Investigation of a n-sided prism.

dylanbui

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Jul 28, 2019
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Currently I am working on a project with optimization of prisms with a regular polygon base. It is similar to the investigation detailed here:https://ibmathsresources.com/2017/05/21/optimization-of-area-an-investigation/. In my project, a surface area of 100 is , and I'm trying to find the maximum volume a n-sided prisms will have. Anyways the problem is I end up with only two equation with three variables, and I am trying to find the optimal volume.

Sorry if this explanation is confusing, the attachment should explain it better.

image0.jpeg
 

Dr.Peterson

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You haven't quite stated what your goal is. Is it, first, to find the largest possible volume for a regular n-gonal prism with surface area 100? I see only two variables there (s and h, taking n as fixed), so two equations should be all you need. It looks like you've expressed V as a function of s alone; find the maximum.

Then you can consider what happens when n changes.
 

LCKurtz

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To simplify a bit what you have written, let's call \(\displaystyle A_b\) the area of the base, \(\displaystyle S\) the total surface area, \(\displaystyle n\) the number of sides, \(\displaystyle s\) the length of a side, and \(\displaystyle h\) the height. So you have \(\displaystyle A_bh = 100\) and \(\displaystyle S=2A_b + nsh\). You can eliminate the \(\displaystyle A_b\) between these two and write \(\displaystyle S = \frac {200} h + nsh\). But \(\displaystyle n\) and \(\displaystyle s\) depend on each other, and both depend on the radius, which you have called \(\displaystyle d\), of the circumscribing circle. Is a fixed radius given or is it allowed to vary? The relation between \(\displaystyle d\) and \(\displaystyle s\) is \(\displaystyle d = \frac{s}{2\sin\frac \pi n}\).
 

dylanbui

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You haven't quite stated what your goal is. Is it, first, to find the largest possible volume for a regular n-gonal prism with surface area 100? I see only two variables there (s and h, taking n as fixed), so two equations should be all you need. It looks like you've expressed V as a function of s alone; find the maximum.

Then you can consider what happens when n changes.
Yes, that is my goal. I am trying to find an equation that gives the optimal height and side for a given n.
I am a bit confused about n being fixed.
I have expressed V as a function of s and n.
 

dylanbui

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To simplify a bit what you have written, let's call \(\displaystyle A_b\) the area of the base, \(\displaystyle S\) the total surface area, \(\displaystyle n\) the number of sides, \(\displaystyle s\) the length of a side, and \(\displaystyle h\) the height. So you have \(\displaystyle A_bh = 100\) and \(\displaystyle S=2A_b + nsh\). You can eliminate the \(\displaystyle A_b\) between these two and write \(\displaystyle S = \frac {200} h + nsh\). But \(\displaystyle n\) and \(\displaystyle s\) depend on each other, and both depend on the radius, which you have called \(\displaystyle d\), of the circumscribing circle. Is a fixed radius given or is it allowed to vary? The relation between \(\displaystyle d\) and \(\displaystyle s\) is \(\displaystyle d = \frac{s}{2\sin\frac \pi n}\).
I am not sure what you mean about being inscribed in a circle, but d can vary based on n and s.
It is the surface area that equals 100, so S=100
 

Dr.Peterson

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Yes, that is my goal. I am trying to find an equation that gives the optimal height and side for a given n.
I am a bit confused about n being fixed.
I have expressed V as a function of s and n.
Find the optimal volume with n=3, then with n=4, and so on. Most of the work will be done in the general case (leaving n as a variable, but treating it as a constant).
 
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