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Math Help - Maximum area of an area within y=x^2

  1. #1
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    Maximum area of an area within y=x^2

    I've got this problem that I'm trying to solve, I've been twisting my brain for several hours now.


    An area 'A' is limited by the curve y=x^2, the x-axel and the line x=2

    In this area is a rectangle with the area R, as seen in the figure.

    Calculate the maximum relation between R and A in exact form.


    If someone could take a look at this and help me, I would appreciate it very very much!
    (sorry for my bad english)
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  2. #2
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    if you make the left hand bottom corner of the rectangle point a then it follows that

    A = L\times W

    subing in some values gives

     A = f(a)\times (2-a)

     A = a^2\times (2-a)

     A = 2a^2-a^3

    for a maximum you must find where \frac{dA}{da}=0

    \frac{dA}{da}=4a-3a^2

    0=4a-3a^2

    0=a(4-3a)

    a=0,\frac{4}{3}

    probably should discard zero as a solution at this point as it implies zero area.

    now f\left(\frac{4}{3}\right)= \frac{9}{16}

    Maximum area should be simple to find from here using

    A = L\times W

     A = f(a)\times (2-a)
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