Building out a van

Question

I need insulation for my van. 110' x 69 x 5'11 high. How much insulation is needed? It will be between 1/2" thick. I know it's optimization but that's where I stop! I would want the maximum of the optimization...I think.

Answer 1

This isn't really the right website for these kinds of questions - which is why your question is attracting downvotes - but, since I find the question mathematically interesting, I've given it an upvote and I'll answer your question.

Assumptions. Assume your van is an $a \times b \times c$ rectangular cuboid. Assume your insulation is going to be $k$ units-of-length thick.

Approximate value. The surface area of your van is $2bc+2ac+2ab,$ so you'll need approximately $$2k(bc+ac+ab)$$ worth of insulation. For example, suppose $a=b=c=10$ and $k=1$. Then the approximate formula gives: $$2 \cdot 1 \cdot (100+100+100) = 600$$

But this actually overestimates the amount of insulation you'll need, because we're double and triple counting in places.

Exact value. The exact formula seems to be:

$$2k(bc+ac+ab - 2k(a+b+c)+4k^2)$$

For example, suppose $a=b=c=10$ and $k=1$. Then the exact formula gives:

$$2 \cdot 1 \cdot (100+100+100-2(10+10+10)+4\cdot 1^2) = 488$$

Just replace $a,b,c$ and $k$ with your own numbers in your preferred units to solve the specific problem you're having.