# Intermediate value theorem

heres the question:

At 8 am on Saturday a man begins running up the side of a mountaion to his weekend campsite. On Sunday morning at 8 am he runs back down the mountain.It takes him 20 minutes to run up, but only 10 mins to run down. At some point on the way down, he realizes that he passed the same place at exactly the same time on Saturday. Prove that he is correct. (hint: let s(t) and r(t) be the position functions for the runs up and down, and apply the Intermediate Value Theorem to the function f(t)= s(t) - r(t)).

ok i understand the int. value th. but i cannot figure out how to apply it here or what the point of the hint was (f(t)= s(t) - r(t)). i know that for a given height, h, there is a value of t such that s(t) = h. likewise there is another value of t such that r(t) = h.

but how do you prove that those t's (times) are the same for s and r to show that at the exact same time he was at the same height? im stuck.

NateTG
Homework Helper
What is the value of f(t) at 8 am?
What is the value of f(t) at 8:20 am?

Assume (for a moment) that there is a suitable time.
What would the value of f be at that time?

f(t) at 8 am would be -h since s(t) = 0 and r(t) = h (assuming the hill has a height of h).

i have no idea what it is at 8:20 am.

if there was a suitable time then both s(t) and r(t) would be equal (at some height, whatever it may be) and f(t) = 0.

am i close?

HallsofIvy
Homework Helper
dnt said:
if there was a suitable time then both s(t) and r(t) would be equal (at some height, whatever it may be) and f(t) = 0.

Actually, the point is the other way around! Assuming that s and r are continuous functions (he doesn't have a "transporter" to send him instantaneously from one point to another!), since f(0) is negative and f(20) is positive, by the intermediate value theorem, there MUST be some time t when f(t)= 0. What does THAT tell you?

HallsofIvy
Homework Helper
dnt said:
f(t) at 8 am would be -h since s(t) = 0 and r(t) = h (assuming the hill has a height of h).

i have no idea what it is at 8:20 am.

if there was a suitable time then both s(t) and r(t) would be equal (at some height, whatever it may be) and f(t) = 0.

am i close?
You are told that it took him 20 minutes to run up the hill so s(20)= h. It took him only 10 minutes to run down the hill and since you are only interested in the hill, you can assume he just waited at the bottom for another 10 minutes: r(20)= 0. Yes, f(0)= s(0)- r(0)= -h. f(20)= s(20)- r(20)= h- 0= h. One is negative, the other positive. What does that tell you about a value of f some time between 0 and 20?