Did the Stewarts Experience the Same Temperature Simultaneously?

[flylike1]

I can be rather obsessive about finding answers to brain teasers. Most brain teasers I can solve with some reasoning, no problem. This one though, I have thought and thought about, and can't come to a conclusion that satisfies me enough to say "this is the answer." The question/prompt is:

There is a bit of trivia about the author of your textbook, Dr. James Stewart, that very few people know. He has an evil twin sister named Onad. Although he loves his sister dearly she dislikes him, and tries to be different from him in all things.

Last winter, they both went on vacation. Dr. Stewart went to Hawaii. Onad had planned on going to Aruba, but decided against it. She hates her brother so much that she was afraid there would be a chance that they might be experiencing the same temperature at the same time, and that prospect was distasteful to her. So she decided to vacation in northern Alaska.

After a few days, Dr. Stewart received a call: "this is Onad. I am very cold and uncomfortable here. That's good, since you are undoubtedly warm and comfortable, and I want us to be different. But I'm not sure why I should be the one in northern Alaska. I think we should switch places for the last half of our trip."

"It is only fair," he agreed.

So they each traveled again. Dr. Stewart took a trip from Hawaii to Alaska, while Onad took a trip from Alaska to Hawaii. They each traveled their own different routes, perhaps stopping at different places along the way. Eventually, they had reversed locations. Dr. Stewart was shivering in Alaska, Onad in Hawaii, warm and happy. She received a call from her brother.

"Hi, Onad. Guess what? At some time during our travels, we were experiencing exactly the same temperature at the same time. So HA!"

Is Dr. Stewart right? Has Good triumphed over Evil? He would try to write out a proof of his statement, but his hands are too frozen to grasp a pencil. Help him out. Either prove him right, or prove him wrong, using a mathematical argument.


Can you help me find the answer?

 

[bob1182006]

Hm...The only thing I can think of this is say...:

Temp in Alaska: -10* F?
Temp in Hawaii: 80* F?

so while Onad went form -10* to 80*, Stewart went from 80* to -10* so somewhere in there they both had the same temp maybe 45*?.

Thinking...
$$\frac{1}{80-(-10)}\int_{-10}^{80} T(x) dx = \frac{1}{-10-80}\int_{80}^{-10} T(x) dx$$

could be totally off tho >< that is a toughie...

 

[HallsofIvy]

Use the intermediate value theorem. Let U(t) be the temperature experienced by Dr. Stewart at time t, V(t) the temperature experienced by Onad at time t. Let F(t)= U(t)- V(t). Is F(t) a continuous function? (Are U(t) and V(t) continuous or is that a reasonable assumption?) What can you say about F(0) and F(T) where 0 is the time Dr. Stewart left Hawaii and T is the time he arrived in Alaska? What does the intermediate value theorem tell you about what happens at some time between 0 and T?