Constrained Optimization IA farmer (orange dot) is trying to put out a small fire at his house. He has a bucket at the orange dot, and needs to fill it with water from the lake (above). What is the shortest path he can use to put out the fire? You may have a different solution to this problem, but try this one. Adjust the total path distance to see all the possible routes he can take that total that fixed distance. Increase the total distance until you find the shortest route that can be taken. What is the shape of the points possible so that the total path distance is 7? What if the total path distance is 8? Think about this for a moment... The answer... the shape is an ellipse! The ellipses formed when the total distance is constant are the level curves of some function (which?) What is the relationship between the ellipse and the lake when point A is the optimal point for gathering water?
Elisha Peterson, May 9, 2007, Created with GeoGebra |