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Math Help - I could not be more confused!!

  1. #1
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    Unhappy I could not be more confused!!

    Hi I need serious help on this maths problem solving question I got the other day. It's long so bare with me
    A Primary School wants to hold an athletics day, but there is only enough room for a 200m track. Calculations will have to be made to determine where the lane markings must be made. All races over 200m have to be run in seperate lanes, and the track has to be made up of two straights connected by two semicircles.
    Since everyone runs the same distance, those running in the outer lanes of the curved part of the track would run an extra distance if all competitors started next to each other.
    If each straight is 40m long, and each lane must have a width of 1.25m, calculate the distance required between each staggered start for the first two lanes.

    If you are anything like me you would be looking like right now, but I hope your still able to help me!!
    Hope to hear from anyone soon. BYE!
    Last edited by Math Help; June 8th 2005 at 12:27 AM.
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  2. #2
    Site Founder Math Help's Avatar
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    Track problem

    Most importantly let us make sure that you understand the problem.
    I have attached an illustration to help you understan the question.
    As you see, I have drawn three tracks; 1 is the inside track and 3 is the outside track. The straight aways for all three are equal length. Here I am talking about the horizontal black lines that are between the vertical red lines.

    However the semicircles on the end are all of different radius. The perimeter of a semicircle is \pi*radius so the distances for the three lanes are different.

    Because the length of each track is different but we want the finish line for all runners to be at the same point we must have those on the inside lane start further back than the rest of the runners.

    Make sense?
    Attached Thumbnails Attached Thumbnails I could not be more confused!!-track.jpg  
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  3. #3
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    Okay...

    Okay, I understand that bit now. So do I need to find the perimiter of each semicircle to help calculate the distance in starting points?? Does the distance between each lane have anything to do with the answer, or is that there just to confuse me. Thankyou for your help so far though, i really appreciate it.
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  4. #4
    Site Founder Math Help's Avatar
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    solution

    Yes you are correct.

    Here is the solution:

    Track is 200m
    Each straight is 40m long
    Each lane is 1.25mwide

    calculate the distance required between each staggered start for the first two lanes.

    Distance of track for inner runner = 200m = 40 + 40 + \pi*r + \pi*r + sd
    Distance of track for outer runner = 200m = 40 + 40 + \pi*(r+1.25m) + \pi*(r+1.25m)

    40+40+\pi*r+\pi*r+sd =40+40+\pi*(r+1.25m)+\pi*(r+1.25m)

    \pi*r + \pi*r + sd = \pi*(r+1.25m) + \pi*(r+1.25m)

    \pi*r + \pi*r + sd = \pi*r+\pi*1.25m + \pi*r+\pi*1.25m

    sd = \pi*1.25m + \pi*1.25m

    staggard distance = pi*2.5m
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