up with the Sun (that is, so that the Sun would appear to remain in the same position in the sky)?

Katelyn Chen

Numerade Educator

Problem 122

A nautical mile equals the length of arc subtended by a central angle of 1 minute on a great circle $^{\dagger}$ on the surface of Earth. See the figure on the next page. If the radius of Earth is taken as 3960 miles, express 1 nautical mile in terms of ordinary, or statute, miles.

Sheryl Ezze

Numerade Educator

Problem 123

Eratosthenes of Cyrene $(276-195 \mathrm{BC})$ was a Greek scholar who lived and worked in Cyrene and Alexandria. One day while visiting in Syene he noticed that the Sun's rays shone directly down a well. On this date 1 year later, in Alexandria, which is 500 miles due north of Syene he measured the angle of the Sun to be about 7.2 degrees See the figure. Use this information to approximate the radius and circumference of Earth.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 124

For a 60 -foot Little League Baseball field, the distance from home base to the nearest fence (or other obstruction) in fair territory should be a minimum of 200 feet.The commissioner of parksand recreation is making plans for a new 60 -foot field. Because of limited ground availability, he will use the minimum required distance to the outfield fence. To increase safety, however, he plans to include a 10 -foot-wide warning track on the inside of the fence. To further increase safety, the fence and warning track will extend both directions into foul territory. In total, the arc formed by the outfield fence (including the extensions into the foul territories) will be subtended by a central angle at home plate measuring $96^{\circ},$ as illustrated.
(a) Determine the length of the outfield fence.
(b) Determine the area of the warning track.
[Note: There is a $90^{\circ}$ angle between the two foul lines. Then there are two $3^{\circ}$ angles between the foul lines and the dotted lines shown. The angle between the two dotted lines outside the 200 -foot foul lines is $\left.96^{\circ} .\right]$

Katelyn Chen

Numerade Educator

Problem 125

Two pulleys, one with radius $r_{1}$ and the other with radius $r_{2},$ are connected by a belt. The pulley with radius $r_{1}$ rotates at $\omega_{1}$ revolutions per minute, whereas the pulley with radius $r_{2}$ rotates at $\omega_{2}$ revolutions per minute. Show that
$$\frac{r_{1}}{r_{2}}=\frac{\omega_{2}}{\omega_{1}}$$

Katelyn Chen

Numerade Educator

Problem 126

Do you prefer to measure angles using degrees or radians? Provide justification and a rationale for your choice.

Katelyn Chen

Numerade Educator

Problem 127

What is 1 radian? What is 1 degree?

Katelyn Chen

Numerade Educator

Problem 128

Which angle has the larger measure: 1 degree or 1 radian? Or are they equal?

Katelyn Chen

Numerade Educator

Problem 129

Explain the difference between linear speed and angular speed.

Katelyn Chen

Numerade Educator

Problem 130

For a circle of radius $r,$ a central angle of $\theta$ degrees subtends an arc whose length $s$ is $s=\frac{\pi}{180} r \theta .$ Discuss whether this statement is true or false. Defend your position.

Abhijith V

Numerade Educator

Problem 131

Discuss why ships and airplanes use nautical miles to measure distance. Explain the difference between a nautical mile and a statute mile.

Katelyn Chen

Numerade Educator

Problem 132

Investigate the way that speed bicycles work. In particular, explain the differences and similarities between 5 -speed and 9-speed derailleurs. Be sure to include a discussion of linear speed and angular speed.

Katelyn Chen

Numerade Educator

Problem 133

In Example $6,$ we found that the distance between Dallas, Texas and Sioux Falls, South Dakota is approximately 744 miles. According to mapquest.com, the distance is approximately 850 miles. What might account for the difference?

Katelyn Chen

Numerade Educator

Problem 134

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the zero of $f(x)=3 x+7$.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 135

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the domain of $h(x)=\frac{3 x}{x^{2}-9}$.

Kian Manafi

Numerade Educator

Problem 136

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Write the function that is finally graphed if all of the following transformations are applied to the graph of $y=|x|$.
(a) Shift left 3 units.
(b) Reflect about the $x$ -axis.
(c) Shift down 4 units.

Jaymie Irwin

Numerade Educator

Problem 137

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the horizontal and vertical asymptotes of $R(x)=\frac{3 x^{1}-12}{x^{2}-5 x-14}$

Jaymie Irwin

Numerade Educator