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.

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?
