Final answer:
To find the horizontal distance between two descending hot air balloons, we set up right triangles using the heights of the balloons as the opposite sides and the angles of depression to calculate the horizontal distances (adjacent sides) using tangent trigonometric ratios, and then find the difference between those distances.
Step-by-step explanation:
The question given involves using trigonometry to find the horizontal distance between two descending hot air balloons spotting the landing pad at different angles of depression. To solve this problem, we can use the concept of angles of depression and trigonometric ratios.
For the first balloon at 400 m high with an angle of depression of 62°, we can set up a right triangle where the opposite side is the altitude of the balloon, and the angle at the balloon is the complement of the angle of depression, which is 28° (since 90° - 62° = 28°). Using the tangent function:
Tan(28°) = Opposite/Adjacent
Adjacent = Opposite / Tan(28°)
Adjacent = 400 m / Tan(28°)
Adjacent ≈ 732.4 m
For the second balloon at 1000 m with an angle of depression of 41°, we again set up a right triangle and use the complement of the angle of depression, which is 49° (90° - 41°). Thus:
Tan(49°) = Opposite/Adjacent
Adjacent = Opposite / Tan(49°)
Adjacent = 1000 m / Tan(49°)
Adjacent ≈ 836.7 m
The horizontal distance between the two balloons is the difference between these two distances:
Distance between balloons = 836.7 m - 732.4 m
Distance between balloons ≈ 104.3 m