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!
[Math] Optimization triangular prism
optimization
Best Answer
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 $$10.3226\cdot1.75 = 18,06455.$$