9. At 8:00 am on a Saturday, a hiker begins running up the side of a mountain to
ID: 2885470 • Letter: 9
Question
9. At 8:00 am on a Saturday, a hiker begins running up the side of a mountain to his te. On Sunday mornin g at 8:00 am, he runs back down the mountain. It takes him 20 minutes to run up the mountain, but only 10 minutes to run down. At some point on the way down, he realized that he passed the same place at exactly the same time on Saturday. Use the Intermediate Value Theorem to prove he is correct. [Hint: let s() and r(t) be the position functions for the runs up and down, respec tively; consider f(t) s(t) -r(t).)Explanation / Answer
Considering position functions s(t) and r(t) for runs up and down respectively and f(t)= s(t)- r(t), running upwards at t=0, the starting position would be 0 and after 20 minutes s(20) would indicate the campsite.
Similarly the position function r(t) would indicate the starting position, at t=0 for running down from the campsite and after 10 minutes r(10) would indicate the site from where running upwards starts.
Hence the two ends of up and down journey indicate a closed time interval. Clearly the value of function f(t) at the two ends of this interval are not equal. The function f(t) is continuous, in this interval, as there is no time gap while running. Running is continuous both ways.
The essential conditions for the applicability of Internmediate value theorem are thus satisfied. Hence according to Intermediate value theorem, there would be a point 'k' between the starting point and the campsite which would be reached for some value of 't' , say t=c, so that f(c)=s(c)-r(c)= k ,while going up or coming down,
*****
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.