Question

A helicopter hovers 1250 feet

above a small island. The figure
shows that the angle of
depression from the helicopter
to point P is 47º. How far off
the coast, to the nearest foot, is
the island?

Answer 1

The student's question involves using trigonometry to find the horizontal distance to an island given a helicopter's altitude and the angle of depression. The tangent function is used to calculate this distance from the height of 1250 feet and the 47-degree angle.

Step-by-step explanation:

The student is asking a math-related question that involves using trigonometry to find the distance from a helicopter to a point of interest on the ground (point P), given the altitude of the helicopter and the angle of depression. The helicopter hovers 1250 feet above the island, and the angle of depression to point P is 47°.

To solve this, we need to use trigonometric ratios. The angle of depression is equal to the angle of elevation from point P to the helicopter because these angles are alternate interior angles formed by a transversal cutting two parallel lines (the line representing the horizon and the line from the helicopter straight down to its shadow directly below it on the ground).

The distance to the island (x) can be found by using the tangent function, which is the ratio of the opposite side to the adjacent side in a right-angled triangle. Therefore, we write the equation:

tangent(47°) = opposite / adjacent

tangent(47°) = 1250 / x

To find x, we calculate:

x = 1250 / tangent(47°)

Using a calculator with trigonometric functions, we can find the value of x to the nearest foot. This will give us the horizontal distance from the helicopter directly above the island to point P, effectively how far off the coast the island is.

Answer 2

Create a right triangle.

The height of the leg will be 1250 feet. We are given an angle of depression of 47°, so we can draw an imaginary line parallel to the base of the triangle.

By the AIA Theorem, the angle of the bottom right side of the right triangle will be 47 degrees.

We want to find the distance between the coast and the island. In other words, find the length of the base of the triangle. We can do so with trigonometric ratios.

tan47 = 1250/x

x = 1166.