The above coordinates are for parking near the cache site.MP#3 continues on the MP series on which there is MP#1 and MP#2.
FORWARD
We are suggesting in this series that the Sport of Geocaching is closely linked to the subject of Analytical Geometry. This is easy to demonstrate from a purely practical viewpoint by relating the longitiude - latitude readout on your GPS device to the x and y coordinates on a Cartesian Grid. Thus, if you walk directly towards a sighting, you will be following a straight line of the form: y = mx + b. If, on the other hand, you decide to make a slight diversion while travelling towards your target and veer off in an arc, then you will be tracking out the relationship: y = ax**2 + bx +c. These two types of motion were studied in exercises MP1 and MP2. Now we will introduce you to the more realistic route that you would follow when you encounter an obstruction. This is the zig - zag path where the motion now takes the form of an S bend. In this case the track will be prescribed by the function: y = ax**3 + bx**2 + cx + d.
THE PROBLEM
A special property of cubic equations is that they may possess an inflection. This is defined as the point where the concave section of the S bend straightens before becoming convex. To delve more fully into the properties of points of inflection consult google (points of inflection)
The four points I, II, III and IV are prescribe waypoints on your journey. These points have the coordinates :
(x,y)
I (7,8)
II (9,9)
III (11,11)
IV (13,12)
ANALYSES
Use the regression methods of MP#2 to establish the parameters a, b, c, and d that define the cubic equation.
Find the zig-zag point (point of inflection) that marks the obstacle. This is the location of the MP3 (math phobia #3) geocache. The special property of a point of inflection is that the second derivative vanishes: d^2(y)/d(x)^2 = 0.
Using the Google search engine specifying (regression java cubic or quadratic) is helpful.
For step two, and searching again with Google (differentiate function) for helpful techniques.
Once you have the x- and y-coordinates of the inflection point a minor mathematically operation is needed to locate the cache.
1. Multiply the x-coordinate by 0.015 and add it to:
N43 08.389
2. Multiply the y-coordiante by 0.012 and add it to:
W079 27.287
Have fun locating the cache!