volume of triangular prism without height

#Volume of triangular prism without height how to
#Volume of triangular prism without height trial

The same question, applied to rectangular prisms, was meant to lead the students to the realization that the maximum volume would be achieved by creating a cube.

#Volume of triangular prism without height how to

The assignment was meant to see that students know how to use the formulae for both surface area and volume of rectangular and triangular prisms. Again, many thanks!Īs the assigning teacher, let me add a bit of clarification. Thanks again for confirming my feeling that this is indeed far beyond the grade 8 classroom within which it was assigned. Not entirely trusting my original answer, i attempted the trials and discovered my original answer was a bit off and that the sides of the triangle were in fact larger than the height of the prism.

#Volume of triangular prism without height trial

I arrived at a similar answer as you did following some trial and error after solving it based solely on my instinct that all sides should be equal in order to maximize the volume. Unfortunately the math required is far beyond her level and current capabilities. My niece, however, recognized that the proper way to answer the question would be by following some sound logic and mathematical reasoning. My guess is that she expects the students to find the largest volume they can (not necessarily THE largest possible). I can only assume it was created and assigned by a teacher who really had no understanding of how complex the question actually was. 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. Do they have those? The algebra still gets a little ugly, but a good algebra student could handle it.Īs for you, if you want advice on doing the calculus, it's a standard optimization problem, apart from the ugliness. One way non-calculus students could solve this is with a graphing calculator. 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?Īfter 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. I can't imagine any way short of calculus that would be valid. My big question, before I go through my work again, is what this is doing as an 8th grade assignment. It's not too hard in principle, but my quick attempt just now gave a different answer than yours. I would certainly use calculus as you suggest. (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.

(They might instead assume it's a right triangle, just to make some things simpler, but that's unlikely to be true.)Īssuming the height is also the same is reasonable as a first guess (given what we know about cubes), but far less justifiable. If this makes sense as an 8th grade problem at all, the assumption that it's equilateral is natural and probably necessary. If you were a student of multivariable calculus, I would expect you to prove everything. This is not to earn a mark for a course, but rather to satisfy my own curiosity. I have searched quite a bit to find an example of a similar question, but cannot seem to find one on the internet. I am also fairly certain I need to apply some calculus and find a derivative somewhere along the line to solve it properly.

(I believe the answer is very close to a volume of 5336 cubic centimetres. (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 solved and determined a side length of 22.74cm presuming an equilateral triangle and an equal height for the prism. I am aware that critical information is missing from the question, but that is all that was provided. I have seen the assignment page, but it is not currently with me. 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. 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.