Posted by Brian on Wednesday, April 4, 2012 at 11:45pm.
Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 96 in. Find the dimensions of a rectangular package that has a square cross section and the largest volume that may be sent via priority mail. Hint: The length plus the girth is 4x + h (see the figure below).

Math  Steve, Thursday, April 5, 2012 at 11:15am
well, we know
4x+h = 96, so h=964x
v = hx^2
= x^2(964x)
= 96x^2  4x^3
= 4(24x^2  x^3)
dv/dx = 4(48x  3x^2)
= 12x(16  x)
dv/dx = 0 when x = 0 or 16
so, h = 964*16 = 32
So, if the package is 16x16x32
v = 8192 in^3
Now, technically, 4x+h <= 96, but it should be clear that max volume occurs when 4x+h is as large as possible.
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