Answer:
- 15.8 ft/s, 71.6°
- 6209 m
Step-by-step explanation:
You want the speed and direction of travel of a hot-air balloon rising at 15 ft/s in a horizontal wind at 5 ft/s, and you want the distance between lighthouses on a N-S line if one sees a ship at 3742 m distant on a bearing of 130°, and the other sees the ship on a bearing of 37°.
1. Balloon
A diagram of the problem is shown in the first attachment. The speed of the balloon is the hypotenuse of a triangle with side lengths 15 m/s and 5 m/s, so will be ...
s² = 15² +5² = 250
s = √250 = 5√10 ≈ 15.8 . . . . ft/s
The direction of travel can be found from the tangent function.
Tan = Opposite/Adjacent
tan(θ) = (15 ft/s)/(5 ft/s) = 3
θ = arctan(3) ≈ 71.6°
The balloon is rising at a rate of 15.8 ft/s at an angle of 71.6° from the horizontal.
2. Ship
The second attachment shows a model of the problem. The 130° bearing is effectively an exterior angle to the triangle, so is the sum of the opposite interior angles. That means the angle at the ship between the two lighthouses is ...
130° -37° = 93°
Now we have enough information to solve the problem using the Law of Sines:
s/sin(S) = b/sin(B)
s = b·sin(S)/sin(B) = 3742·sin(93°)/sin(37°) ≈ 6209
The distance between lighthouses is about 6209 meters.
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Additional comment
The third attachment shows the calculator input and output for these problems. It is able to translate the balloon vector components to a magnitude and angle in one step. It can be helpful to learn to use your calculator for vector math like this. (Note: the calculator is set to Degrees mode.)
