Results 1 to 2 of 2

Thread: Decorate My Christmas Tree

  1. #1
    MHF Contributor
    Joined
    Dec 2013
    From
    Colombia
    Posts
    1,788
    Thanks
    581

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,700
    Thanks
    2395

    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:

    $\ell = \dfrac{4 \pi a b \sqrt{4 a^2 \left(\pi ^2 b^2+h^2+1\right)+b^2 h^2}+\left(4 a^2 \left(h^2+1\right)+b^2 h^2\right) \left(\ln \left(4 a^2 \left(h^2+1\right)+b^2 h^2\right)-2 \ln \left(\sqrt{4 a^2 \left(\pi ^2 b^2+h^2+1\right)+b^2 h^2}-2 \pi a b\right)\right)}{16 \pi a b h}$
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum