We might as well get the hard stuff out of the way first!
The first assumption (or leap of faith) we have to make is that our horns are cones that are missing their tips. Cones that have been truncated are called circular cone frustums. Rather than trying to determine the dimensions of the entire horn we’ll cut across the cone again by removing the neck and measure only that. The tasks we have at hand are:
- Think through the problem and get some idea of what math we’re going to need
- Determine the dimensions and taper of the neck
- Derive the dimensions of the missing cone from the dimensions of the neck
- Calculate the volume of the missing cone
- Measure the volume of a mouthpiece to see if it conforms to theory
I’ve already done most of item 1 and drawn the following:
NO! NO!! Don’t run away! I know it looks like all the horrid stuff from high school geometry but what we’re trying to accomplish is fairly straight forward. We can measure the openings in each end of the neck easily enough (R and r). The actual length of the neck (L) is a bit harder . . . in fact bloody hard. So instead we’ll measure the length of the side of the neck (S).
We in fact don’t care about the volume of the neck – what we really need is the angle e between lines L’ and S. There’s an easy formula for that and once we have it we can figure out all the angles in the upper cone (a by definition is 90 degrees, b, and c). When we have that information a couple more simple formulas will give us the volume of the theoretical mouthpiece.
NEXT: Measure Twice