What is the angle of depression from the top of a cliff to the bottom of a lake, if the height of the cliff is 50 meters and the horizontal distance from the top of the cliff to the edge of the lake is 100 meters?
A. 30°
B. 45°
C. 60°
D. 75°
To find the angle of depression, we need to visualize a right triangle formed by the cliff, the lake, and a line of sight from the top of the cliff to the bottom of the lake. The angle of depression is the angle formed between the horizontal line and the line of sight.
In this case, the height of the cliff is 50 meters, and the horizontal distance from the top of the cliff to the edge of the lake is 100 meters. This horizontal distance is the adjacent side of the right triangle, and the height of the cliff is the opposite side.
We can use the tangent function to find the angle of depression:
tan(angle of depression) = opposite / adjacent
Since the opposite side is the height of the cliff (50 meters) and the adjacent side is the horizontal distance (100 meters), we have:
tan(angle of depression) = 50 / 100
To find the angle of depression, we can use the inverse tangent function (also known as arctangent) to solve for the angle:
angle of depression = arctan(50 / 100)
Using a calculator or trigonometric table, we can find that arctan(50 / 100) is approximately 26.57 degrees.
Therefore, the angle of depression from the top of the cliff to the bottom of the lake is approximately 26.57 degrees.
Since none of the given answer options match exactly, the closest option is 30° (option A).