Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30deg and 45deg respectively. If the lighthouse is 100 m high, the distance between the two ships is:
Question
Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30deg and 45deg respectively. If the lighthouse is 100 m high, the distance between the two ships is:
Solution
To solve this problem, we need to use the concept of trigonometry.
Step 1: Let's denote the distances from the ships to the lighthouse as A and B for the ships that see the lighthouse at angles of 30 degrees and 45 degrees respectively.
Step 2: We know that the tangent of an angle in a right triangle is equal to the ratio of the opposite side to the adjacent side. So, we can write the following equations using the given angles and the height of the lighthouse (100m):
tan(30) = height / A tan(45) = height / B
Step 3: Solve these equations for A and B.
For A, we get: A = height / tan(30) = 100 / tan(30) = 100 * sqrt(3) = 173.2 m
For B, we get: B = height / tan(45) = 100 / tan(45) = 100 m
Step 4: The distance between the two ships is the sum of A and B, which is 173.2 m + 100 m = 273.2 m.
So, the distance between the two ships is approximately 273.2 meters.
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