Question

Starting at 9 a.m. on Monday, a hiker walked at a steady pace from the trailhead up a mountain and reached the summit at exactly 3 p.m. The hiker camped for the night and then hiked back down the same trail, again starting at 9 a.m. On this second walk, the hiker walked very slowly for the first two hours, but walked faster on other parts of the trail and returned to the starting point in exactly six hours. Prove that there is some point on the trail that the hiker passed at exactly the same time on the two days.

Answer 1

Using the Intermediate Value Theorem, we can prove that there is some point on the trail that the hiker passed at exactly the same time on the two days.

Step-by-step explanation:

To prove that there is some point on the trail that the hiker passed at exactly the same time on the two days, we can use the Intermediate Value Theorem. Let's assume that the hiker starts at 9 a.m. and reaches the summit at 3 p.m. on the first day, and starts at 9 a.m. and returns to the starting point at 3 p.m. on the second day.

On the first day, the hiker spent 6 hours on the trail (from 9 a.m. to 3 p.m.). On the second day, the hiker also spent 6 hours on the trail (from 9 a.m. to 3 p.m.). Since the hiker took the same amount of time on both days, there must be a point on the trail where the hiker passed at the same time on both days.

Therefore, we can conclude that there is some point on the trail that the hiker passed at exactly the same time on the two days.