Honestly, this is a grade 8 question that was given to my niece and I (an intermediate math teacher) am trying to figure out the answer. There is literally no other information included in the question aside from the total surface area and the fact that it is to be a triangular prism. I have seen the assignment page, but it is not currently with me. I am aware that critical information is missing from the question, but that is all that was provided. I solved and determined a side length of 22.74cm presuming an equilateral triangle and an equal height for the prism. (Yes, a guess, but I did actually do the work to determine this.) Some quick trial and error have shown me that, while my answer isn't ridiculously far off, it is not the correct one. (I believe the answer is very close to a volume of 5336 cubic centimetres. I am also fairly certain I need to apply some calculus and find a derivative somewhere along the line to solve it properly. I have searched quite a bit to find an example of a similar question, but cannot seem to find one on the internet. This is not to earn a mark for a course, but rather to satisfy my own curiosity.
Okay, context can make a big difference. If you were a student of multivariable calculus, I would expect you to prove everything. If this makes sense as an 8th grade problem at all, the assumption that it's equilateral is natural and probably necessary. (They might instead assume it's a right triangle, just to make some things simpler, but that's unlikely to be true.)
Assuming the height is also the same is reasonable as a first guess (given what we know about cubes), but far less justifiable. (And making a trial guess like that as a basis for checking a final answer, or even hoping it might be demonstrably optimal, is quite reasonable. There's nothing wrong in doing that.)
I would certainly use calculus as you suggest. It's not too hard in principle, but my quick attempt just now gave a different answer than yours. That may well be my fault.
My big question, before I go through my work again, is what this is doing as an 8th grade assignment. I can't imagine any way short of calculus that would be valid. While I work on it again, can you confirm that the problem, more or less exactly as stated, is really an assignment given at that level? Might it have been an extreme challenge problem, or only asking for a guess? What topics have been covered that it might be intended to use?
EDIT:
After fixing my errors, I find the maximum volume (under the assumption that the base is equilateral) to be 5339.6 cm
3, with the base edge at 27.7 cm and height about 16 cm.
One way non-calculus students could solve this is with a graphing calculator. Do they have those? The algebra still gets a little ugly, but a good algebra student could handle it.
As for you, if you want advice on doing the calculus, it's a standard optimization problem, apart from the ugliness. You'll write a constraint equation for the surface area, and then maximize the volume by solving the constraint for h and putting that in the volume formula.