What values for the dimensions of the package lead to maximum volume given a constraint on length + girth? Suppose postal requirements are that the maximum of the length plus the girth (cross...

What values for the dimensions of the package lead to maximum volume given a constraint on length + girth?

Suppose postal requirements are that the maximum of the length plus the girth (cross sectional perimeter) of a rectangular package that may be sent is 275 inches. Find the dimensions of the package with square ends whose volume is to be maximum.

Square side:

Length:

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If `a` is the length of one of the square sides and `l` is the length of the package, we have that the required girth `G` satisfies

` ``G = l +4a`

We want the girth to be as big as possible as this will lead to a bigger volume, so we have that `G=275` (the maximum girth allowed).

The volume `V` of the package satisfies

`V = la^2`

To maximise `V` with...

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