The Water Boy and the River

(Thanks to Jim Steiner, St. Paul Central, for this example)

There once was a young man named Nathan. Nathan's job was to each morning go to the river near his home with a bucket, fill the bucket with water and then bring the bucket to his father's workshop in the village.  Now, Nathan was a smart boy and he thought that he should not work any harder at this task then was absolutely necessary. Thus, he decided that he should determine the route to take that would minimize the total distance that he would have to walk each day. For simplicity sake, he assumed that he would walk in a straight line from his house to the river and then in another straight line from the river to his father's workshop. The problem was how to minimize the sum of these two distances.

Let us help Nathan with his problem.  We will assume that in the figure below that the river is the line AB, Nathan's home is at point D, he fills the bucket at point C, and his father's workshop is at point E.

 
 water1
 
Thus, what we need to do is analyze the sum of the distances from D to C and from C to E.  We first use the "Distance" menu option under the Measure menu to measure these distances. Then, we use the Calculator to calculate the sum of these two distances, evaluate this sum, and add the new measure to the main window.  Now, we want to compare the total distance to some variable that measures where on the river we dip our bucket. An easy way to record this variable is by finding the distance from point A to point C.  This will be 0 if we dip the bucket at point A, and will increase as we move along the river.  It will also be useful to measure the angles made by the paths we take in relation to the point C. Measure angles B,C,E  and  D,C,A  (remember that angles must be measured with respect to an orientation). Then, calculate the difference in these two angles and add that to the main window. Since the difference in the angles can be quite large, we also calculate the difference in the nagles divided by 20 and add that to the main window.

Now we are ready to compare our variables.  Plot the distance from A to C  and the total distance (D to C and C to E) as an (x,y) pair on the coordinate axes. This is point F on the graph below.  Also, plot the distance from A to C and the angle difference divided by 20 as another (x,y) pair. (Point G on the graph.)

As we move point C, the two graphed points will also move correspondingly. If we put a trace on points F and G (Use the Trace options under the View menu)  we see quite quickly that point F travels in what appears to be a parabolic path and G travels along a line.  We also notice an interesting relationship between where F reaches a minimum and the value of G. What does this tell you about the solution to Nathan's problem? 
 

water2