Question

Problem:
Find dimensions of a box that provide the greatest volume when the length plus the girth cannot exceed six feet.

I tried 2w+2h+l=6, l=6-2w-2h, but couldn't get to w=1, h=1, l=2, which is the correct answer.

Answer 1

2w+2h+l<6

volume=lwh
dv=whdl+hldw+lwdh=0

from the constraint
2dw+2dh+dl=0

so that is two equations, need one more. I don't see one. I suspect the problem meant for the girth to be a square, thus, h=w. In that case, two equations will suffice. dw=dh, and h=w

2dw+2dw+dl=0 or 4dw=-dl
and
w^2dl+2wldw=0

dl/dw=-4
dl/dw=-2l/w

or settint them equal
2l=4w or l=2w and then l=2h
but
2w+2w+l<=6
2w+2w+2w<=6 or w=1 and you can proceed from there to the solution

Answer 2

84 total volumn inches