Is the given shape a parabola? What is the formula?

Specifically, are those golden arches in the exact or near to exact form of a parabola?

To use Digitizer , we import the McDonald's image first:

Then, we scale the picture and set the coordinates and choose the first batch of points.

Following, these points are copied from the data window, pasted into Excel and parsed into two columns.

We employ the familiar formula for a parabola parallel to the -axis given by

Thus the constant is determine from any ordered pair of data by taking the ratio

After obtaining our coordinates along the "suspected" circle we compute the value for each

We confirm that the curve in questionmay be a parabola if all the values are about the same. Our data gives the following values.

Except for the first value, for which the value is very close to zero and for which much error is possible, all these values seem to be about the same. However, if we take points on the side of the same arch and also the other arch, we obtain a different story. The new digitized image is

Our spreadsheet is

In both these cases it is clear that the values are not nearly equal, and we make exactly the opposite conclusion. The MacDonald's arches are not parabola. This is a curious case where "two little" data is just not enough to generate the desired results. (Of course, for the second set of data we used the base formula, . Why?)

In summary, here are the steps followed.

• First import the image into Digitizer . Scale as needed. :

• We eventually considered both arches. Moreover, we'll check just the outer shape. Define an origin (the red cross) and a point on the -axis (blue cross). Specify its value. In the present case we selected the value to be 10. The exact value is not needed because we are determining the shape, not any particular measurements.

• Now digitize points along the circle and show the data.

One may also rotate the image and use a different form for the parabola.

In this case we employ the familiar formula for a parabola parallel to the -axis given by

Thus the constant is determine from any ordered pair of data by taking the ratio

After obtaining our coordinates along the "suspected" circle we compute the value for each

We confirm that the curve in question is a circle if all the values are about the same.

Here's are the results.

• 
  
  

• The final step is the computation of the ( ) for each of the data points. Note that this method does not require multiple data points for a single computation.
  
  As is evident the data does not support the hypothesis that the curve is a parabola.

In summary, we required mathematics to use a proper model to test our hypothesis, the digitizer to make coordinates, and a spreadsheet to analyze the data.

Questions

1. Discuss the value of the data above, which comes from an value close to zero. What general conclusions can we draw?

2. What conclusions can you make about the selection of data?

3. What can we conclude are about the golden arches? For example, are the two arches the same?

4. Looking carefully, you can see that the picture is on a slight angle. What effect might this have on the outcomes?

5. Here is a link to the St. Louis Arch. Perform a similar analysis on this famous arch. (http://arteagaphotos.com/archprint.htm). Clearly the outer edge is different from the inner edge. Can this effect the outcome?