Analysis Page 2

Law of Sines, Cosines

  1. An aircraft is spotted by two observers who are 1000 feet apart. As the airplane passes over the line joining them, each observer takes a sighting of the angle of elevation to the plane. Person A (who is on the left) measures his sighting to be 40degrees, whereas person B (who is on the right) has a sighting of 35degrees. How high is the airplane?
  1. A gold mine is dug into the face of a hill that slopes upward from the horizontal by 15 deg. If a 70-ft shaft is dug into the hill so that the shaft makes an angle of 14 deg. down from the horizontal, how far is the end of the shaft vertically from the surface?
  1. A sign 46 feet high is placed on top of an office building. From a point on the sidewalk level with the base of the building, the angle of elevation to the top of the sign and the angle of elevation to the bottom of the sign are 40 degrees and 32 degrees respectively. Find the height of the building to the nearest foot.
  1. Coast Guard Station Able is located 150 miles due south of Station Baker. A ship at sea sends an SOS call that is received by each station. The call to Station Able indicates that the ship is located N55E; the call to Station Baker indicates that the ship is located S60E.

a)  How far is each station from the ship?

b)  If a helicopter capable of flying 200 miles per hour is dispatched from the nearest station to the ship, how long will it take to reach the ship?

  1. The course for a bike race follows a triangular shaped trail. The first leg is a straight 6 kilometer ride. After that, the biker must turn 283 degrees to the left and follow the train to the next turn. Turning 269 degrees again to the left will take the biker to the finish line, which is exactly where he started the race. How long is the entire race?
  1. Two observers, who are two miles apart on a horizontal plane, observe a balloon the same vertical plane with themselves. The angles of elevation are 50' and 65' respectively. Find the height of the balloon

a)  if it is between the observers

b)  if it is on the same side of both of them.

  1. Two airplanes leave an airport, and the angle between their flight paths is 40º. An hour later, one plane has traveled 300 miles while the other has traveled 200 miles. How far apart are the planes at this time?
  1. Two ships leave port at 4 p.m. One is headed at a bearing of N 38 E and is traveling at 11.5 miles per hour. The other is traveling 13 miles per hour at a bearing of S 47 E. How far apart are they when dinner is served at 6 p.m.?

9.  A radio tower 500 feet high is located on the side of a hill with an inclination to the horizontal of 5 degrees. How long should two guy wires be if they are to connect to the top of the tower and be secured at two points 100 feet directly above and directly below the base of the tower?

10.  In a rhombus with a side length of 24, the longer diagonal is 36 inches. Find the measure of the largest angle of the rhombus.

11.  Three sides of a triangle measure 20”, 30” and 40”. Find the largest angle of the triangle.

12.  A boat race runs along a triangular course marked by buoys A, B, and C. The race starts with the boats headed west for 3700 meters. The other two sides of the course lie to the north of the first side, and their lengths are 1700 meters and 3000 meters. Draw a figure that gives a visual representation of the problem, and find the bearings for the last two legs of the race.