Let's work through this problem step by step.
1. Understanding the Problem:
- An airplane is flying at an altitude of 6000 meters.
- The angle of depression from the plane to the coastline is 14°.
- We need to find the horizontal distance the airplane has to fly to be directly above the coastline.
2. Visualizing the Problem:
- Imagine a right triangle where:
- The altitude (6000 meters) forms the opposite side relative to the angle of depression.
- The horizontal distance to the coastline we need to find forms the adjacent side.
- The angle of depression (14°) is between the hypotenuse (line of sight) and the horizontal ground.
3. Trigonometric Relationship:
- Use the tangent function, which relates the angle of a right triangle to the lengths of the opposite side and the adjacent side:
[tex]\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Here, the opposite side is the altitude (6000 meters), and the adjacent side is the distance we seek.
4. Setting Up the Equation:
- Substitute the known values into the tangent function:
[tex]\[
\tan(14^\circ) = \frac{6000}{\text{distance to coastline}}
\][/tex]
5. Solving for the Distance:
- Rearrange the equation to solve for the distance:
[tex]\[
\text{distance to coastline} = \frac{6000}{\tan(14^\circ)}
\][/tex]
6. Result Interpretation:
- The calculated distance to the coastline is approximately 24064.685601215067 meters.
7. Rounding to the Nearest Meter:
- After rounding, this distance is 24065 meters.
Therefore, the airplane has to fly approximately 24065 meters farther before it is directly above the coastline.
