Question

A rescue team is deployed via helicopter to rescue a struggling swimmer. The helicopter locates the swimmer, who is several feet from a tethered buoy. The helicopter hovers 400 feet over the buoy as its team prepares to rescue the swimmer. Initially, the swimmer sees the helicopter at an angle of elevation of x degrees. The swimmer then swims 5 feet closer to the buoy, and now sees the helicopter at an angle of y degrees. Which trigonometric equation correctly models the distance the swimmer swam toward the buoy in terms of x and y?

Answer 1

To find the distance the swimmer moves toward the buoy in terms of angles x and y, use the tangent trigonometric function and solve the equations tan(x) = 400 / (d + 5) and tan(y) = 400 / d to find d.

Step-by-step explanation:

The question involves trigonometric concepts that can be applied to real-world scenarios. In this scenario, a swimmer is moving closer to a buoy while a helicopter hovers 400 feet directly above the buoy. The distances the swimmer moves are related through trigonometric functions corresponding to the angles of elevation before and after swimming closer to the buoy.

We can use the tangent trigonometric function to model the situation. Initially, the swimmer sees the helicopter at an angle of elevation of x degrees and then moves 5 feet closer to the buoy, seeing the helicopter at an angle of y degrees. The trigonometric equation involving the tangent of the angles x and y can be used to find the distance the swimmer swam toward the buoy.

Before moving, the swimmer is at a distance d + 5 feet from the buoy, and after moving 5 feet closer, the swimmer is at a distance d feet from the buoy. The relationship can be expressed as follows:

• tan(x) = 400 / (d + 5)
• tan(y) = 400 / d

Thus, to find the distance the swimmer moved in terms of angles x and y, you solve these two equations for d.