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Tricky Question

  1. Feb 18, 2009 #1

    JPC

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    1. The problem statement, all variables and given/known data

    Hello
    Its a pretty tricky question ! Help me find the answer so i can finally sleep (i only get the answer when i hand in the question sheets in 3 weeks) :D

    ->>>>

    You walk for 1 hour, and do 5 miles during that hour

    Question : show that there exist a time interval of length 30minutes during which you have walked 2.5 miles

    2. Relevant equations

    3. The attempt at a solution

    I thought about writing :

    dD = Vi * dt
    where D : distance
    t = time
    (Vi = dP / dt)

    And integrating dD between 0 and 1hour. and saying that the whole is equal to 5 miles
    and then separate the integral into 3 ones where one is from 0 to 30 minutes

    but it doesn't lead to much conclusions

    Any ideas would be appreciated, thank you :)
     
  2. jcsd
  3. Feb 18, 2009 #2

    Dick

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    Homework Helper

    Call M(t) the distance you walk between time t and time t+30 minutes. Think about M(0) and M(30). Is it possible they are both less than 2.5? Is it possible they are both greater than 2.5? M(t) is continuous as t goes from 0 to 30 minutes.
     
  4. Feb 22, 2009 #3

    JPC

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    sorry for the late reply, i had an exam, 1 oral test, 2 big homeworks in between :D

    Hum, yes we have M(0) + M(30) = 5
    one is necessarily bigger than the other if they are not equal. But then, if one is bigger than the other, how do you justify that there is a 30minutes time interval during which you have walked exactly 2.5 miles ?
     
  5. Feb 22, 2009 #4

    Dick

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    Because M(t) is a continuous function for t in [0,30]. Think about using the intermediate value theorem.
     
  6. Feb 22, 2009 #5

    JPC

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    oh, i understand now, thank you :)

    if M(0) = a
    M(30) = b

    we have a+b = 5
    we suppose 'a' different than 'b'

    M(t) continuous on the interval [0, 30]
    so there exists a 't' belonging to R where M(t) = 2.5
     
  7. Feb 22, 2009 #6
    You still haven't proven anything. What you're saying is equivalent to me saying:
    "if M(0) = a
    M(30) = b

    we have a+b = 5
    we suppose 'a' different than 'b'

    M(t) continuous on the interval [0, 30]
    so there exists a dog named ralph, somewhere"

    Why is this the case?
     
  8. Feb 22, 2009 #7

    JPC

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    oh, this is not what i would write, first of all because i am in France ( i would write it in french), and finally because i just quickly resumed it here.

    I will of course clearly show i have verified all the hypothesis of the intermediate value theorem. And i better do, my maths professor is more rigorous than my calculator :D
     
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