Questions: A helicopter hovers 1150 feet above a small island. The figure shows that the angle of depression from the helicopter to point P is 41°. How far off the coast is the island? The island is off the coast. (Round to the nearest whole number as needed.)

A helicopter hovers 1150 feet above a small island. The figure shows that the angle of depression from the helicopter to point P is 41°. How far off the coast is the island?

The island is off the coast. (Round to the nearest whole number as needed.)

Transcript text: A helicopter hovers 1150 feet above a small island. The figure shows that the angle of depression from the helicopter to point P is $41^{\circ}$. How far off the coast is the island? The island is $\square$ $\square$ off the coast. (Round to the nearest whole number as needed.)

Solution

Solution Steps

Step 1: Visualize the problem

The helicopter is 1150 feet above the island. The angle of depression from the helicopter to point P is 41°. We are asked to find the distance 'd' from the island to the coast, which is the horizontal distance. This forms a right triangle where 1150 ft is the side opposite to the angle of 41°, and 'd' is the side adjacent to it.

Step 2: Set up the trigonometric equation

We can use the tangent function to relate the angle, the opposite side, and the adjacent side.
tan(41°) = opposite/adjacent = 1150/d

Step 3: Solve for d

d = 1150 / tan(41°)

Step 4: Calculate

d ≈ 1150 / 0.8391 ≈ 1370.58