Final answer:
The airplane's speed passes through 610 miles per hour on two occasions due to the Intermediate Value Theorem, which states that any continuous transition from a higher speed to a lower speed (or vice versa) must pass through all intermediate speeds.
Step-by-step explanation:
Understanding Airplane Speed Changes
When the airplane initially encounters a headwind that slows its speed from 620 miles per hour to 608 miles per hour, and then to 614 miles per hour, it's because of the varying intensity of the wind resistance it faces. As the intensity of the headwind changes, the plane must adjust its thrust to maintain or alter its speed. This fluctuation in wind strength means that at two different moments during the flight, the plane's speed must pass through 610 miles per hour - once when the speed reduces from 620 to 608 miles per hour and again when it increases from 608 to 614 miles per hour.
To illustrate further, consider that anytime a variable quantity (in this case, the speed of the plane) decreases and then increases (or vice versa), it must pass through every intermediate value at least once. This is a consequence of the Intermediate Value Theorem, a fundamental concept in calculus which states that for any continuous function, if the function takes on a particular value at one point and a different value at a second point, then it must take on any intermediary value between the two points at some stage. Hence, the plane's speed of 610 miles per hour will occur at least twice if the speed changes continuously, once when decreasing from 620 to 608 mph, and again when increasing from 608 to 614 mph.