The dimensions of the package of maximum volume that can be sent are 20 inches by 40 inches.
Let us represent the length of the square cross-section of the postal package with x, and the width of the package with y.
So, the perimeter is given by the equation -
y + 4x = 120
y = 120-4x -----------(1)
the volume will be
V = -----------(2)
Now, substitute (1) in (2), and we get,
V = (120-4x)
V = 120 - 4
Differentiating with respect to x, we get,
V' = 240x - 12
V'=0
hence, 240x - 12 = 0
12 = 240x
dividing by 120x on both sides,
hence, x=20
Now, we know that,
y = 120 - 4x
y = 120-4(20)
y = 120-80
y=40
Hence, the dimensions that maximize the volume of the package are 20 inches by 40 inches.
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