A boat is heading towards a lighthouse, whose beacon-light is 139 feet above the water. The boat’s crew measures the angle of elevation to the beacon, 5degrees ∘ . What is the ship’s horizontal distance from the lighthouse (and the shore)? Round your answer to the nearest hundredth of a foot if necessary.
Question
A boat is heading towards a lighthouse, whose beacon-light is 139 feet above the water. The boat’s crew measures the angle of elevation to the beacon, 5degrees ∘ . What is the ship’s horizontal distance from the lighthouse (and the shore)? Round your answer to the nearest hundredth of a foot if necessary.
Solution
To solve this problem, we can use the tangent of the angle of elevation, which is the ratio of the opposite side (height of the lighthouse) to the adjacent side (distance from the lighthouse).
The formula for tangent is:
tan(θ) = opposite/adjacent
We know the opposite side (height of the lighthouse) is 139 feet and the angle of elevation (θ) is 5 degrees. We want to find the adjacent side (distance from the lighthouse).
Rearranging the formula to solve for the adjacent side gives us:
adjacent = opposite/tan(θ)
Substituting the given values into the formula gives us:
adjacent = 139/tan(5)
Using a calculator to find the tangent of 5 degrees and then divide 139 by that number gives us the distance from the lighthouse.
Make sure your calculator is set to degrees, not radians. The tangent of 5 degrees is approximately 0.0875.
So, the calculation is:
adjacent = 139/0.0875 ≈ 1588.57 feet
So, the boat is approximately 1588.57 feet away from the lighthouse. If we round this to the nearest hundredth of a foot, we get 1588.57 feet.
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