Trigonometry/ProOneSoftware-Trigonometry-Win31.iso/trig/chapter2.4r

Text File | 1995-04-09 | 6KB | 174 lines

 172
à 2.4ïNavigation Problems

ä Please solve the following problems involving navigation.

âïAn airplane is flying with instruments indicating a heading of
215° and a speed of 190 mph.ïA 30 mph wind is blowing from 115°.ïFind
the actual speed and heading of the airplane relative to the ground.ïBy
the Law of Cosines the actual speed relative to the ground is 197.43
mph, and by the Law of Sines the actual heading is 223.61°.ïPlease see
the Details for an explanation.
éSêêëFirst note that the angle at C is 100°.ïIn
êêêïthe figure, the actual speed relative to the
êêêïground is the length of side c in triangle ABC.
êêêïSince two sides and the included angle are known,
êêêïthe Law of Cosines can be used to find the length
êêêïof side c.
@fig2401.bmp,15,25
#êêêêcì = 30ì + 190ì - 2∙30∙190∙cos 100°
êêêêêè cï≈ï197.43 mph
To find the actual heading, we can use the Law of Sines to find the
angle at B in triangle ABC.
êêê30/sin Bè=è197.43/sin 100°
êêêèsin Bè≈è.14964
êêêêBè≈è8.61°
Thus, the actual heading is 215° + 8.61°ï=ï223.61°.
 1ïA lighthouse keeper, while working at the top of his light-
house, spots a boat in distress at an angle of depression of 27.3° from
the top of the lighthouse.ïIf he knows that the lighthouse is 100.2
feet high, how for out is the boat?
êêA)ï194.13 ftêêèB)ï176.3 ft

êêC)ï255 ftêêë D)ïå of ç
üêêë The angle at A is 27.3° since alternate inte-
êêêïrior angles are equal.ïYou can use the tan for-
êêêïmula in this right triangle ABC.
@fig2402.bmp,100,600
êêêêëtan 27.3°è=è100.2 ft/x
êêêêêë xè≈è194.13 ft
êêêïThus, the boat is 194.13 feet off shore.
Ç A
 2ïThe bearing from ship A to a distant ship is N39.3°E.ïFrom
ship B, which is 500 yards due east of ship A, the bearing to the dis-
tant ship is N15°W.ïHow far is ship A from the distant ship?

êêA)ï623 ydsêêëB)ï796.3 yds

êêC)ï594.72 ydsêêïD)ïå of ç
üêêë First, angle A is 50.7°, and angle B is 75° in
êêêïtriangle ABC.ïTherefore, the angle at C is 54.3°.
êêêïThe Law of Sines can be used to find side b.
@fig2403.bmp,100,600
êêêêè b/sin 75°è=è500/sin 54.3°
êêêêêëbè≈è594.72 yds
êêêïThus, ship A is 594.72 yds from the distant ship.
Ç C
 3ïAn airplane flies at a heading of 210° for 20 minutes at a
speed of 320 mph.ïHe then turns and flies at a heading of 265° for 40
minutes at the same speed.ïHow far is the plane from the starting point
if the wind is not a factor?
êêA)ï280 miêêë B)ï288.08 mi

êêC)ï296.3 miêêè D)ïå of ç
üêêë First, angle C in triangle ABC is seen to be
êêêï125°.ïThe distance AC is 2/3∙ 320 mph = 213.33
êêêïmi.ïAlso, the distance BC is 1/3∙320 mph =
êêêï106.67 mi.ïUse the Law of Cosines to find "c".
#êêêïcì = (106.67)ì+ (213.33)ì-2∙106.67∙213.33∙cos125°
êêêêêècè≈è288.08 mi
@fig2404.bmp,100,600
êêêïThus, the plane is 275.9 miles from point B.
Ç B
 4ïA small plane flies in a southeasterly direction for three
hours.ïIf it is 120 miles east and 230 miles south of its starting
point, what is the heading of the airplane?

êêA)ï110.6°êêë B)ï97.3°

êêC)ï152.45°êêëD)ïå of ç
üêêë In the figure, triangle ABC is a right tri-
êêêïangle. The tangent formula can be used to find
êêêïthe angle Θ.
êêêêê tan Θè=è230/120
êêêêêëΘè≈è62.45°
êêêïThus, the heading is 90° + 62.45°ï=ï152.45°.
@fig2405.bmp,100,600
Ç C
 5ïFrom a 100 ft high sand dune overlooking Croatan Sound, an
observer watches two ships pass directly in line with his view of a
point at Mann's Harbour.ïAt the instant the ships pass, their angles of
depression are 28.7° and 35.2°.ïHow far apart are the ships at that
moment?êA)ï97.2 ftêêëB)ï40.89 ft

êêC)ï86.3 ftêêëD)ïå of ç
üêêë In the figure, triangles ABC and DBC are right
êêêïtriangles.ïFirst, find AC using triangle ABC,
êêêïthen find DC using triangle DBC.
êêêètan 28.7° = 100/ACêïtan 35.2° = 100/DC
êêêëACï≈ï182.65 ftêè DCï≈ï141.76 ft
êêêïThus, the distance between the ships at the moment
@fig2406.bmp,100,600
êêêïthey pass is 182.65 - 141.76ï=ï40.89 feet.
Ç B
 6ïA fishing boat leaves Bogue Inlet at 5:00 AM with a bearing
of S25°E and a speed of 12 mph.ïA second boat leaves Beaufort Inlet, 18
miles due east, traveling at 24 mph with the intention of intercepting
the first boat.ïWhat bearing should the second boat take to catch the
first boat?èA)ïS63.05°Wêêè B)ïN20°W

êêC)ïS33.6°EêêëD)ïå of ç
üêêë First, note that angle A in triangle ABC is
êêêï65°.ïLet distance AC be 12∙t and distance BC be
êêêï24∙t.ïThen use the Law of Sines.
êêêêè24∙t/sin 65°è=è12∙t/sin B
êêêêêèsin Bè≈è.4532
êêêêêêBè≈è26.95°
@fig2407.bmp,100,600
êêêïThus, the bearing should be S63.05°W.
Ç A
 7ïA fishing boat leaves Bogue Inlet at 5:00 AM with a bearing
of S25°E and a speed of 12 mph.ïA second boat leaves Beaufort Inlet, 18
miles due east, traveling at 24 mph with the intention of intercepting
the first boat.ïHow long will it take the second boat to catch the
first boat?èA)ï1 1/2 hrêêè B)ï2 hr 20 min

êêC)ï.68 hrêêë D)ïå of ç
üêêë First, note that angle B was found to be
êêêï26.95° in Problem 6.ïThen, use the Law of Sines.
êêêï(side AC is 12∙t and side BC is 24∙t)
êêêê 18/sin 88.05°è=è12∙t/sin 26.95°
êêêêêë tè≈è.68 hr
êêêïThus, it should take the second boat .68 hours
@fig2407.bmp,100,600
êêêïto meet the first boat.
Ç C
 8ïAn airplane's instruments indicate a heading of 136.5° and a
speed of 210 mph.ïA 20 mph wind is blowing from 275°.ïFind the actual
speed of the airplane relative to the ground.

êêA)ï214.6 mphêêèB)ï225.37 mph

êêC)ï235.3 mphêêèD)ïå of ç
üêêë First, observe that the angle at B is 138.5°.
êêêïThen, since two sides and the included angle are
êêêïknown, use the Law of Cosines.
#êêêè bì = 210ì + 20ì - 2∙210∙20∙cos 138.5°
êêêêêbè≈è225.37 mph
êêêïThus, the actual speed of the airplane relative
@fig2408.bmp,100,600
êêêïto the ground is 225.37 mph.
Ç B
 9ïAn airplane's instruments indicate a heading of 136.5° and a
speed of 210 mph.ïA 20 mph wind is blowing from 275°.ïFind the actual
heading of the airplane relative to the ground.

êêA)ï201.4°êêë B)ï198.3°

êêC)ï206.63°êêëD)ïå of ç
üêêë First, side b was found to be 225.37 in Pro-
êêêïblem 8.ïYou can use the Law of Sines to find
êêêïangle A in triangle ABC.
êêêêè20/sin Aè=è225.37/sin 138.5°
êêêêê Aè≈è3.37°
êêêïThus, the actual heading of the airplane relative
@fig2408.bmp,100,600
êêêïto the ground is 210° - 3.37°ï=ï206.63°.
Ç C