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Math Help - Specially for you GodelProof!

  1. #1
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    Specially for you GodelProof!

    X[A,B,C].............................k.................... .............[M]Y

    At X: 3 walkers A, B, C ; speeds a=2, b=8, c=10 respectively.
    At Y: motorcyclist M, speed m=14.

    Distance XY = k : k is an integer.

    All is as in your problem: A,B and C arrive at Y at same time.
    The time is minimum, and is an integer.

    Also, M's "pickups" are at integer times (you're welcome!)

    So find k and the total time.
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  2. #2
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    Re: Specially for you GodelProof!

    Quote Originally Posted by Wilmer View Post
    X[A,B,C].............................k.................... .............[M]Y

    At X: 3 walkers A, B, C ; speeds a=2, b=8, c=10 respectively.
    At Y: motorcyclist M, speed m=14.

    Distance XY = k : k is an integer.

    All is as in your problem: A,B and C arrive at Y at same time.
    The time is minimum, and is an integer.

    Also, M's "pickups" are at integer times (you're welcome!)

    So find k and the total time.
    minimum time plan is like this:

    1) m meets c at t1=k/24, when a and c(m) are at a distance L1=k-14k/24-2k/24=k/3.

    2) m drives c backwards t2 hours and drop him in between a and b. now a and c(m) are at distance L2=L1-t2(14+2)=k/3+16t2.

    3) it takes t3=L2/16=k/48-t2 to meet a, when a, b and c are at (t1+t2+t3)*2=k/8, (t1+t2+t3)*8=k/2, and k-14k/24-14t2+10t1=5k/6-14t2 distance from X respectively, i.e., they are L3=7k/8, L4=4k/8 and L5=k/6+14t2 away from Y.

    4) m picks up a and drives him forward to Y, and arrive at Y when b and c also just arrive at Y (This takes t4=L3/14=k/16.). This means L3:L4:L5=14:8:10. L3:L4=14:8 is always satisfied. L4:L5=(k/2): (k/6+14t2)=8:10 \Rightarrow 336{t}_{2}=11k

    -----------------------------------------------------------------------
    So the answer:

    1) If we don't require integer solutions, then let k \rightarrow 0 and and the minimum time can be as small as we wish! (This is a direct result from "property 2" from here:http://www.mathhelpforum.com/math-he...rd-183226.html)

    2) For integer solutions, we need 2k/24 (pickup point for c), 2k/16 (pickup point for a) ,k and t=t1+t2+t3+t4=k/8 to be integers. Hence k=48 and total time T=48/8=6.

    -------------------------------------------------------
    Second time edit: Red letters
    Last edited by godelproof; June 18th 2011 at 08:43 PM.
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  3. #3
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    Re: Specially for you GodelProof!

    Quote Originally Posted by godelproof View Post
    2) For integer solutions, we need 2k/24 (pickup point for c), 2k/16 (pickup point for a) and k to be integers. Hence k=24 and total time T=t1+t2+t3=24/16=3/2.
    NO ! 3/2 is not an integer!
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  4. #4
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    Re: Specially for you GodelProof!

    Quote Originally Posted by Wilmer View Post
    The time is minimum, and is an integer.
    Sorry i missed "integer"
    Now i edit it and all should be right
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  5. #5
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    Re: Specially for you GodelProof!

    Agree.
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