Final answer:
Using trigonometry with the angles of elevation and the constant altitude, we calculate the horizontal distances from the observer to the airplane at two points and find the difference to determine the distance traveled by the airplane. The airplane traveled approximately 18059.9 feet from point A to point B.
Step-by-step explanation:
To find the distance the airplane traveled from point A to point B, we can use trigonometry. Since the airplane maintains a constant altitude of 7475 feet, we have two right-angle triangles where the altitude is the same but the angles of elevation are different.
For the initial measurement at point A with an angle of elevation of 16 degrees, we can write the following equation based on the tangent of the angle:
tan(16°) = altitude / distance_A
distance_A = altitude / tan(16°)
Similarly, for the measurement at point B with an angle of elevation of 43 degrees:
tan(43°) = altitude / distance_B
distance_B = altitude / tan(43°)
The distance the airplane traveled from point A to point B is the difference between distance_B and distance_A:
distance_traveled = distance_B - distance_A
Substituting the values, we get:
distance_A = 7475 feet / tan(16°)
distance_B = 7475 feet / tan(43°)
Calculating these using a calculator:
distance_A = 7475 / 0.28675 ≈ 26075.8 feet
distance_B = 7475 / 0.93252 ≈ 8016 feet
Now, we find the distance traveled:
distance_traveled = 26075.8 - 8016 ≈ 18059.8 feet
Rounding to the nearest tenth, the airplane traveled approximately 18059.9 feet from point A to point B.