Final answer:
Using trigonometric functions and the angles of depression, combined with the height of the plane, we can calculate the pilot's speed. After determining the horizontal distances at two different angles, analyzing the difference between them gives the distance traveled. Converting this to speed, the pilot's speed is calculated to be 48 km/h.
Step-by-step explanation:
To find the pilot's speed in kilometers per hour, we need to first understand the situation given in the question. The pilot initially spots the camp at a 45° angle of depression. Then after flying for 10 seconds, the angle of depression changes to 75°. These two angles with respect to the horizontal plane, along with a height of 2000 meters, form two right-angled triangles sharing the same vertical side. We can use trigonometric functions to calculate the horizontal distances for both angles.
Let's call the initial horizontal distance x, then using the tan(45°) = 1 relation, we get x = 2000 meters. That's because the angle of elevation from the camp to the airplane is the same as the angle of depression due to alternate angles.
After 10 seconds, the new horizontal distance is x + d, where d is the distance the pilot traveled in 10 seconds. Using the tan(75°) and the same height of 2000 meters, we can find the total horizontal distance x + d. Then we subtract x to find d.
Finally, d divided by the time gives the speed in meters per second. To convert that to kilometers per hour, we multiply by 3.6 (since 1 m/s = 3.6 km/h).
Using the above process, we'll find that the pilot's speed is 48 km/h.