An observer in a lighthouse is 45 meters above the surface of the water. The observer sighted two ships due east. The angle of depression to ship A is 51 degrees and the angle of depression to ship B is 24 degrees. Which ship is closer to the base of the lighthouse?
Question
An observer in a lighthouse is 45 meters above the surface of the water. The observer sighted two ships due east. The angle of depression to ship A is 51 degrees and the angle of depression to ship B is 24 degrees. Which ship is closer to the base of the lighthouse?
Solution
To solve this problem, we can use the concept of trigonometry, specifically the tangent of an angle in a right triangle.
The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the lighthouse (45 meters), and the adjacent side is the distance from the base of the lighthouse to the ship.
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For ship A, we have:
tan(51 degrees) = 45 meters / distance to ship A
Solving for the distance to ship A gives:
distance to ship A = 45 meters / tan(51 degrees)
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For ship B, we have:
tan(24 degrees) = 45 meters / distance to ship B
Solving for the distance to ship B gives:
distance to ship B = 45 meters / tan(24 degrees)
Now, we can compare the distances to ship A and ship B. The ship with the smaller distance to the lighthouse is the closer ship.
Note: The tangent of an angle decreases as the angle decreases, so the larger the angle of depression, the closer the ship is to the lighthouse. Therefore, without doing any calculations, we can say that ship A is closer to the lighthouse than ship B.
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