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Math Help - Distance problems?

  1. #1
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    Distance problems?

    y=x^(1/2). what's the shortest distance from that curve to the point (1/2,0)?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Distance problems?

    Let \left(x,x^{\frac{1}{2}} \right) be an arbitrary point on the curve. It will be simpler to minimize the square of the distance between this arbitrary point and the given fixed point. Can you state the square of this distance?
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  3. #3
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    Re: Distance problems?

    would this use the distance formula? or am i way off?
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: Distance problems?

    You are spot on! Since our objective function is a distance, then that's a great place to start!
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  5. #5
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    Re: Distance problems?

    so how do i use that? what do i plug in?
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  6. #6
    MHF Contributor MarkFL's Avatar
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    Re: Distance problems?

    We have two points:

    \left(x,x^{\frac{1}{2}} \right) and \left(\frac{1}{2},0 \right).

    Now, the distance formula tells us the distance d between the two points P_1(x_1,y_1) and P_2(x_2,y_2) is:

    d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

    So, choose either of our two points to be P_1 and the other as P_2 and plug into the equivalent:

    d^2=(x_2-x_1)^2+(y_2-y_1)^2

    Then simplify and optimize by differentiation. Be sure to demonstrate you have minimized by using an appropriate test.
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