Package Size

What is the largest package volume that can be shipped and the package dimensions at the maximum volume?

The limit on package size is that the girth plus the longest dimension cannot exceed 180 inches. Consider two kinds of packages, a rectangular prism shape with square ends and a cylindrical shape with the cylinder height greater than the diameter (girth equals circumference).

Determine for each shape the largest package volume and the dimensions at the max volume.

Let the prism have square ends with area w^2, and length l (letter L), where l ≥ w. The constraint is l + 4w = 180 ==> l = 180 - 4w, and the volume V = lw^2 = (180 - 4w) w^2 = 180 w^2 - 4 w^3.

dV/dw = 360 w - 12 w^2 = 0 ==> 12 w = 360 ==> w = 30 (Answer)

l = 180 - 4w = 60 (Answer)

V = 30 x 30 x 60 = 54,000 cu. in. (Answer)

For the cylinder, let the diameter be d, and the height h, with h > d. The constraint is h + πd = 180 ==> h = 180 - πd. The volume is V = h (πd^2)/4 = (π/4) (180 - πd) d^2 = 45πd^2 - π^2 d^3 / 4

dV/dd = 90πd - (3π^2 / 4) d^2 = 0 ==> πd = 120 ==> d = 120/π (Answer)

h = 180 - 120 = 60 (Answer)

You figure out the volume of the cylinder. The cylinder volume ought to be greater than 54,000 cu. in.

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