Use the Intermediate Value Theroem to solve

Question

A monk leaves the monastery at 7:00 am and takes his usual path to the top of the mountain, arriving at 7:00 pm. the following morning, he starts at 7:00 am at the top and takes the same path back arriving at 7:00 pm. Use the intermediate value theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.

Answer 1

If the position of the monk down the path is Down(t), and the position of the monk up is Up(t) where t is the time of day and the month walks continuously then both Down() and Up() are continuous functions. 

 

The difference between the positions, Diff(t) = Up(t) - Down(t) {measured from the monastery} is also a continuous function. 

 

If L is the length of the path measured from the monastery, then Down(t₀) = 0 and Up(t₀) = L

 

Since Up(t₀) > Down(t₀) where t₀ is the start of the respective days, then Diff(t₀) = L which is greater than 0. 

 

At the end of the day, t_f, Up(t_f) = 0, Down(t_f) = L and Diff(t_f) = 0 - L.

Up(t_f) < Down(t_f) and Diff(t_f) is negative because the monk has finished the journey.

 

Since Diff(t) is continuous, by the intermediate value theorem, if Diff(t₀) > 0 and Diff(t_f) < 0, it must pass through a point where Diff(t) = 0.  At this point Down(t) = Up(t).  At time t the monk is at the same point along the path on both days.