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Math Help - Hyperbola Problem

  1. #1
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    Hyperbola Problem

    Listening post are at A, B, and C. Point A is 2000ft north of point B, and point C is 2000ft east of B. The sound of a gun reaches A and B simultaneously one second after it reaches C. Show that the coordinates of a gun's position are approximately (860,1000), where the x - axis passes through B and C and the origin is midway between B and C. Assume that sound travels 1100ft/sec
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  2. #2
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    Listening post are at A, B, and C. Point A is 2000ft north of point B, and point C is 2000ft east of B. The sound of a gun reaches A and B simultaneously one second after it reaches C. Show that the coordinates of a gun's position are approximately (860,1000), where the x - axis passes through B and C and the origin is midway between B and C. Assume that sound travels 1100ft/sec
    The coordinates of the points A, B and C are: (-1000, 2000), (-1000,0), (1000,0).

    Let the gun's position be (x,y), the we are effecticly told that (as the
    distances of the gun from A and from B are equal):

    (-1000-x)^2 + (2000-y)^2 = (-1000-x)^2 + y^2.

    Which fixes y=1000.

    Also we are told that as the sound reaches B 1 second after it reaches C:

    sqrt[(-1000-x)^2 + y^2] = sqrt[(1000-x)^2 + y^2] + 1100

    This is a curve with a single branch and so if y=1000, has a single root,
    and this can be found numericaly to be at x=858.0.

    (you can substitute the suggested x=860 y=1000 into the above equation
    and show that the two sides are approximatly equal)

    RonL
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  3. #3
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    Listening post are at A, B, and C. Point A is 2000ft north of point B, and point C is 2000ft east of B. The sound of a gun reaches A and B simultaneously one second after it reaches C. Show that the coordinates of a gun's position are approximately (860,1000), where the x - axis passes through B and C and the origin is midway between B and C. Assume that sound travels 1100ft/sec
    HI,

    CaptainBlack types twice as fast as I can do, so I don't reply to your problem.

    If you like to get some further information see here: LORAN - Wikipedia, the free encyclopedia

    EB
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  4. #4
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    Hello, ^_^Engineer_Adam^_^!

    I have no idea why they placed the origin that way . . . very inconsiderate!


    Listening post are at A, B, and C.
    Point A is 2000ft north of point B, and point C is 2000ft east of B.
    The sound of a gun reaches A and B simultaneously one second after it reaches C.
    Show that the coordinates of a gun's position are approximately (1860,1000),
    where the origin is at B. .Assume that sound travels 1100ft/sec
    Code:
        A *
          |*
          | *
          |  *
          |   * d+1100
          |    *
     2000 |     *
          |      *  G
          |       *(x,y)
          |     *     *    d+1100
          |   *d          *
          | *                 *
          * - - - - - - - - - - - *
          B         2000          C

    The gun is at G(x,y).
    Let GB = d. .Then GA = GC = d + 1100

    Since G is equidistant from A and C, it lies on the 45-line: x = y

    The distance from G(x,x) to B(0,0) is: .√(2x) .= .d

    The distance from G(x,x) to C(2000,0) is: .√[(x - 2000) + x] .= .d + 1100


    And we have this system of equations:

    . . . . . . . . . . . 2x .= .d

    . . (x - 2000) + x .= .(d + 1100)


    Good luck!

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