Distance Traveled by Bird

by Hector Gonzalez

Submitted : Fall 2016

In this problem, the cable is wrapped around a cone from the bottom and tightly wound all the way up. In order to figure out how far the bird must fly to unravel the cable, I had to make an assumption of the birds travel since it wasn’t expressly given to me and I had the direction from my problem to make my own assumptions. My physics T.A. James Kruczek provided me with this problem and we discussed finding the surface area of the cone and making the assumption that the surface area would have to match the cable length since there are no bald spots. I would then assume the bird flew perpendicular to the cone, creating a path parallel to the ground, and use the Pythagorean Theorem to find the distance the bird must travel relative to the cone’s placement on the ground. In simpler terms, the outstretched cable would be the hypotenuse and the distance from the cone would be the x-component of this proposed right triangle

While James and I thought this could work out, it didn’t seem to satisfy the requirement of using calculus III material. My mathematics advisor, Dr. McClendon, suggested using a cylindrical equation to find a helix that would measure the cable and that would be the distance the bird traveled. I’d also have to use parametric equations and integrals, so that satisfied my calculus III requirement. This also assumes that the bird unravels the cord by flying straight up. The conclusion I drew from this problem is that while it was very difficult, it was really interesting to not only be able to obtain a formula for any dimensions of the situation given (which will be shown later), but also see how just changing the assumption of the bird’s flight could extremely change the makeup of the problem. 

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