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Question
"at 8:00 am on saturday a man begins running up the side of a mountain to his weekend campsite. On sunday morning at 8:00 am he runs back down the mountain. it takes him 20 minutes to run up, but only 10 min 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)."
To prove it exists and also find it.
Since they both start at 8 am, imagine there are two people, one running up and one running down.
The one running down goes twice as fast, so for every 2 yards he goes down, the once coming up has only made a yard. Looking at it so far, the ratio is 2:1, and the total of 2 and 1 is 3.
That means they will meet when the one from below has gone up 1/3 of the way
and the one from the top has gone down 2/3 of the way.
Going up it would be at 2/3 of 10 minutes and going down it would be at 1/3 of 20 minutes.
Both of these values are 6 2/3 minutes from the starting point.
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Scotto
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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.
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