To find the height of the airplane, we can use trigonometric functions, specifically the tangent function. The angle of depression corresponds to the angle made with the horizontal line from the airplane to the atoll.
1. The scenario can be visualized as a right triangle where:
- The angle of depression is 10 degrees.
- The horizontal distance from the airplane to the atoll is the adjacent side of the right triangle, which is 5172 meters.
- The height of the airplane is the opposite side of the triangle.
2. Using the tangent function:
tan(angle) = opposite / adjacent
Substituting our values:
tan(10 degrees) = height / 5172 meters
3. Rearranging the equation to solve for height:
height = tan(10 degrees) * 5172 meters
4. Now we calculate:
tan(10 degrees) ≈ 0.1763 (using a calculator)
height ≈ 0.1763 * 5172 ≈ 912.13 meters
Thus, the height of the airplane when it sights the atoll is approximately 912.13 meters.