Optimizing Chocolate Packaging with Equilateral Triangular Prisms

  • Context: Undergrad 
  • Thread starter Thread starter decibel
  • Start date Start date
  • Tags Tags
    Optimization
Click For Summary

Discussion Overview

The discussion revolves around optimizing the dimensions of an equilateral triangular prism package for chocolate, specifically aiming to minimize the surface area while maintaining a fixed volume of 400 cm³. Participants explore mathematical formulations and approaches related to this optimization problem.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant identifies the need to express the volume of the prism as a function of its dimensions, proposing that the volume constraint leads to a relationship between the side length of the triangle and the height.
  • Another participant suggests that minimizing the surface area is equivalent to minimizing the amount of material used, and outlines a method to express surface area as a function of one variable by substituting the volume constraint.
  • A third participant provides a detailed breakdown of the surface area calculation, emphasizing the relationship between the height of the triangle and its base, and introduces the concept of using the method of Lagrange multipliers for optimization.
  • There is a request for further assistance, indicating that the initial poster is struggling with the formulation and calculations.

Areas of Agreement / Disagreement

Participants generally agree on the need to minimize surface area while adhering to the volume constraint. However, there are differing approaches to how to formulate the problem and apply optimization techniques, indicating that the discussion remains unresolved.

Contextual Notes

Participants have not reached a consensus on the best method for optimization, and there are unresolved mathematical steps regarding the relationships between the dimensions of the prism.

Who May Find This Useful

This discussion may be useful for students or individuals interested in optimization problems, particularly in the context of geometry and packaging design.

decibel
Messages
107
Reaction score
1
A choclate manufacturer uses an equilateral trianglular prism package. if the volume of chocklate to be contained in the package is 400 cm ^3 . what dimenesions of the package will use the minumum amount of materials?

i'm having trouble putting the formulas together, I am thinking of the front and back of the prism as being A=2(1/2bh) then i don't know wut to do with the 3 rectangular peices in the middle...can someone push me in the right lane here?...this is urgent, i need to do this in the next couple of hours, thanks in advance.
 
Physics news on Phys.org
I understand that this is a triangular prism.

Since the triangular is equilateral, this triangular prism is defined by only two parameters: the length of the side of the triangle (call it L) and the length of the sides connecting the two triangles (call it M).

Since the volume of the triangular prism has to equal 400. We get one equation, namely the volume as a function of L and M, Volume(L,M) = 400. That is our restraint equation.

Now we want to minimize the amount of material, which I interpret as pretty much the same as the surface area (unless someone else thinks of something else better). So we take the surface area as a function of L and M, SurfaceArea(L,M). We plug in stuff we know from the restraint to get L in terms of M or vice versa. Then we will get surface area as a function of only one variable SurfaceArea(L) or SurfaceArea(M). Then minimize the function with respect to the variable. Then use the restraint equation to figure out the other variable.

Hope all this makes sense.
 
You are starting out correctly: the two ends each have area (1/2)bh so the total area of the two ends will have area bh. But what is h? Since you are told that this is an "equlateral triangular prism", if you divide the end triangle into two right triangles, each has hypotenuse of length b, one leg of length (1/2)b and one leg of h. Now use the Pythagorean formula to find h in terms of bl: h= (sqrt(3)/2)b. Of course, each rectangular side has area bw where w is the length of the package. The total area is
sqrt(3)b+ 3bw while the volume is (1/2)bhw= sqrt(3)b2w= 400. Since we are not given any relation between w and b, we will need to treat this as a function of two independent varialbles and use the "Laplace multiplier" method. The gradient of the area function is the vector <sqrt(3)+ 3w, b> while the gradient of the condition is <2sqrt(3)bw, &radic;(3)b2>. Since one must be a multiple of the other, we must have 2√(3)bw= &lambda;(√(3)+ 3w) and sqrt(3)b2= &lambda; b.
 
Last edited by a moderator:
help appreciated thanks
 

Similar threads

Undergrad Cylinder vs Rectangular Prism: Which is More Economical for Juice Packaging?
  • · Replies 2 ·
Replies
2
Views
3K