# Math Help - Help solving and reasoning a word problem.

1. ## Help solving and reasoning a word problem.

I need to know how to solve this problem. This is the information I was able to write down. Hopefully, someone has heard of it before and can tell me how to solve it.

A priest takes a pilgrimage to a hill every morning. He steps on the path at 6:00 am and stays on the path until he reaches the top of the hill. Six hours later, he arrives and reaches the top of the hill at 12:00 noon.
Then, he turns back and starts again, but at some point in the journey, he will reach a specific point at exactly the same time and same place both trips. What goes on from 6:00 am to noon time is up to the individual.

How can this take place?

2. Originally Posted by Ash
I need to know how to solve this problem. This is the information I was able to write down. Hopefully, someone has heard of it before and can tell me how to solve it.

A priest takes a pilgrimage to a hill every morning. He steps on the path at 6:00 am and stays on the path until he reaches the top of the hill. Six hours later, he arrives and reaches the top of the hill at 12:00 noon.
Then, he turns back and starts again, but at some point in the journey, he will reach a specific point at exactly the same time and same place both trips. What goes on from 6:00 am to noon time is up to the individual.

How can this take place?
Let $f(t).\ t \in [6,12]$ be the function giving the priests position as a function of
time as he ascends, and $g(t),\ t \in [6,12]$ be the corresponding function for
the descent. These are both continuous functions taking values on the same
interval, as is $h(t)=f(t)-g(t)$, but $h(6)=0$, and $h(12)=0$, so the Intermediate
Value Theorem tells us that there is a time $\tau \in (6,12)$ such that $h(\tau)=0$.

RonL