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Solution

“surface area equals 6000”
2xy + 2xz + 2yz = 3000
“(sum of lengths of all edges) is 400”
4x + 4y + 4z = 400
“It can be deduced that the box is not a cube”
Yes, it is obvious.
“let x represent a side with x does not = y and x does not = z”
OK

xy + xz + yz = 6000
x + y + z = 60

x = 60 – y – z

(60 – y – z)y + (60 – y – z)z + yz = 6000
(y + (z – 60)/2)² = (z – 60)²/4 - 1000 - z² + 60z

The left side is a square, so the right side is non-negative. It helps us to find the range od z.

(z – 60)²/4 - 1000 - z² + 60z 0
3z² - 120z + 400 0
20 - 206/3 z 20 + 206/3
3.67007 z 36.32993
That’s the range of z. Because of symmetry the range of y is the same.

(y + (z – 60)/2)² = (z – 60)²/4 - 1000 - z² + 60z
y = 30 - z/2 + (-0.75z² + 30z - 100)
y = 30 - z/2 - (-0.75z² + 30z - 100)

The volume of the box is
V = xyz = (60 – (30 - z/2 + (-0.75z² + 30z - 100)) – z) (30 - z/2 + (-0.75z² + 30z - 100)) z
or
V = xyz = (60 – (30 - z/2 - (-0.75z² + 30z - 100)) – z) (30 - z/2 - (-0.75z² + 30z - 100)) z
(because of symmetry both formulas are equivalent).

To find the maximum and maximum volume need some more routine work. The result is:

The maximum volume is V(max)=5088.66211 for z=11.835.
The minimum volume is V(min)=2911.33789 for z=28.165.