Largest Rectangle Under the curve y=-.0064x^2+36?

What is the largest rectangle you can fit under this equation? the rectangle is bound by the equation and the x-axis

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  • 1 decade ago
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    I assume this is for a calculus class.

    given:

    y = -0.0064x² + 36

    for (x, y) > 0

    so

    y = height of rectangle

    2x = width of rectangle

    see

    http://i584.photobucket.com/albums/ss282/JimPfoss/...

    so

    A(x) = area of rectangle

    A(x) = 2x(-0.0064x^2 + 36)

            = -0.0128x³ + 72x

    max area is where A'(x) = 0, so:

    A'(x) = -0.0384x² + 72

    0.0384x² = 72

    x² = 1875

    x = 25√(3)

    so

    width = 50√(3) ≈ 86.6

    height = 24

    area = 1200 √(3) ≈ 2079 units²

    ♣♦

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