Precalc

# 1
a special window has the shape o a rectangle surmounted by an eqilatera tiangle. ifthe perimeter o the winow is 16 ft, what dimensions will admit te most light (area of triangle= suarre root of 3/4 times x2.

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  1. let the length of the rectangle be 2x
    then the sides of the equilateral triangle are also 2x
    and the height of the triangle is √3 x
    let the height of the rectangle be y

    2y + 6x = 16
    y + 3x = 8
    y = 8-3x

    area = 2xy + (1/2)√3 x(2x)
    = 2x(8-3x) + √3 x^2
    = 16x - 6x^2 + √3 x^2

    d(area)/dx = 16 - 12x + 2√3 x = 0 for a max/min
    16 = 12x - 2√3x)
    8 = x(6 - √3)
    x = 8/(6-√3)
    2x = 16/(6-√3) = appr 3.75
    y = appr .377

    check my arithmetic

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