From the top of a lighthouse 210 feet high, the angle if depression to a boat is 27 degress. Find the distance from the boat to the food of the lighthouse. The lighthouse was built at sea level.
Solved under a previous post.
To find the distance from the boat to the foot of the lighthouse, we can use trigonometry. Specifically, we can use the tangent function, which relates the angle of depression to the opposite and adjacent sides of a right triangle.
In this case, the opposite side is the height of the lighthouse, 210 feet, and the angle of depression is 27 degrees. We want to find the adjacent side which represents the horizontal distance from the boat to the foot of the lighthouse.
Using the tangent function, we have:
tan(θ) = opposite/adjacent
tan(27°) = 210/adjacent
To solve for the adjacent side (distance from the boat to the foot of the lighthouse), we can rearrange the equation:
adjacent = 210/tan(27°)
Using a calculator, we can find the tangent of 27 degrees:
tan(27°) ≈ 0.509
Now we can substitute this value back into the equation:
adjacent = 210/0.509
adjacent ≈ 411.77 feet
Therefore, the distance from the boat to the foot of the lighthouse is approximately 411.77 feet.