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Math Question

  1. Feb 25, 2004 #1
    I'm in IB math methods, working on a project and I can't seem to figure out what to do with this problem.

    A radio transmitter sends signals to a railway which run along a straight track. When a set of coodinate axes is used to represent this system, the transmitter is at R(1,0), and the track T is a line w/ equation 2x+y=30, where the units are kilometers.
    - Points A and be are the points of T that cross the x and y axis
    - The engine receives the strongest signal at C, which is the point on T closest to R.
    - P(x,y) is a general point on T

    My problem is this:
    let s(x) represent the distance between R(1,0) and P(x,y). Show that
    s(x)= sq rt (5x^2-122x+901).

    Help! I don't even know where to begin. I said that point C is
    (15x-15, -60x + 900). Since RP is (x-1)i + (-2x+30)j, does the 901 in the thing I'm supposed to show have to do with... oh never mind I'm REALLY confused. Please help me. Thanks!
     
  2. jcsd
  3. Feb 25, 2004 #2
    poof

    First, all we have to do is show that s(x) = sqrt (5x^2 - 122x + 901). It seems like your making this out to be alot more complicated then it really is. Forget vectors or anything other then good old algebra for a minute.

    Recall the distance formula: sqrt[ (x2-x1)^2 + (y2-y1)^2 ]

    We plug in our values for (x1,y1) (x2,y2) from the points given.

    Therefore:

    s(x) = sqrt [ (x-1)^2 + y^2 ]

    Now we need to change that y^2 into x..we are in luck because they have given us the equation of the line on which these points are located. Solve 2x + y = 30 for y... we get: y = 30 - 2x

    Substitute this back into the equation we just obtained with the distance formula and multiply it out.

    s(x) = sqrt [ (x-1)^2 + (30-2x)^2 ]

    s(x) = sqrt [ x^2 - 2x +1 + 900 - 120x + 4(x)^2 ]

    simplify:

    s(x) = sqrt [ 5(x)^2 - 122x + 901 ]

    There you have it.
     
  4. Feb 25, 2004 #3
    Thanks!

    Thank you so much! I can't believe how simple that was. I have such a huge problem when I am asked to prove things. I just never seem to find the way to do it.
     
  5. Feb 26, 2004 #4
    Another Problem

    In the above problem, they want me to draw an arc with the center R with a radius of 28 and then find the length of the portion of T that will be within the range of the arc. The drawing the arc was no problem, but I don't know how to find it.
     
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