A boat heading out to sea starts out at Point AA, at a horizontal distance of 1159 feet from a lighthouse/the shore. From that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon-light from that point to be 16degrees ∘ . At some later time, the crew measures the angle of elevation from point BB to be 8degrees ∘ . Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.
Question
A boat heading out to sea starts out at Point AA, at a horizontal distance of 1159 feet from a lighthouse/the shore. From that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon-light from that point to be 16degrees ∘ . At some later time, the crew measures the angle of elevation from point BB to be 8degrees ∘ . Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.
Solution
To solve this problem, we can use the tangent of the angles of elevation, which is the ratio of the opposite side (the height of the lighthouse) to the adjacent side (the horizontal distance from the lighthouse).
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First, we can find the height of the lighthouse using the first angle of elevation (16 degrees) and the initial horizontal distance (1159 feet).
tan(16) = height / 1159 height = tan(16) * 1159
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Next, we can find the new horizontal distance from the lighthouse to point B using the second angle of elevation (8 degrees) and the height of the lighthouse.
tan(8) = height / distance_B distance_B = height / tan(8)
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Finally, we can find the distance from point A to point B by subtracting the initial horizontal distance from the new horizontal distance.
distance_AB = distance_B - 1159
Let's calculate these steps:
- height = tan(16) * 1159 ≈ 326.7 feet
- distance_B = 326.7 / tan(8) ≈ 2342.3 feet
- distance_AB = 2342.3 - 1159 ≈ 1183.3 feet
So, the distance from point A to point B is approximately 1183 feet when rounded to the nearest foot.
Similar Questions
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