Applied Calculus

If an open box is made from a tin sheet 7 in. square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made. (Round your answers to two decimal places.)

Height:
Length:
Width:

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  1. base = 7- 2x
    height = x

    volume = x(7-2x)^2
    = 49x - 28x^2 + 4x^3
    d(volume)/dx = 49 - 56x + 12x^2
    = 0 for a max volume

    12x^2 - 56x + 49 = 0
    x = (56 ± √784)/24
    = (56 ± 28)/24 = 3.5 or 7/6 or 1.1666...
    but clearly x < 3.5 or we have cut the whole base away.

    base is 7 - 2(7/6) = 14/3 by 14/3
    and the height is 7/6

    round to your required decimals

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  2. new length = 7 - 2 h
    height = h
    volume = (7-2h)(7-2h)(h)
    v = (49 -28 h + 4 h^2)h
    so
    v = 4 h^3 -28 h^2 + 49 h
    dv/dh = 0 for max or min

    dv/dh = 12 h^2 -56 h + 49 = 0

    (6h -7)(2 h -7) = 0
    h = 7/6 or h = 7/2
    if h = 7/2, the box has zero bottom
    so the answer is h = 7/6
    7 - 2(7/6) = 7-7/3 = 14/3 = length and width

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