Final answer:
Using trigonometry and the tangent function, we determine that Plane B, which takes off at a lower 9° angle, reaches a greater horizontal distance from the airport when it reaches an altitude of 10,000 feet.
Step-by-step explanation:
To determine which plane reaches a greater horizontal distance from the airport when reaching an altitude of 10,000 feet, we can use trigonometry. For each plane, the altitude forms the opposite side of a right-angled triangle, and the angle of ascent is given. We can find the horizontal distance (the adjacent side of the triangle) by using the tangent function, where tan(θ) = opposite/adjacent, which yields adjacent = opposite/tan(θ). We can then calculate the distances for both planes.
For Plane A, with a 13° angle:
- adjacent = 10,000 / tan(13°)
For Plane B, with a 9° angle:
- adjacent = 10,000 / tan(9°)
Without calculating these values exactly, we can already tell that Plane A, with the steeper angle, will have a shorter horizontal distance compared to Plane B, because the tangent function is an increasing function. This means that as the angle decreases, the tangent value decreases, the denominator decreases and therefore the adjacent side (the horizontal distance) increases.
Therefore, Plane B reaches a greater horizontal distance from the airport when it reaches an altitude of 10,000 feet.