# Thread: Decorate My Christmas Tree

1. ## Decorate My Christmas Tree

I bought a new Christmas tree this weekend, and I need to know how many lights to buy.

The tree is triangular in silhouette, $a$ metres tall and the lower branches are $b$ metres long. I'm going to attach one end of the string of lights to the top of the tree and wind them round the ends of the branches in spiral so that each revolution round the tree drops the string a constant vertical distance $\Delta h$. The other end of the lights will be fixed exactly at the end of one of the lower branches.

The question is, for a given $\Delta h$, how long does the string of lights need to be. Or, equivalently, for a string of lights of $l$ metres, what will $\Delta h$ be.

2. ## Re: Decorate My Christmas Tree

I get a closed form answer that's fairly complicated so it makes me feel like I'm missing something.
Spoiler:

$\ell = \dfrac{4 \pi a b \sqrt{4 a^2 \left(\pi ^2 b^2+h^2+1\right)+b^2 h^2}+\left(4 a^2 \left(h^2+1\right)+b^2 h^2\right) \left(\ln \left(4 a^2 \left(h^2+1\right)+b^2 h^2\right)-2 \ln \left(\sqrt{4 a^2 \left(\pi ^2 b^2+h^2+1\right)+b^2 h^2}-2 \pi a b\right)\right)}{16 \pi a b h}$