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4 March, 09:45

A poster of area 24,000 cm2 has blank margins of width 10 cm on the top and bottom and 6 cm on the sides. find the dimensions that maximize the printed area. (let w be the width of the poster, and let h be the height.)

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  1. 4 March, 10:19
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    The total poster area is:

    A = (l + 20) (w + 12)

    where l and w is the length and width of the printed area

    24000 = (l + 20) (w + 12)

    l = [24000 / (w + 12) ] - 20

    The printed area is simply:

    a = l w

    substituting l:

    a = ([24000 / (w + 12) ] - 20) w

    a = (24000w) (w + 12) ^-1 - 20 w

    Taking the first derivative da/dw:

    da/dw = (24000) (w + 12) ^-1 - (24000w) (w + 12) ^-2 - 20

    Set da/dw = 0:

    (24000) / (w + 12) - (24000w) / (w + 12) ^2 - 20 = 0

    Multiply everything by (w + 12) ^2:

    24000 (w + 12) - 24000w - 20 (w + 12) ^2 = 0

    24000w + 288000 - 24000w - 20 (w^2 + 24w + 144) = 0

    -20 w^2 - 480 w + 285120 = 0

    w^2 + 24 w = 14256

    Completing the square:

    (w + 12) ^2 = 14256 + 12^2

    w + 12 = ±120

    w = 108 cm

    l = [24000 / (w + 12) ] - 20

    l = [24000 / (108 + 12) ] - 20

    l = 180 cm

    Therefore the dimensions of the whole poster are:

    w + 12 = 120 cm

    l + 20 = 200 cm

    The poster should be 200 cm x 120 cm
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