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Math Help - Calculating the shortest time possible

  1. #1
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    Calculating the shortest time possible

    Hi there I need some help with this problem
    sorry if this is in the wrong place, Im a new member here

    A bug is crawling on the coordinate plane from (5,9) to (-15,-7) at constant
    speed one unit per second except in the second quadrant where it travels at 1/2
    units per second. What path should the bug take to complete its journey in as
    short a time as possible?

    So far I drew a picture of it and I was assume the shortest amount of time would be to draw a diagonal line connecting the points together and then calculating the time, but it doesn't feel correct to me. I hope someone can help/guide me through this problem, so I can understand how to do this, thanks.
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Calculating the shortest time possible

    Check if avoiding 2nd quad with 1/2 speed is faster via A-O-D:



    Crawling times along:
    AOD : 26.849/s = less time. (be double checked)
    ABCD: 33.616/s
    Attached Thumbnails Attached Thumbnails Calculating the shortest time possible-crawler-bug.png  
    Last edited by MaxJasper; October 12th 2012 at 07:41 PM.
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  3. #3
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    Re: Calculating the shortest time possible

    Hey there Jasper thanks for helping me out with this problem, I looked over your solution and doubled checked it. What I've gotten for the total time for ABCD was 21.612 seconds and for AOD it was 26.849 seconds. I've included the 1/2 speed from B to C and discover that avoiding the 2nd quadrant still takes longer than going straight to the destination

    I was wondering if there was a proof to determine if going a straight line was the shortest time to the destination or avoiding it.
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  4. #4
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Calculating the shortest time possible

    Crawling times along:
    AOD : 26.849/s = less time. (be double checked)
    ABCD: 33.616/s
    Last edited by MaxJasper; October 12th 2012 at 07:42 PM.
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  5. #5
    MHF Contributor MarkFL's Avatar
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    Re: Calculating the shortest time possible

    I find, using differential calculus and help from the computer finding the critical point, that the quickest route is from (5,9) to (0,0) to (-15,-7), giving a time of about 26.85 s. If the bug crawls along the straight line between the two given points, then the time is about 42.08 s. We must in effect, double the distance in the 2nd quadrant, since the speed there is 1/2.
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  6. #6
    Senior Member MaxJasper's Avatar
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    Re: Calculating the shortest time possible

    Yes, quickest route is AOD
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  7. #7
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    Re: Calculating the shortest time possible

    Oh right sorry I made a mistake, I understand now thanks a lot everyone!
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