1. ## Intermediate Value Theorem

A manatee leaves his home at 7a.m. and swims along a well travelled path until he reaches
a friend's home 10 hours later (where an all night manatee party ensues). The next day
the manatee leaves his friend's house at 7a.m. and returns home in the same amount of
time taking the same path.
If f(t) represents the distance from his home on the rst day and g(t) represents the
distance from his home on the second day and t represents the time in hours modulo 24
(i.e. a particular t value represents the same time on any given day), prove that there is
at least one time where the manatee was at the exact same spot on both days. That is,
prove that there is a t such that f(t) = g(t). You may assume that manatees cannot
teleport.

My professor told us to use intermediate value theorem to solve this.
I have no idea how to set up the two equations f(t) and g(t) into one
so I can apply the IVT.

2. ## Re: Intermediate Value Theorem

First, I recommend draw possible graphs of f(t) and g(t).

Originally Posted by nubshat
I have no idea how to set up the two equations f(t) and g(t) into one
so I can apply the IVT.
f and g are not equations; they are functions. You are saying you don't know how to combine f and g into one function. Have you tried arithmetic operations? Does any of them have a geometric meaning on the graph of f and g?

3. ## Re: Intermediate Value Theorem

You want to show that, for some t, f(t)= g(t). That is the same as f(t)- g(t)= 0.

4. ## Re: Intermediate Value Theorem

I have f(t) -g(t), but I need to find a value for t such that f(t)-g(t) <0 and a value where f(t)-g(t) > 0, what could I sub in to the function to make this true?