An observatory is 150 feet high. When a person is standing in the observatory looking at an island, the angle of depression is 25 degrees
. What is the approximate horizontal distance from the observatory to the island?
The picture shows a right-angled triangle. The length of the opposite side is 150 ft. A dotted horizontal line (Observatory) touches the top vertex. The angle of the left vertex (Island) and the outer angle of the top vertex is 25 degrees.
We can use the tangent function to determine the horizontal distance from the observatory to the island.
In the right-angled triangle, the opposite side is 150 ft. Let's denote the horizontal distance as x.
The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, the angle of depression is 25 degrees, so we have:
tan(25°) = opposite/adjacent
tan(25°) = 150/x
To solve for x, we can rearrange the equation:
x = 150/tan(25°)
Using a calculator, we can find that tan(25°) is approximately 0.4663. Plugging this value into the equation, we get:
x = 150/0.4663
x ≈ 321.45 ft
Therefore, the approximate horizontal distance from the observatory to the island is 321.45 feet.