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Math Help - Optimization Problem

  1. #1
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    Optimization Problem

    Hi, I had a problem that I just had no idea how to do. I was told it was an optimization but I am still confused. The problem states :
    Let S denote the set of points with the coordinates (x,y) such that 0≤x≤15 and y^2 = 150- 10x. Find the coordinates of the point(s) of S nearest to the origin. Also find the coordinates of the point(s) of S farthest from the origin. Justify your answer.
    All i have gotten is y= squareroot(150-10x) and that is not much.
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  2. #2
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    Quote Originally Posted by calc_help123 View Post
    Hi, I had a problem that I just had no idea how to do. I was told it was an optimization but I am still confused. The problem states :
    Let S denote the set of points with the coordinates (x,y) such that 0≤x≤15 and y^2 = 150- 10x. Find the coordinates of the point(s) of S nearest to the origin. Also find the coordinates of the point(s) of S farthest from the origin. Justify your answer.
    All i have gotten is y= squareroot(150-10x) and that is not much.
    remember the distance formula ? distance of the point (x,y) from the origin is ...

    d = \sqrt{(x - 0)^2 + (y - 0)^2}

    d = \sqrt{x^2 + y^2}

    substitute for y^2

    d = \sqrt{x^2 - 10x + 150}

    let d = \sqrt{z}

    if z is maximized/minimized, so is d ...

    z = x^2 - 10x + 150

    \frac{dz}{dx} = 2x - 10

    2x - 10 = 0

    x = 5

    \frac{d^2z}{dz^2} = 2 > 0

    x = 5 will yield a minimum distance

    the endpoints, x = 0 and x = 15 , will be maximums ... calculate which yields the absolute maximum distance.
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