From what you have in your SA equation, the container has no lid! If it has a lid, then

Therefore, .

Now you have to find the critical point of this multivariable function.

Therefore,

and

Now, set and . Observe that

Therefore,

But this implies that . Therefore or .

If then

Therefore, or . Since length can't be negative, we have .

As a result, . Therefore, .

Therefore, the rectangular box that has maximum volume and SA = 96 sq. meters is a cube with dimensions meters.

You can verify that this is a maximum by applying the second derivative test at the point .

Does this make sense?

(I'm sure you could have applied Lagrange Multipliers here to with the restriction (which could cut back on the calculations that are required), but this is what came to my mind first.)