From the top of a lighthouse 210 feet high, the angle of 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.
please some sugesstions.
Check 4-18-11,1:42pm post for solution.
To find the distance from the boat to the foot of the lighthouse, we can use trigonometry specifically the tangent function.
In this problem, the angle of depression is given as 27 degrees, which means it is the angle between the line of sight from the top of the lighthouse to the boat and a horizontal reference line.
Let's represent the height of the lighthouse as h, the angle of depression as θ, and the distance from the boat to the foot of the lighthouse as d.
To find d, we can use the tangent function, which relates the opposite side (height of the lighthouse) to the adjacent side (distance from the boat to the foot of the lighthouse):
tan(θ) = opposite / adjacent
In this case, the opposite side is the height of the lighthouse (h) and the adjacent side is the distance from the boat to the foot of the lighthouse (d). Therefore, the equation becomes:
tan(27 degrees) = h / d
Now we can substitute the given value of h as 210 feet and solve for d:
tan(27 degrees) = 210 / d
To isolate d, we can rearrange the equation:
d = 210 / tan(27 degrees)
Using a scientific calculator, evaluate the tangent of 27 degrees:
tan(27 degrees) ≈ 0.5095
Now substitute this value back into the equation:
d ≈ 210 / 0.5095
Calculating this expression, we find:
d ≈ 411.9205 feet
Therefore, the distance from the boat to the foot of the lighthouse is approximately 411.9205 feet.