Maximum volume of triangular prism given the surface area

Cyd

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How do I calculate the maximum volume of a triangular prism with a total surface area of 2000 square centimetres? I think I'm right that the triangular base will be equilateral, and I have solved it assuming all sides, including the height, are equal. However, I am certain that while this has got me closer to the maximum volume possible, that this is not the complete answer/the correct way to solve it. Hopefully someone can help. Thanks!
 
How do I calculate the maximum volume of a triangular prism with a total surface area of 2000 square centimetres? I think I'm right that the triangular base will be equilateral, and I have solved it assuming all sides, including the height, are equal. However, I am certain that while this has got me closer to the maximum volume possible, that this is not the complete answer/the correct way to solve it. Hopefully someone can help. Thanks!

So far, you seem to be doing nothing but guessing (assuming). Were you told anything more than what you say in the first sentence? I suppose in your context it might be understood that you mean a right prism, not an oblique prism; but you would have to be told if is it assumed to be equilateral, or else prove it.

In order to help you actually solve it, we'll have to know what you have learned. For example, are you learning one-variable calculus, multivariable calculus, or something else? Also, is this a specific assignment, or just a question of your own? Have you been given any similar examples?
 
So far, you seem to be doing nothing but guessing (assuming). Were you told anything more than what you say in the first sentence? I suppose in your context it might be understood that you mean a right prism, not an oblique prism; but you would have to be told if is it assumed to be equilateral, or else prove it.

In order to help you actually solve it, we'll have to know what you have learned. For example, are you learning one-variable calculus, multivariable calculus, or something else? Also, is this a specific assignment, or just a question of your own? Have you been given any similar examples?

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.
 
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 cm3, 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.
 
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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 cm3, 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.

Thank you very much. It is indeed a grade 8 assignment. I can only assume it was created and assigned by a teacher who really had no understanding of how complex the question actually was. My guess is that she expects the students to find the largest volume they can (not necessarily THE largest possible). My niece, however, recognized that the proper way to answer the question would be by following some sound logic and mathematical reasoning. Unfortunately the math required is far beyond her level and current capabilities. 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. 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. Thanks again for confirming my feeling that this is indeed far beyond the grade 8 classroom within which it was assigned. Again, many thanks!
 
Thank you very much. It is indeed a grade 8 assignment. I can only assume it was created and assigned by a teacher who really had no understanding of how complex the question actually was. My guess is that she expects the students to find the largest volume they can (not necessarily THE largest possible). My niece, however, recognized that the proper way to answer the question would be by following some sound logic and mathematical reasoning. Unfortunately the math required is far beyond her level and current capabilities. 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. 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. Thanks again for confirming my feeling that this is indeed far beyond the grade 8 classroom within which it was assigned. Again, many thanks!

Good for your niece!

I would love to see what answer the teacher considers "correct" (including work).

Yes, it's quite possible that "the largest you can make" was intended.
 
As the assigning teacher, let me add a bit of clarification...

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. 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. Students were told to begin with any rectangular prism, and use the "guess-and-check" method to get as close as possible to 2000 square centimeter surface area, while attempting to get the maximum volume possible for them. Almost everyone has arrived at the conclusion that the cube would be the required shape.

As far as the triangular prism goes, it was certainly not my expectation that anyone would use calculus to determine the dimensions of the prism. That's well above what ANY Grade 8 student should know. However, it is reasonable to assume that the students can manipulate the side lengths and see how it affects the volume of the prism.

Sorry if this is causing any of the students to fret. I'll be sure to clarify things in class, although I thought it was already fairly clear.
 
As the assigning teacher, let me add a bit of clarification...

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. 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. Students were told to begin with any rectangular prism, and use the "guess-and-check" method to get as close as possible to 2000 square centimeter surface area, while attempting to get the maximum volume possible for them. Almost everyone has arrived at the conclusion that the cube would be the required shape.

As far as the triangular prism goes, it was certainly not my expectation that anyone would use calculus to determine the dimensions of the prism. That's well above what ANY Grade 8 student should know. However, it is reasonable to assume that the students can manipulate the side lengths and see how it affects the volume of the prism.

Sorry if this is causing any of the students to fret. I'll be sure to clarify things in class, although I thought it was already fairly clear.

Thanks for writing.

This is among the reasons we ask students to show the entire problem as given to them, and to explain the context -- so we can be sure what they are being asked to do, and how. That rarely happens, however.

All I can go by is what we were told; and all a parent can go by is what is passed on to them by the child (orally, or on paper). But if the problem actually asked for "the maximum volume of a triangular prism with a total surface area of 2000 square centimetres", then you were giving the students an impossible assignment. You didn't tell them to use calculus, of course, but without that an exact answer is impossible (or nearly so), and that is what the question is asking for (outside of the context that everything you were having them do was guess-and-check). That is going to be frustrating for good students who recognize this. (Poor students would just plow through the problem and not be bothered by it.)

If it was worded more like "What is the largest triangular prism you can find ...", or "Experiment with triangular prisms to see if you can find the largest ...", or otherwise making it clear that what is expected is a guess-and-check approximation rather than an actual maximum, it would be much clearer. Math is an exact subject, unless stated otherwise; so while students should be learning that both approximate methods and exact methods are valid, they should also know the difference.

None of this should be taken as a criticism of you as a teacher, but it does nicely demonstrate the importance (and difficulty) of communicating about math, especially to and through children. I've seen plenty of situations where parents are stymied because what the child brings home does not provide the background needed to see how they are expected to solve a problem, and the child can't tell them.
 
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