algebra

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Amy's project in metal shop class is to make a box (with no lid). Her shop teacher gave her a piece of metal that is 28 cm long and 20 cm wide. The assignment is to form the box by cutting squares out of each corner and bending the sides as shown below. The shop teacher, Mr. Rust, will give extra credit to any student who forms the box that holds the most when the cut-out square is a whole number of centimeters on each side. Amy wants the extra point! How large a square should she cut from each corner to form the box with the greatest volume?

  • algebra -

    Let x = side of square to cut off from each corner
    20 - 2x = width of bottom rectangle
    28 - 2x = length of bottom rectangle
    V = x(20 - 2x)(28 - 2x) = 4x(10 - x)(14 -x) =
    V = 4x(140 - 24x + x²)
    V = 4(140x - 24x² + x³)
    Differentiating and equating to zero
    dV/dx = 4(140 - 48x + 3x²) = 0
    3x² - 48x + 140 = 0
    x = {48 ± √[48² - 4(3)(140)] } /2(3)
    x = {48 ± √624] } /6
    x = {48 ± 4√39] } /6
    x = (24 ± 2√39) / 3
    x = (24 + 2√39) / 3 = 12.16 Discard, > ½20
    x = (24 - 2√39) / 3 = 3.8 inch
    Use x = 4 since whole number of cm is required.
    So each square to cut is 4 cm on the side.

    4 cm squares gives a box of 4 x 12 x 20 = 960 cm^3

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