Analysis
Law of Sines, Cosines
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1. John wants to measure the height of a tree. He walks exactly 100 feet
from the base of the tree and looks up. The angle from the ground to the
top of the tree is 33º. This particular tree grows at an angle of 83º with
respect to the ground rather than vertically (90º). How tall is the tree?
60.60 ft or 71.1 ft
2. A building is of unknown height. At a distance of 100 feet away from the
building, an observer notices that the angle of elevation to the top of the
building is 41º and that the angle of elevation to a poster on the side of
the building is 21º. How far is the poster from the roof of the building?
48.54 ft
3. When the angle of elevation of the sun is 62o, a telephone pole tilted at
an angle of 7o away from the sun casts a shadow 30 feet long on the
ground. To tenths place, approximate the length of the phone pole.
46.18 ft
4. To find the distance between two points A and B that are on opposite
sides of a river, a surveyor measures a distance on the same side of the
river as point A. The distance to this point is 240 feet and call it point C.
He then measures the angles from A to B as 62o and measures the angle
from C to B as 55o. To tenths place, approximate the distance from A to
B.
220.65 ft
5. A flagpole is situated on a hill that makes an angle of 18° with the
horizontal. How tall is the flagpole if its shadow is 14 meters when the
angle of elevation of the sun is 31°?
3.67 ft
6. The angle of elevation to the top of a building is 21°. At a point 80 feet
closer, the angle of elevation to the top of the same building is 33°.
Approximate the height of the building.
75.1 ft
Analysis
Law of Sines, Cosines Problem Set
Page 2
7. You are taking a flight from Daytona Beach to St. Louis. There is a
stopover in Atlanta. The bearing from Daytona Beach to Atlanta is
N 38.3 W. The bearing from Daytona straight to St. Louis is N 43.4 W.
The distance of the first leg of the trip from Daytona to Atlanta is
approximately 599.8 kilometers. From Atlanta to St. Louis is another
748.4 kilometers.
a. What is the total length of the trip with the stopover?
2692.12 miles
b. How much shorter would it be to fly straight to St. Louis from
Daytona Beach without a stopover?
4.28 miles
8. An observer is near a river and wants to calculate the distance across the
river. He measures the angle between his observations of two points on
the shore, one on his side and one on the other side, to be 28º. The
distance between him and the point on his side of the river can be
measured and is 300 feet. The angle formed by him, the point on his side
of the river, and the point directly on the opposite side of the river is
128º. What is the distance across the river?
581.2 ft
9. 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?
195.13 ft
10. Two people are walking toward each other on a path through the park.
The path runs east and west. A hot air balloon is directly above the path
between them. One of the walkers, a female, sees the balloon when
looking east at an angle of elevation of 46°. The other walker, a male,
sees the balloon looking west at an angle of elevation of 72°. If the
walkers are 55 yards apart, how far is the balloon from the male
observer?
44.81 yards
Analysis
Law of Sines, Cosines Problem Set
Page 3
11. To approximate the length of a lake, a surveyor starts at one end of the
lake and walks 245 yards. He then turns 110 and walks 270 yards until he
arrives at the other end of the lake. Approximately how long is the lake?
296.1 yards
12. 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.?
36.18 miles
Analysis
Law of Sines, Cosines Problem Set
Page 4
13. You are heading to Beech Mountain for a ski trip. Unfortunately, state
road 105 in North Carolina is blocked off due to a chemical spill. You have to
get to Tynecastle Highway which leads to the resort at which you are
staying. NC-105 would get you to Tynecastle Hwy in 12.8 miles. The detour
begins with a 18 veer off onto a road that runs through the local city.
After 6 miles, there is another turn that leads to Tynecastle Hwy. Assuming
that both roads on the detour are straight, how many extra miles are you
traveling to reach your destination?
.53 m,iles
14. The distance on a map from the airport in Miami, FL to the one in Nassau,
Bahamas is 295 kilometers due east. Bangor, Maine is northeast of both
cities; its airport is 2350 kilometers from Miami and 2323 kilometers
from Nassau. What bearing would a plane need to take to fly from
Nassau to Bangor?
N1.64E
Analysis
Law of Sines, Cosines Problem Set
Page 5
15. After the hurricane, the small tree in my neighbor’s yard was leaning. To
keep it from falling, we nailed a 6-foot strap into the ground 4 feet from
the base of the tree. We attached the strap to the tree 3½ feet above
the ground. How far from vertical was the tree leaning?
16.09°
16. Two ships leave a harbor at the same time, traveling on courses that have
an angle of 140 degrees between them. If the first ship travels at 26
miles per hour and the second ship travels at 34 miles per hour, how far
apart are the two ships after 3 hours?
169.34 miles
17. A triangluar parcel of land has sides of length 680 feet, 320 feet, and
802 feet. What is the area of the parcel of land? If land is valued at
$2100 per acre (1 acre is 43560 square feet), what is the value of the
parcel of land?
$5159.36