Question

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).

Answer 1

well, we know

4x+h = 96, so h=96-4x
v = hx^2
= x^2(96-4x)
= 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 = 96-4*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.