Finding a Common Point of Origin

Background

Digit had data from three crime scenes and fit a parabola to each of them. Now her goal was to transfer the details from these three parabolas to a map of the area and see if they all met in a single point. You may recall from the story that they did not meet in a point, but the analysis led authorities to a very small trianglular area in a nearby town...

Preparing the Raw Data

Collecting the data wasn't easy in the story. Let's take a look at the data taken at the barn:

   

The three data points are not directly measurable, so she did the next best thing and found their locations in space, relative to what she called a baseline, the line in front of the barn in the drawings. Point A, for example, is located at the mark on the baseline, but shifted back from the baseline to A', as are the other two points, B and C.

The coordinates of the three points on the trajectory labeled A, B, and C are used to fit a parabolic curve. We fit a parabola to each set of data, from each of the three crime scenes.

Finding the Location on a Map

These three sets of raw data give us a parabolic curve for each trajectory, and each curve is defined by a quadratic function. If we assume that the terrain is flat, we can solve for the X coordinate of the origin of each trajectory, which is the distance from the crime scene to the location of the cannon. We plot each distance as a straight line on the map, at an angle to magnetic North that was recorded at each crime scene. Given the unavoidable errors in measurement, the three points will probably not coincide, but they should be close to one another. As in the story, they will define a small triangle, within which is the actual location of the cannon.

This tiny triangle (solid yellow) is a lot easier to search than the (dashed) triangle defined by the three crime scenes! That's how powerful Algebra can be, and even High School and early College students can put that power to good use.

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