In a previous post, I described two different approaches to solving the Burning Tent optimization problem. Now I’d like to offer a related problem that I assigned many years ago to my pre-service mathematics teachers at New York University.
A cowgirl wants to give her horse some food and water before returning to her tent. She starts at point C and decides to travel first to the pasture, then to the river, and then back to her tent. What path should she take to minimize her riding distance? The websketch below (and here) shows one possibility. You can drag points A and B to change the path.
As with the Burning Tent problem, there is a solution to this Cowgirl problem whose key insight involves reflecting points across the pasture and the river. See if you can use the tools on page 1 of the websketch to model this approach.
The reflection technique was, in fact, the only solution I expected to see when I assigned the problem to my class. But one student, Steven Harris, having used an ellipse tool to solve the Burning Tent problem, realized that it was also possible to employ ellipses when solving the Cowgirl problem. Can you find a way to do so on page 2 of the websketch?
If you’d like some help, you can watch the video below.