Problem

A hiker sets out at dawn to walk over a mountain and down the other side to a lake, a distance of x kilometers, and he reaches his destination exactly 12 hours later. He camps, then leaves at the same time the next morning and travels the same trail back, arriving at his starting place 12 hours later. Use the Intermediate Value Theorem to show that at some specific time, he was at the same place on his hike on both days.

Solution

Let d₁(t) be his distance traveled after t hours on the first day. Let d₂(t) be his distance remaining after t hours of travel on the second day. Let f(t) = d₁(t) – d₂(t).

At dawn (t = 0), the value of f(t) is negative, since at dawn on the first day he has not traveled at all, and at dawn the second day he is the full length of the trail (x km) from his starting place.

At t = 12, f(t) is positive: at the end of the first day he has traveled x km, and at the end of the second day he has returned to his starting place.

Therefore the Intermediate Value Theorem states that at some time t₀ between 0 and 12 hours, f(t₀) = 0. This implies

so he must be at the same point on the trail at the time t₀ on both days.