1. Trina and Mazaheer are standing on the same side of a red maple tree. The angle of elevation from Mazaheer to the tree top is 67° and the angle of elevation from Trina to the tree top is 53°. If Mazaheer and Trina are 9.3 feet apart and Mazaheer is closer to the tree than Trina, how tall is the tree?
2. Two roads separate from a village at an angle of 37°. Two cyclists leave the village at the same time. One travels 7.5 km/h on one road and the other travels 10.0 km/h on the other road. How far apart are the cyclists after 2 hours?
2. Two roads separate from a village at an angle of 37°. Two cyclists leave the village at the same time. One travels 7.5 km/h on one road and the other travels 10.0 km/h on the other road. How far apart are the cyclists after 2 hours?
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1. You have an oblique triangle in which
T = Trina's position, M = Mazaheer's position, P = peak of tree
Angle TMP = 180 deg - 67 deg = 113 degrees
Angle MTP = 53 degrees
Therefore angle MPT = 180 deg - 166 deg = 14 degrees
Side MT = 9.3 feet
You are trying to determine EITHER of the other two sides
in order to find the height of the tree.
I'll go for side MP, using the Law of Sines:
sin(53)/(side MP) = sin(14)/(side MT)
side MP = (9.3 feet) sin(53) / sin(14) = 30.70 feet
Hence, the height of the tree is (30.70 feet)*sin(67) = 28.3 feet
#2. You have an oblique triangle with a 37-degree angle
included between two sides whose lengths are 15.0 km and 20.0 km.
The third side is found from the Law of Cosines:
c^2 = 15^2 + 20^2 - 2(15)(20) cos(37 deg) = 145.82 km^2
c = 12.1 km
T = Trina's position, M = Mazaheer's position, P = peak of tree
Angle TMP = 180 deg - 67 deg = 113 degrees
Angle MTP = 53 degrees
Therefore angle MPT = 180 deg - 166 deg = 14 degrees
Side MT = 9.3 feet
You are trying to determine EITHER of the other two sides
in order to find the height of the tree.
I'll go for side MP, using the Law of Sines:
sin(53)/(side MP) = sin(14)/(side MT)
side MP = (9.3 feet) sin(53) / sin(14) = 30.70 feet
Hence, the height of the tree is (30.70 feet)*sin(67) = 28.3 feet
#2. You have an oblique triangle with a 37-degree angle
included between two sides whose lengths are 15.0 km and 20.0 km.
The third side is found from the Law of Cosines:
c^2 = 15^2 + 20^2 - 2(15)(20) cos(37 deg) = 145.82 km^2
c = 12.1 km