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Math Help - Optimization Problem4

  1. #1
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    Optimization Problem4

    Find the dimension of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola
    y=8-x^2
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  2. #2
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    Quote Originally Posted by camherokid View Post
    Find the dimension of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola
    y=8-x^2
    Hello,

    I've attached an image of the parabola and the rectangle.

    One point on the parabola has the coordinates P(s, 8-s). The the area of the rectangle is:
    a = 2s * (8-s) = 16s - 2s

    Calculate the first derivative:

    a'(s) = 16 - 6s

    You'll get an extremum (maximum or minimum) if a'(s) = 0. Thus s = √(8/3)

    Therefore the dimensions of the rectangle are: l = 2√(8/3), w = 16/3, a_(max) = 32/3*√(8/3) ≈ 17.42

    With this solution I assumed that the rectangle is above the x-axis. Otherwise the area of the rectangle is unlimited.
    Attached Thumbnails Attached Thumbnails Optimization Problem4-rechtck_parabel.gif  
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