Question

an observatory is 150 feet high. when a person is standing in the observatory looking at an island, the angle of depression is 25°. what is the approximate horizontal distance from the observatory to the island?
image of a right triangle with observatory at the top right vertex, vertical side labeled 150 ft, horizontal side from island to the right angle, and angles of 25° at the island and at the observatory (angle of depression).
a. 70 feet
b. 322 feet
c. 355 feet
d. 166 feet

Explanation:

Step1: Use the tangent function

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 25^{\circ}\), the opposite side is the height of the observatory (\(h = 150\) ft), and the adjacent side \(x\) is the horizontal distance we want to find. So, \(\tan(25^{\circ})=\frac{150}{x}\).

Step2: Solve for \(x\)

We can rewrite the equation as \(x=\frac{150}{\tan(25^{\circ})}\). Since \(\tan(25^{\circ})\approx0.4663\), then \(x=\frac{150}{0.4663}\approx322\) ft.

Answer:

B. 322 feet