Now you have to find the critical point of this multivariable function.
Now, set and . Observe that
But this implies that . Therefore or .
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.)