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Math Help - Minimum Distance from Point to Curve

  1. #1
    Junior Member asemh's Avatar
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    Minimum Distance from Point to Curve

    Hi
    I have been attempting this question, but i do no know how to continue...

    Determine the minimal distance from the point (0,4) to the curve y = 8 - x^2

    I know the distance formula is  D = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2}

    after plugging in the x1 and y1 i got  D  = \sqrt{x^2 + (y-4)^2}

    What do i do next? Am i doing a step wrong?

    Thanks
    Last edited by asemh; March 25th 2010 at 05:51 PM.
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  2. #2
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    Quote Originally Posted by asemh View Post
    Hi
    I have been attempting this question, but i do no know how to continue...

    Determine the minimal distance from the point (0,4) to the curve y = 8 - x^2

    I know the distance formula is D = sqrt((x2-x1)^2 + (y2 - y1)^2)

    after plugging in the x1 and y1 i got = sqrt(x^2 + (y-4)^2)

    What do i do next? Am i doing a step wrong?

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

    (x_2,y_2) = (x,8-x^2)

    (x_1,y_1) = (0,4)

    d = \sqrt{(x-0)^2 + [(8-x^2) - 4]^2}

    clean up the expression underneath the radical.

    since you posted this in the precalculus and not the calculus section, I assume you are not familiar with using the methods of calculus for optimization.
    what method(s) have you been using in class for problems like this?
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  3. #3
    Junior Member asemh's Avatar
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    We usually get a formula, derive it, then solve for x
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  4. #4
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    Quote Originally Posted by asemh View Post
    We usually get a formula, derive it, then solve for x
    do you mean take the derivative?
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  5. #5
    Junior Member asemh's Avatar
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    Yes

    BTW, When i cleaned up the equation i got

     d = \sqrt{ x^4 - 7x^2 + 16}

    is that correct?
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  6. #6
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    Quote Originally Posted by asemh View Post
    Yes

    BTW, When i cleaned up the equation i got

     d = \sqrt{ x^4 - 7x^2 + 16}

    is that correct?
    fine ... note that if you minimize (x^4-7x^2+16) , you also minimize its square root.
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  7. #7
    Junior Member asemh's Avatar
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    Do you mean when i derive it?

    So it will become like 0.5(x^4 - 7x^2 + 16)^-0.5 (4x^3 - 7) ?
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  8. #8
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    Quote Originally Posted by asemh View Post
    Do you mean when i derive it?

    So it will become like 0.5(x^4 - 7x^2 + 16)^-0.5 (4x^3 - 7) ?
    finish it.
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  9. #9
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    Quote Originally Posted by asemh View Post
    Do you mean when i derive it?

    So it will become like 0.5(x^4 - 7x^2 + 16)^-0.5 (4x^3 - 7) ?
    Skeeter also said "note that if you minimize x^4- 7x^2+ 16 , you also minimize its square root.[quote]. Its derivative is 4x^3- 14x. Where is that 0?
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