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I am working on a math project on optimisation and have a triangular prism (tent) with a given volume of 2.8 cubic meters. The "top" of the tent is out of a water-resistant material for 2.25 dollars per square meter, and the floor material costs 1.75 dollars per square meter. I am supposed to find the dimensions of the tent while minimizing the costs. I just spent these past two hours trying to figure it out without much outcome. I would appreciate any help! Thank you!

Conny Schneider
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  • Do you mean triangular pyramid? A triangular prism has a triangle for a top and a bottom, which does not look like a tent. – kccu May 18 '17 at 02:12
  • @kccu Turn the prism on it’s side, like a pup tent with a floor. – amd May 18 '17 at 05:34
  • I see, I was picturing a tent with one point at the top. – kccu May 18 '17 at 17:52
  • So, is the tent a prism, as you’ve written, or a tetrahedron (pyramid)? The answer you accepted solves the problem for the latter. – amd May 19 '17 at 02:40

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This boils down to an optimization problem of a function of 3 variables subject to a constraint:

You can vary 3 parameters (height, base width, base length), for which you have a "Price Function" which is the sum of all the material times its respective cost (this is a trivial problem in geometry). Thus you wish to minimize the "price function" c(h,w,l)

You are constrained by the fact that the volume (also a function of h,l,w) is constrained.

This is the textbook application of Lagrange Multipliers. If you are unfamiliar with them, I would check out Paul's Online or Khan Academy, as they both provide great explanations. I hope this helps!

Andrew Murphy
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Let $a,b,c$ are the sides of base and $h$ is the height of prizm. Then the area of base is $$S_b = {1\over4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)},$$ the volume is $$V = S_bh$$ and required quantity of material is $$S = S_b + (a+b+c)h.$$ Using the condition $V=2.8,$ we have $h={2.8\over S_b},$ so we have to minimize $$S = S_b + 2.8\,{a+b+c\over S_b},$$ $$S={1\over4}\sqrt{a+b+c}\sqrt{(-a+b+c)(a-b+c)(a+b-c)}$$ $$+{11.2\,\sqrt{a+b+c}\over\sqrt{(-a+b+c)(a-b+c)(a+b-c)}}.$$ for $a, b, c >0.$

Using Wolfram Alfa, one can get $$S=10.3226\text{ at }(a, b, c)\approx(2.81892, 2.81892, 2.81892),\ h\approx 0,27125$$ with the price $$10.3226\cdot1.75 = 18,06455.$$

Yuri Negometyanov
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