Investigation of a n-sided prism.

dylanbui

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