Saturday, October 22, 2011

The Golden Arches ® Exposed!

On the blog dy/dan, there was a recent discussion about the shape of a bridge.  One commenter joked about the Golden Arches of McDonald's as being the most famous parabolas in America.  As I am teaching quadratic modeling in College Algebra right now, I though I would find an equation for these parabolas.  The problem is that the Golden Arches are not parabolas.

To find the equation, I scanned the Golden Arches from a placemat that I found at McDonald's tonight.  I took the scanned picture and imported it into GeoGebra.  I picked five points on the outside of one arch, and five points inside of the arch.  The picture is below.
Ten points through the Golden Arch
For the outside of the arch, I chose the points (0,0), (3.73, 11.8), and (6.62, 7.87).  By many calculations, I got the parabola y = -0.68329x^2 + 5.71222x.  After even more calculations to get the focus and directrix, I was able to graph the parabola in GeoGebra.  I used the program Maxima to perform the calculations.  The parabola is below.
The parabola through the points on the outside of the arch
For the inside of the arch, I chose the points (1.63, 0), (3.73, 10.94), and (5.86, 0.86).  The parabola I got was y = -2.35034x^2 + 17.80732x - 22.78133.  The second parabola went much faster because I was able to reuse my Maxima worksheet from the previous parabola.  The graph is below.
The parabola through the points on the inside of the arch.
Just on a lark, I used the conic section tool in GeoGebra to draw the conic sections through the two sets of five points for each arch.  Geogebra came up with two ellipses to pass through the points.  The ellipses look like they match the boundary of the arches pretty well.
Ellipses work well to model the arches