Sam starts from point A at 8:00am and hikes up a mountain path. He arrives at the top, (point B) at 4:00pm. He camps there for the night and the following morning he starts his descent at 9:00am. He travels faster since he goes downhill and arrives at the bottom (point A) at 3:00pm. Now, does there exist a spot along the path that Sam passes at the same time on both days? Please explain your answer.
yes
if f(t) is his elevation at time t ascending,
f(8) = 0
f(16) = h
if g(t) is is elevation at time t descending,
g(9) = h
g(15) = 0
f(t) = 0 + h/(16-8) (t-8)
g(t) = h - h/(15-9) (t-9)
so, we want
0 + h/8 (t-8) = h - h/6 (t-9)
We can, without loss of generality, set h=1, so
t-8+t/8 = 1-(t-9)/6
t=12
So, at noon on both days Sam is at the same height, namely h/2