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Calculus (Optimization)

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The U.S. Post Office will accept a box for shipment only if the sum of the length and girth (distance around) is at most 108 inches. Find the dimensions of the largest acceptable box with square ends.

  • Calculus (Optimization) -

    v = vol = x^2 y
    girth = 4 x
    length = y
    so y + 4x </=108
    since maximizing
    y + 4x = 108
    or
    y = 108 - 4x

    v = x^2 (108-4x)
    v = 108 x^2 - 4 x^3
    dv/dx = 216 x - 12 x^2
    = 0 for max or min
    so
    x(216 - 12x) = 0
    x = 18 for max
    then y = 108 -4(18) = 36

  • Calculus (Optimization) -

    Let the width and the height of the box both be x inches. (That makes the end a square)
    Let the length by y inches
    distance around lengthwise = 2x + 2y
    distance around widthwise = 4x
    (think of a ribbon around a box when you wrap for Christmas)
    distance = 6x+2y= 108
    y = 54 - 3x

    I will assume that by "largest box" you mean greatest volume.
    V = x^2 y
    = x^2(54-3x)
    = 54x^2 - 3x^3
    dV/dx = 108x - 9x^2 = 0 for a max V
    9x^2 = 108x
    x = 12
    then y = 54 - 36 = 18
    the largest box is 18" long, 12" wide, and 12" high.

  • Calculus (Optimization) -

    Reading the question again, I think that I took the wrong interpretation and Damon took the right one.

  • Calculus (Optimization) -

    It is just length + x^2
    not girth in both directions
    http://pe.usps.com/text/qsg300/Q401.htm

  • Calculus (Optimization) -

    I mean length + 4x

  • Calculus (Optimization) -

    Both of you, thank you very much!!! I arrived at the correct answer width = 18 and length = 36, but I just got that answer by chance and wasn't sure how I could prove (mathematically) that it was indeed correct, your explanations helped tremendously!

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