Answer:
- 212°
- S27°E
- 74.2 cm
- 204.9 km, 23.3°
Step-by-step explanation:
Compass directions are generally measured east or west from north or south. S32°W means the direction is 32° west (clockwise) from straight south.
Bearings are measured clockwise from north.
1.
S32°W is clockwise 32° from south, which is already clockwise 180° from north. This direction corresponds to a bearing of 180° +32° = 212°. See the first attachment for a drawing.
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2.
153° is 180° -153° = 27° counterclockwise from south. The compass direction will be expressed as S27°E. See the second attachment for a drawing.
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3.
The vertical distance from the pivot to the end of the pendulum will be L·cos(θ), where θ is the angle from the vertical. Then the height of the end of the pendulum above its rest position will be ...
L -L·cos(θ) = L(1 -cos(θ))
We want that height to be 5.4 cm, so we have ...
5.4 cm = L(1 -cos(22°)) = L(0.072816)
Then the length of the pendulum is ...
L = (5.4 cm)/0.072816 ≈ 74.2 cm
A diagram is shown in the third attachment.
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4.
a)
After 4 hours at 42 km/h, the ship is 4×42 = 168 km from the start on a bearing of 48°. After 2.5 hours at 35 km/h, the ship is 2.5×35 = 87.5 km from the turning point, on a bearing of -30°. The final position in relation to the starting point can be found a couple of ways. Perhaps most directly, it can be found using the law of cosines. A diagram is shown in the fourth attachment.
The distance from the start to the final position is the third side of a triangle with sides 168 and 87.5, and an included angle of 102°. The law of cosines tells you that length is ...
c² = a² +b² -2ab·cos(C)
c² = 168² +87.5² -2(168)(87.5)cos(102°) ≈ 41992.85
c ≈ √41992.85 ≈ 204.9 . . . km
b)
The law of sines can be used to find the internal angle opposite the side of length 87.5.
sin(B)/b = sin(C)/c
B = arcsin(b/c·sin(C)) = arcsin(87.5/204.9·sin(102°)) ≈ arcsin(0.41766)
B ≈ 24.7°
Then the bearing to the ship's final position is 48° -24.7° = 23.3°. This can be written as the compass direction N23.3°E.
