How Can Joe Optimize the Spacing of His Christmas Tree Lights?

[clanijos]

Hello there! I was decorating my Christmas tree recently, and for some strange reason, I thought: "Hrm, I wonder if I could come up with an optimization problem where I have a definite length of lights/garland, and want to have equal space between each strand of lights/garland as they go around. How big would the spaces be?"

So I thought about it, and I can't seem to figure out how I would relate the values I gave myself. I suppose we must have dimensions of the Christmas Tree (conical form), so feel free to use whatever values you feel to be reasonable.

Perhaps the distance isn't even the most interesting thing to evaluate, what other variables can you come up with?

Any help would be appreciated! Happy Holidays!

 

[Stephen Tashi]

If we have a given cone (christmas tree) what the shortest line segment (string of lights) that we can spiral around it such that no point on the surface of the cone is farther than d ( some given distance) from some point on the line segment.

...or would that be too hard to solve?

 

[clanijos]

Yeah, that's perfect! Anyone want to take a stab at solving it?

 

[clanijos]

Anyone care to educate me as to how one does this problem? (Oh yeah, and *bump*)

 

[Stephen Tashi]

It's probably a hard problem unless you add additional conditions. For example you can require the line segment to be a spiral without proving that shape contains the most efficient shape. You can assume that in addition to the spiral there is a circle of lights around the bottom of the tree. Intuitively, on a line going "straight up" the side of the cone, the point that is equidistant between two points where the line that cuts the spiral is the farthest point on that line from the string of lights. So perhaps you can get all points on the surface to be within distance to the midpoint in all such situations is no farther than d as measured along such a line.

I don't really like to solve problems like this. You only asked for one to be make up. So, over to you! - or whoever is good a things like this.

 

[clanijos]

I was thinking something like:

Joe is decorating his christmas tree, he only has one [LENGTH] string of lights. If his tree is [HEIGHT OF TREE] tall, and has a base that is [WIDTH OF BASE] wide, how much space should Joe leave in between each row if he would like to have them spread evenly about the entire tree?

Perhaps the people who inhabit the "Calculus and Beyond" homework help forum could do something about it, I certainly can't see how it should be done.