We can finally calculate the volume of our theoretical mouthpiece! Again we use the Law of Sines to calculate the length of the cone, l. Here’s the original diagram so you don’t have to refer to another page:

We know that angle c is equal to e’ or e both of which are 1.16 degrees, we know the side opposite angle c is r which is 0.660 cm, and we can calculate angle b since we know the value of e’. Side l is opposite angle b so we’re all set to do the arithmetic:

That is one slender cone! 1.32 cm in diameter and 32.6 cm long, but what’s its volume? One more formula is needed:

and we can calculate the volume of our theoretical mouthpiece:

While we may feel a certain amount of satisfaction with this, we still have to compare this volume to the volume of a real mouthpiece. Before we can do that there will be a short physical science review in: The Science Lesson