precalculus

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a rectangular piece of cardboard measuring 14 inches by 27 inches is to be made into a box with an open top by cutting squares of equal size from each corner and golding up the sides . let x represent the length of a side of each square. for what value of x will the volume be a maximum? round to two decimal places.

  • precalculus -

    v = x(14-2x)(27-2x)

    dv/dx = 2(6x^2 - 82x + 189)

    dv/dx=0 when x = (41 ± √547)/6 = 2.93, 10.73

    knowing the shape of cubics, you should have no trouble "differentiating" the max from the min.

  • precalculus -

    length = z = 27 -2x
    width = y = 14 - 2x

    v = (27-2x)(14-2x)x
    v = ( 378 - 82 x + 4 x^2 ) x
    v = 378 x -82 x^2 + 4 x^3

    max when dv/dx = 0
    dv/dx = 378 - 164 x + 12 x^2 = 0
    6 x^2 - 82 x + 189 = 0

    x = [ 82 +/- sqrt (6724-4536) ]/12

    x = [82 +/- 46.8 ]/12

    x = 10.7 too big, negative width
    x = 2.93 inches

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