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Bearing: First-Type

One method of giving direction, or bearing, involves reference to a north–south line using an acute angle. For example, $N 55 ° W$ $N 55 ° W$ means $55 °$ $55 °$ west of north and $S 67 ° E$ $S 67 ° E$ means $67 °$ $67 °$ east of south.

A second-type of bearing that gives directions in degrees from north is covered in Section 6.3.

Example 7 Distance to a Forest Fire.

A forest ranger at point A sights a fire directly south. A second ranger at point B, 7.5 mi east of the first ranger, sights the same fire at a bearing of $S 27 ° 23' W$ $S 27 ° 23' W$ . How far from A is the fire?

Solution

We first find the complement of $27 ° 23'$ $27 ° 23'$ :

\begin{array}{l} B & = & 90 ° - 27 ° 23' & Angle B is opposite side d in triangle BAF . \\ = & 62 ° 3 7^{'} \\ \approx & 62.62 ° . \end{array}

$\begin{array}{l} B & = & 90 ° - 27 ° 23' & Angle B is opposite side d in triangle BAF . \\ = & 62 ° 3 7^{'} \\ \approx & 62.62 ° . \end{array}$

From the figure shown above, we see that the desired distance d is a side of right triangle BAF. We have

\begin{array}{l} \frac{d}{7.5 mi} & \approx & \tan 62.62 ° \\ d & \approx & 7.5 mi \cdot \tan 62.62 ° \approx 14.5 mi . \end{array}

$\begin{array}{l} \frac{d}{7.5 mi} & \approx & \tan 62.62 ° \\ d & \approx & 7.5 mi \cdot \tan 62.62 ° \approx 14.5 mi . \end{array}$

The forest ranger at point A is about 14.5 mi from the fire.

Now Try Exercise 37.

Example 8 U.S. Cellular Field.

In U.S. Cellular Field, the home of the Chicago White Sox baseball team, the first row of seats in the upper deck is farther away from home plate than the last row of seats in the original Comiskey Park, which it replaced. Although there is no obstructed view in U.S. Cellular Field, some of the fans still complain about the distance from home plate to the upper deck of seats. From a seat in the last row of the upper deck directly behind the batter, the angle of depression to home plate is $29.9 °$ $29.9 °$ , and the angle of depression to the pitcher’s mound is $24.2 °$ $24.2 °$ . Find the viewing distance to home plate and the viewing distance to the pitcher’s mound.

Solution

From geometry, we know that $θ_{1} = 29.9 °$ $θ_{1} = 29.9 °$ and $θ_{2} = 24.2 °$ $θ_{2} = 24.2 °$ . The standard distance from home plate to the pitcher’s mound is 60.5 ft. In the drawing, we let $d_{1} = the viewing distance to home plate$ $d_{1} = the viewing distance to home plate$ , $d_{2}$ $d_{2}$ the viewing distance to the pitcher’s mound, h the elevation of the last row, and x the horizontal distance from the batter to a point directly below the seat in the last row of the upper deck.

We begin by determining the distance x. We use the tangent function with $θ_{1} = 29.9 °$ $θ_{1} = 29.9 °$ and $θ_{2} = 24.2 °$ $θ_{2} = 24.2 °$ :

\tan 29.9 ° = \frac{h}{x} and \tan 24.2 ° = \frac{h}{x + 60.5}

$\tan 29.9 ° = \frac{h}{x} and \tan 24.2 ° = \frac{h}{x + 60.5}$

h = x \tan 29.9 ° and h = (x + 60.5) \tan 24.2 ° .

$h = x \tan 29.9 ° and h = (x + 60.5) \tan 24.2 ° .$

Then substituting $x \tan 29.9 °$ $x \tan 29.9 °$ for h in the second equation, we obtain

x \tan 29.9 ° = (x + 60.5) \tan 24.2 ° .

$x \tan 29.9 ° = (x + 60.5) \tan 24.2 ° .$

Solving for x, we get

$\begin{array}{l} x \tan 29.9 ° & = & x \tan 24.2 ° + 60.5 \tan 24.2 ° \\ x \tan 29.9 ° - x \tan 24.2 ° & = & x \tan 24.2 ° + 60.5 \tan 24.2 ° - x \tan 24.2 ° \\ x (\tan 29.9 ° - \tan 24.2 °) & = & 60.5 \tan 24.2 ° \\ x & = & \frac{60.5 \tan 24.2 °}{\tan 29.9 ° - \tan 24.2 °} \\ x & \approx & 216.5. \end{array}$ $\begin{array}{l} x \tan 29.9 ° & = & x \tan 24.2 ° + 60.5 \tan 24.2 ° \\ x \tan 29.9 ° - x \tan 24.2 ° & = & x \tan 24.2 ° + 60.5 \tan 24.2 ° - x \tan 24.2 ° \\ x (\tan 29.9 ° - \tan 24.2 °) & = & 60.5 \tan 24.2 ° \\ x & = & \frac{60.5 \tan 24.2 °}{\tan 29.9 ° - \tan 24.2 °} \\ x & \approx & 216.5. \end{array}$

We can then find $d_{1}$ $d_{1}$ and $d_{2}$ $d_{2}$ using the cosine function:

\cos 29.9 ° = \frac{216.5}{d_{1}} and \cos 24.2 ° = \frac{216.5 + 60.5}{d_{2}}

$\cos 29.9 ° = \frac{216.5}{d_{1}} and \cos 24.2 ° = \frac{216.5 + 60.5}{d_{2}}$

\begin{array}{l} d_{1} & = & \frac{216.5}{\cos 29.9 °} & and & d_{2} & = & \frac{277}{\cos 24.2 °} \\ d_{1} & \approx & 249.7 & d_{2} & \approx & 303.7. \end{array}

$\begin{array}{l} d_{1} & = & \frac{216.5}{\cos 29.9 °} & and & d_{2} & = & \frac{277}{\cos 24.2 °} \\ d_{1} & \approx & 249.7 & d_{2} & \approx & 303.7. \end{array}$

The viewing distance to home plate is about 250 ft,* and the viewing distance to the pitcher’s mound is about 304 ft.

Now Try Exercise 25.

6.2 Exercise Set

In Exercises 1–6, solve the right triangle.

In Exercises 7–16, solve the right triangle. (Standard lettering has been used.)

7. $A = 87 ° 43', a = 9.73$ $A = 87 ° 43', a = 9.73$
8. $a = 12.5, b = 18.3$ $a = 12.5, b = 18.3$
9. $b = 100, c = 450$ $b = 100, c = 450$
10. $B = 56.5 °, c = 0.0447$ $B = 56.5 °, c = 0.0447$
11. $A = 47.58 °, c = 48.3$ $A = 47.58 °, c = 48.3$
12. $B = 20.6 °, a = 7.5$ $B = 20.6 °, a = 7.5$
13. $A = 35 °, b = 40$ $A = 35 °, b = 40$
14. $B = 69.3 °, b = 93.4$ $B = 69.3 °, b = 93.4$
15. $b = 1.86, c = 4.02$ $b = 1.86, c = 4.02$
16. $a = 10.2, c = 20.4$ $a = 10.2, c = 20.4$
17. Aerial Photography. An aerial photographer who photographs farm properties for a real estate company has determined from experience that the best photo is taken at a height of approximately 475 ft and a distance of 850 ft from the farmhouse. What is the angle of depression from the plane to the house?
18. Memorial Flag Case. A tradition in the United States is to drape an American flag over the casket of a deceased U.S. Forces veteran. At the burial, the flag is removed, folded into a triangle, and presented to the family. The folded flag will fit in an isosceles right triangle case, as shown below. The inside dimension across the bottom of the case is $21 \frac{1}{2}$ $21 \frac{1}{2}$ in. (Source: Bruce Kieffer, Woodworker’s Journal, August 2006). Using trigonometric functions, find the length x and round the answer to the nearest tenth of an inch.
19. Zip Line. The ZipRider^®, a zip line at Icy Straight Point, Alaska, is 5495 ft long, and has a vertical drop of 1320 ft (Source: www.ziprider.com). Find its angle of depression.
20. Setting a Fishing Reel Line Counter. A fisherman who is fishing 50 ft directly out from a visible tree stump near the shore wants to position his line and bait approximately $N 35 ° W$ $N 35 ° W$ of the boat and west of the stump. Using the right triangle shown in the drawing, determine the reel’s line counter setting, to the nearest foot, to position the line directly west of the stump.
21. Framing a Closet. Sam is framing a closet under a stairway. The stairway is 16 ft 3 in. long, and its angle of elevation is $38 °$ $38 °$ . Find the depth of the closet to the nearest inch.
22. Loading Ramp. Charles needs to purchase a custom ramp to use while loading and unloading a garden tractor. When down, the tailgate of his truck is 38 in. from the ground. If the recommended angle that the ramp makes with the ground is $28 °$ $28 °$ , approximately how long must the ramp be?
23. Longest Escalator. The longest escalator in the world is in the subway system in St. Petersburg, Russia. The escalator is 1084.6 ft long and drops a vertical distance of 195.8 ft. What is its angle of depression?
24. Cloud Height. To measure cloud height at night, a vertical beam of light is directed on a spot on the cloud. From a point 135 ft away from the light source, the angle of elevation to the spot is found to be $67.35 °$ $67.35 °$ . Find the height of the cloud to the nearest foot.
25. Mount Rushmore National Memorial. While visiting Mount Rushmore in Rapid City, South Dakota, Landon approximated the angle of elevation to the top of George Washington’s head to be $35 °$ $35 °$ . After walking 250 ft closer, he guessed that the angle of elevation had increased by $15 °$ $15 °$ . Approximate the height of the Mount Rushmore memorial, to the top of George Washington’s head. Round the answer to the nearest foot.
26. Golden Gate Bridge. The Golden Gate Bridge has two main towers of equal height that support the two main cables. A visitor on a tour boat passing through San Francisco Bay views the top of one of the towers and estimates the angle of elevation to be $30 °$ $30 °$ . After sailing 670 ft closer, he estimates the angle of elevation to this same tower to be $50 °$ $50 °$ . Approximate the height of the tower to the nearest foot.
27. Inscribed Pentagon. A regular pentagon is inscribed in a circle of radius 15.8 cm. Find the perimeter of the pentagon.
28. Height of a Weather Balloon. A weather balloon is directly west of two observing stations that are 10 mi apart. The angles of elevation of the balloon from the two stations are $17.6 °$ $17.6 °$ and $78.2 °$ $78.2 °$ . How high is the balloon?
29. Height of a Building. A window washer on a ladder looks at a nearby building 100 ft away, noting that the angle of elevation to the top of the building is $18.7 °$ $18.7 °$ and the angle of depression to the bottom of the building is $6.5 °$ $6.5 °$ . How tall is the nearby building?
30. Height of a Kite. For a science fair project, a group of students tested different materials used to construct kites. Their instructor provided an instrument that accurately measures the angle of elevation. In one of the tests, the angle of elevation was $63.4 °$ $63.4 °$ with 670 ft of string out. Assuming the string was taut, how high was the kite?
31. Quilt Design. Nancy is designing a quilt that she will enter in the quilt competition at the State Fair. The quilt consists of twelve identical squares with 4 rows of 3 squares each. Each square is to have a regular octagon inscribed in a circle, as shown in the figure. Each side of the octagon is to be 7 in. long. Find the radius of the circumscribed circle and the dimensions of the quilt. Round the answers to the nearest hundredth of an inch.
32. Rafters for a House. Blaise, an architect for luxury homes, is designing a house that is 46 ft wide with a roof whose pitch is $11 / 12$ $11 / 12$ . Determine the length of the rafters needed for this house. Round the answer to the nearest tenth of a foot.
33. Rafters for a Medical Office. The pitch of the roof for a medical office needs to be $5 / 12$ $5 / 12$ . If the building is 33 ft wide, how long must the rafters be?
34. Angle of Elevation. The Millau Viaduct in southern France is the tallest cable-stayed bridge in the world (Source: www.abelard.org/france/viaduct-de-millau.php). What is the angle of elevation of the sun when a pylon with height 343 m casts a shadow of 186 m?
35. Distance Between Towns. From a hot-air balloon 2 km high, the angles of depression to two towns in line with the balloon and on the same side of the balloon are $81.2 °$ $81.2 °$ and $13.5 °$ $13.5 °$ . How far apart are the towns?
36. Distance from a Lighthouse. From the top of a lighthouse 55 ft above sea level, the angle of depression to a small boat is $11.3 °$ $11.3 °$ . How far from the foot of the lighthouse is the boat?
37. Lightning Detection. In extremely large forests, it is not cost-effective to position forest rangers in towers or to use small aircraft to continually watch for fires. Since lightning is a frequent cause of fire, lightning detectors are now commonly used instead. These devices not only give a bearing on the location but also measure the intensity of the lightning. A detector at point Q is situated 15 mi west of a central fire station at point R. The bearing from Q to where lightning hits due south of R is $S 37.6 ° E$ $S 37.6 ° E$ . How far is the hit from point R?
38. Length of an Antenna. A vertical antenna is mounted atop a 50-ft pole. From a point on level ground 75 ft from the base of the pole, the antenna subtends an angle of $10.5 °$ $10.5 °$ . Find the length of the antenna.
39. Lobster Boat. A lobster boat is situated due west of a lighthouse. A barge is 12 km south of the lobster boat. From the barge, the bearing to the lighthouse is $N 63 ° 20' E$ $N 63 ° 20' E$ . How far is the lobster boat from the lighthouse?

Skill Maintenance

Find the distance between the points. [1.1]

40. $(- 9, 3)$ $(- 9, 3)$ and (0, 0)
41. $(8, - 2)$ $(8, - 2)$ and $(- 6, - 4)$ $(- 6, - 4)$
42. Convert to a logarithmic equation: $e^{4} = t$ $e^{4} = t$ . [5.3]
43. Convert to an exponential equation: $\log 0.001 = - 3$ $\log 0.001 = - 3$ . [5.3]

Synthesis

44. Diameter of a Pipe. A V-gauge is used to find the diameter of a pipe. The advantage of such a device is that it is rugged, it is accurate, and it has no moving parts to break down. In the figure, the measure of angle AVB is $54 °$ $54 °$ . A pipe is placed in the V-shaped slot and the distance VP is used to estimate the diameter. The line VP is calibrated by listing as its units the corresponding diameters. This, in effect, establishes a function between VP and d.
1. a) Suppose that the diameter of a pipe is 2 cm. What is the distance VP?
2. b) Suppose that the distance VP is 3.93 cm. What is the diameter of the pipe?
3. c) Find a formula for d in terms of VP.
4. d) Find a formula for VP in terms of d.
45. Find h, to the nearest tenth.
46. Sound of an Airplane. It is common experience to hear the sound of a low-flying airplane and look at the wrong place in the sky to see the plane. Suppose that a plane is traveling directly at you at a speed of 200 mph and an altitude of 3000 ft, and you hear the sound at what seems to be an angle of inclination of $20 °$ $20 °$ . At what angle $θ$ $θ$ should you actually look in order to see the plane? Consider the speed of sound to be $1100 ft / \sec$ $1100 ft / \sec$ .

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