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A rectangle has its base on the x-axis and its upper two vertices on the parabola
y=12−x2 .

What is the largest area that the rectangle can have

  • calculus -

    let the points of contact be (x,y) and (-x,y)
    so the base is 2x and the height is y
    but y=12-x^2
    so the
    Area = 2xy
    = 2x(12-x^2)
    = 24x - 2x^3
    d(Area)/dx = 24 - 6x^2
    = 0 for a max Area

    solve 2x^3 - 24 = 0
    sub that back into the Area = 2x(12-x^2)

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