4.4.2: Homework

Reading Questions

1. What is the difference between an angle of elevation and an angle of depression?
2. How is the angle of inclination of a line defined?
3. How is the slope of a line, \(m\), related to its angle of inclination, \(\alpha\)?
4. In navigation, what is a "heading"? From what direction and in what manner is it measured?
5. What are the three components of a navigational "bearing"?
6. What is the primary difference between how a heading is measured and how a bearing is measured?
7. In a multi-part navigation problem, what is a helpful drawing strategy to use at each "stop-over" point?
8. What defines a "compound right triangle application"?
9. In a compound right triangle problem, what is the key to solving the problem?

Homework

Vocabulary Check

1. The angle between the horizon and the line of sight above the horizon is called the angle of ___, also known as the angle of ___.

2. A(n) ___ is an angle measured clockwise from due north.

3. A(n) ___ is a direction given in the form north or south by a certain number of degrees east or west (denoted, for example, by \( \mathrm{S} \, 13^{ \circ } \, \mathrm{W} \)).

Concept Check

4. If the angle of inclination of a line is greater than \(45^{\circ}\), what can you say about its slope?

5. What is the bearing that shares a terminal side with the standard position angle \( \theta = 45^{ \circ } \)?

6. What is the bearing that shares a terminal side with the standard position angle \( \theta = 120^{ \circ } \)?

7. What is the bearing that shares a terminal side with the standard position angle \( \theta = -150^{ \circ } \)?

True or False? For the following exercises, determine if the statement is true or false. If true, cite the definition or theorem stated in the text supporting your claim. If false, explain why it is false and, if possible, correct the statement.

8. An angle of depression is measured from due south to an individual's line of sight below the horizon.

9. The angle given in a bearing must be acute.

10. The angle given in a heading must be acute.

11. The angle \( \theta = 120^{ \circ } \) in standard position has a terminal side shared with the heading angle \( 120^{ \circ } \).

12. The angle \( \theta = 120^{ \circ } \) in standard position has a terminal side shared with the heading angle \( -120^{ \circ } \).

13. The angle \( \theta = 120^{ \circ } \) in standard position has a terminal side shared with the heading angle \( 60^{ \circ } \).

14. The angle \( \theta = 120^{ \circ } \) in standard position has a terminal side shared with the heading angle \( -60^{ \circ } \).

15. The angle \( \theta = 120^{ \circ } \) in standard position has a terminal side shared with the heading angle \( 240^{ \circ } \).

16. The angle \( \theta = 120^{ \circ } \) in standard position has a terminal side shared with the heading angle \( -240^{ \circ } \).

Basic Skills

For the following exercises, sketch the situation using a right triangle and label each angle.

17. The angle of elevation from the ground to the top of a power pole is \( 50^{ \circ } \).

18. The angle of depression from the eye of a hiker on a cliff to the bottom of the canyon is \( 85^{ \circ } \).

19. Two people are looking up at a hot air balloon. They are positioned on opposite sides of the ballon. The angle of elevation to the ballon for one person is \( 57^{ \circ } \), and the angle of elevation to the ballon for the other person is \( 30^{ \circ } \).

For the following exercises, sketch a diagram illustrating the given bearing.

20. The bearing from Sacramento to South Lake Tahoe is \( \mathrm{N} \, 72^{ \circ } \, \mathrm{E} \).

21. The bearing from New York City to Philadelphia is \( \mathrm{S} \, 48^{ \circ } \, \mathrm{W} \).

22. The bearing from the Southern Terminus of the Pacific Crest Trail to the Northern Terminus is \( \mathrm{N} \, 10^{ \circ } \, \mathrm{W} \).

23. The bearing from Lincoln Airport to Placerville Airport is \( \mathrm{S} \, 69^{ \circ } \, \mathrm{E} \).

For the following exercises, find the angle of inclination of the line.

24. \(y=\dfrac{5}{4} x-3\)

25. \(y=6+\dfrac{2}{9} x\)

26. \(y=-2-\dfrac{3}{8} x\)

27. \(y=\dfrac{-7}{2} x+1\)

For the following exercises, find an equation for the line passing through the given point with an angle of inclination \(\alpha\).

28. \((3,-5), \alpha=28^{\circ}\)

29. \((-2,6), \alpha=67^{\circ}\)

30. \((-8,12), \alpha=112^{\circ}\)

31. \((-4,-1), \alpha=154^{\circ}\)

Synthesis Questions

32. The slope of a line is a function of its angle of inclination, \(m = f(\alpha)\). Complete the table and sketch a graph of the function.

    \(\alpha\)

    \(0^{\circ}\)

    \(15^{\circ}\)

    \(30^{\circ}\)

    \(45^{\circ}\)

    \(60^{\circ}\)

    \(75^{\circ}\)

    \(90^{\circ}\)

    \(105^{\circ}\)

    \(120^{\circ}\)

    \(135^{\circ}\)

    \(150^{\circ}\)

    \(165^{\circ}\)

    \(180^{\circ}\)

    \(m\)

    1. What happens to the slope of the line as \(\alpha\) increases toward \(90^{\circ}\)?

    2. What happens to the slope of the line as \(\alpha\) decreases toward \(90^{\circ}\)?

     

33. The angle of inclination of a line is a function of its slope, \(\alpha = g(m)\). Complete the table and sketch a graph of the function.

    \(m\)	-20	-10	-5	-2	-1	-0.75	-0.5	-0.25	0
    \(\alpha\)

    \(m\)	0.25	0.5	0.75	1	2	5	10	20
    \(\alpha\)

    1. What happens to the angle of inclination as the slope increases toward infinity?

    2. What happens to the angle of inclination as the slope decreases toward negative infinity?

     

Applications

Angles of Elevation and Angles of Depression

For the following exercises,
(a) Sketch a right triangle that illustrates the situation. Label your sketch with the given information.
(b) Choose the appropriate trigonometric ratio, write an equation, and solve the problem.

34. Estimating Height. The birdhouse in Phuong's front yard is 12 feet tall, and its shadow at 4 P.M. is 15 feet 4 inches long. What is the angle of elevation of the sun at 4 P.M.?

35. Estimating the Angle of Elevation. If a 65.2-foot sculpture casts a shadow 42.1 feet long, find the sun's angle of elevation to the nearest tenth of a degree.

36. Air Traffic Control. To measure the height of cloud cover, airport controllers fix a searchlight to shine a vertical beam on the clouds. The searchlight is 120 yards from the office. A technician in the office measures the angle of elevation to the light on the cloud cover at \(54.8^{\circ}\). What is the height of the cloud cover?

37. Salvage Crew. A salvage ship is searching for the wreck of a pirate vessel on the ocean floor. Using sonar, they locate the wreck at an angle of depression of \(36.2^{\circ}\). The depth of the ocean at their location is 260 feet. How far should they move to be directly above the wrecked vessel?

38. Volcanology. Cinder cones are the most common type of volcano in the world. They may look like an idealized depiction of a volcano as they are steep, conical hills that usually have a prominent crater at the top. Maha decides to hike to the top of Capulin Volcano, a cinder cone in New Mexico. Once at the rim, she measures the angle of depression to the surrounding plain to be \( 26.3^{ \circ } \). Her pedometer shows that her hike up to the top of Capulin Volcano covered 2,939 feet. Assuming the hike up the volcano was along a straight path, what was her elevation gain?

39. Meteorology. The radio signal from a weather balloon indicates that it is 1500 meters from a meteorologist on the ground. The angle of elevation to the balloon is \(48^{\circ}\). What is the balloon's altitude?

40. Ancient Warfare. According to Chinese legend, around 200 BC, the general Han Xin used a kite to determine the distance from his location to an enemy palace. He then dug a secret tunnel which emerged inside the palace. When the kite was directly above the palace, its angle of elevation was \(27^{\circ}\) and the string to the kite was 1850 feet long. How far did Han Xin’s troops have to dig?

41. Ocean Rescue. A hot air balloon was forced far off course, and the occupants had to make a forced landing at sea. They sent out a distress signal before taking the balloon down, and, very luckily, a United States Lifesaving Association boat was nearby. The distress signal the boat receives gives the balloon's altitude before descending as 1,128 feet. Immediately, an observer on the boat could see the balloon and measure the angle of elevation to the balloon as \( 22.1^{ \circ } \). How far would the boat travel to reach any survivors if the balloon went directly (vertically) down and crashed into the water?

42. Airplane Navigation. An airplane begins its descent when its altitude is 10 kilometers. The angle of descent should be \(3^{\circ}\) from horizontal.

    1. How far from the airport (measured along the ground) should the airplane begin its descent?

    2. How far will the airplane travel on its descent to the airport?

43. Estimating Height. Linda is on the upper floor of a tall building. She measures the angle of elevation to the building across the street to be \( 38^{ \circ } \) and the angle of depression to the base of that building as \( 65^{ \circ } \). If she knows the two buildings are 50 feet apart, how tall is the building across the street?

44. Height of a Flagpole. To estimate the height of a flagpole, Kim positions herself due south of the pole, and Matt stands due east of the pole. If they are the same distance from the pole and 50 feet from each other, find the height of the pole if the angle of elevation from the ground to the top of the pole at each person’s position is \( 49^{ \circ } \).

45. Estimating Height - D&D Edition. Deitrich Wyvernfang bravely stands in defense against a massive stone giant. He is waiting for the precise moment when the creature steps within reach of his trusty glaive; however, as with most stone giants, the beast is being measured in its approach and purposefully keeps its distance. In the intervening moments, Hoot de'La Temp, Deitrich's trusty but enigmatic companion, is overwhelmed with a momentary curiosity. In his studies of all creatures magical and curious, Hoot had never read of a stone giant this tall.
    
    Always taking advantage of the moment, Hoot places his hand on Deitrich's shoulder, casually saying, "Hold the giant off for me. This is a rare opportunity to measure the stature of such a creature in its non-deceased form." Deitrich's surprised, wide-eyed glance at Hoot is all the wizard needs to know that his friend must share his excitement about such an opportunity. Kneeling, Hoot quickly utters the arcane words for his calculate spell and measures the angle of elevation to the stone giant's head to be \( 52^{ \circ } \, 34^{\prime} \). Standing, he nods at Deitrich (who, as Hoot thinks about it, now looks disgusted for some reason) and says, "Keep him still for just a few moments longer."
    
    Hoot walks directly away from the stone giant, keeping Deitrich between himself and the dangerous creature. Each step he takes is a measured 3-foot distance. He smiles warmly as his friend decides to help him by barking "But... But... But..." in time with his steps - what a great companion. After ten such steps, Hoot turns, kneels, and casts his calculate spell again. This time, the angle of elevation to the now confused giant's head measures \( 23^{ \circ } \, 31^{\prime} \). Standing, the wizard shouts victoriously, "Got it! That beast is precisely ___ feet tall! Well, to the nearest tenth of a foot, that is. You may slay him now."
    
    What is the number of feet that Hoot estimated the height of this stone giant to be?

46. Neighborly Dispute. Nam's neighbors have built a massive pile of junk in their yard topped with a standard door frame. He would like to report them to the county for the eyesore, but the county will only give the neighbor a citation if the structure is more than 30 feet tall. Unfortunately, Nam's neighbors are not allowing him to measure the height of the junk pile on their property, so he resorts to his mathematics training. From a point on the ground (level with the base of the junk pile), Nam measures the angle of elevation to the top of the entire pile to be \( 37^{ \circ } \). From that same point, he measures the angle of elevation to the base of the door frame (on the top of the junk pile) to be \( 32^{ \circ } \). Assuming the door frame is made to industry standards, it should be 80 inches tall. How tall is the entire pile of junk (in feet)? Can the county cite the neighbor?

47. Estimating Distance. To find the distance across a ravine, Roy takes some measurements from a small airplane. When he is a short distance from the ravine at an altitude of 500 feet, he finds that the angle of depression to the near side of the ravine is \(56^{\circ}\), and the angle of depression to the far side is \(32^{\circ}\). What is the width of the ravine? (Hint: First find the horizontal distance from Roy to the near side of the ravine.)

48. Radius of the Earth. A communications satellite is in a low earth orbit (LOE) at an altitude of 400 km. From the satellite, the angle of depression to the earth’s horizon is \(19.728^{\circ}\). Use this information to calculate the radius of the planet.

49. Sunlit Area. The window in Nhat’s office is 4 feet wide and 5 feet tall. The bottom of the window is 3 feet from the floor. When the sun is at an angle of elevation of \(64^{\circ}\), what is the area of the sunny spot on the floor?

Heading Problems

For the following exercises, a ship sails from the seaport on the given heading for the given distance.
(a) Make a sketch showing the ship’s current location relative to the seaport.
(b) How far east or west of the seaport is the ship’s present location? How far north or south?

50. \(36^{\circ}, 26\) miles

51. \(124^{\circ}, 80\) km

52. \(230^{\circ}, 120\) km

53. \(318^{\circ}, 75\) miles

54. \(285^{\circ}, 32\) km

55. \(192^{\circ}, 260\) miles

56. A private plane flies 2.5 hours at 110 mph on a heading of \( 45^{ \circ } \). It then turns and continues another 1.5 hours at the same speed but on a bearing of \( 135^{ \circ } \).

    1. How far is the plane from its starting point at the end of this time?

    2. What is the heading from the pilot's starting location to her ending location?

Bearing Problems

57. Ship Navigation. A ship travels on a course of bearing \( \mathrm{N} \, 62^{ \circ } \, \mathrm{E}\) for 83.1 miles. How many miles north and how many miles east has the boat traveled?

58. Ship Navigation. A ship leaves port and travels 20 miles in the direction \( \mathrm{S} \, 29^{ \circ } \, \mathrm{W} \). It then turns \( 90^{ \circ } \) and travels another 14.4 miles in the direction \( \mathrm{N} \, 61^{ \circ } \, \mathrm{W} \). At that time, how far is the ship from port, and what is its bearing back to the port?

59. Hiking Navigation. Mike parks his car on a road that runs east-west. From this road, he sets out on a straight trail at a bearing of \( \mathrm{N} \, 65^{ \circ } \, \mathrm{E} \) and hikes for 25 miles to a water cache. After a short break, Mike turns onto another straight trail with a bearing of \( \mathrm{S} \, 25^{ \circ } \, \mathrm{E} \) for an additional 5 miles to a lunch spot.

    1. To the nearest tenth of a mile, how far was Mike from the east-west road when he reached the water cache?

    2. By the time Mike reaches the lunch spot, what is the bearing from the lunch spot back to his car?

    3. How far away (in terms of direct distance) is Mike's car from the lunch spot?

60. Locate Object - D&D Edition. The Bandits, as the heroes are known, are searching for a wand of water control that has been lost to the overgrowth of the Amedio Jungle. Hoot de'La Temp, the group's enigmatic wizard, casts a locate object spell multiple times to find the missing wand. Unfortunately, this spell only reveals the direction of an object - not its distance (another issue is that the spell can only find an object once the object is within 1,000 feet of the wizard). Frustrated and tired, the Bandits begin trekking back to town; however, after a short time, Hoot decides to give the spell one more try.
    
    "Wait! I sense the wand - it's that way," Hoot exclaims, pointing deeper into the jungle.
    
    Anlie, the Bandit's gruff ranger quickly snaps, "How far? Not all the way back to that group of gnolls, I hope."
    
    "I don't know. This magic only gives me direction - not distance," Hoot responds, frustratedly shaking his head. "This particular issue concerns the bending of the Gygaxian weaves with those of the natural weaves present in the surrounding region. I have often thought there might be a better..."
    
    Cutting the wizard off, Anlie tersely states, "I don't need a magic lesson, Hoot. A simple, 'I don't know' works."
    
    The ranger looks up through the jungle canopy, searching for signs only someone with her skill could observe. "Okay, you're pointing roughly north by \( 45^{ \circ } \) east." She looks around, spotting a path with less vegetation, "How much time do you have left on that spell?"
    
    Hoot, unruffled by the interruption, shrugs, "Seven minutes." Pausing, he adds, "And 36 seconds?"
    
    Rolling her eyes, Anlie quickly moves down the trail. "Follow me."
    
    After a few minutes of hiking through the jungle, Anlie turns to Hoot. "Now point toward the wand."
    
    Furrowing his brow and giving the ranger an appraising look, Hoot absentmindedly raises his arm to the side, pointing his index finger in the perceived direction of the magic item.
    
    Anlie looks at the wizard accusingly, "Are you sure?"
    
    Feigning a look of innocence, the wizard responds, "Nobody can be absolutely certain about these things, but I have the utmost confidence that it's roughly out there," wiggling his finger.
    
    Sighing, Anlie looks around, spending a minute looking for those hidden signs only she seems to see. "Okay, you're pointing north by \( 45^{ \circ } \) west. It's 1,050 feet back to where we started in the direction north by \( 78^{ \circ } \) west."
    
    "Are you sure," the wizard asks with a grin.
    
    Giving him a flat stare, Anlie sighs again, "Yes, and I know how far away the wand is."
    
    How far away is the wand of water control from their current position?

61. Distance. From two observation points 25 miles apart on a straight shoreline running north and south, the bearings to a boat at sea are \( \mathrm{N} \, 26^{ \circ } \, \mathrm{W} \) and \( \mathrm{S} \, 56^{ \circ } \, \mathrm{W} \). What is the shortest distance from the boat to the shore?