Tuesday, May 5, 2009

solution: burning tent

In the burning tent problem, we have a starting point (the camper at the moment of realization) and a finishing point (the burning tent). We want to minimize the distance from the camper to the tent via the river. A first step may be to realize that this task is not the same as minimizing the first leg or the second leg of the trip. That is, going straight to the river does not give us the shortest route, and neither does going to the point of the river closest to the tent. The answer will involve a happy medium. As such, calculus is an option.

The clever way to solve this problem is to imagine that the point along the river bank has been chosen and we are watching the reflection of the camper across the line that is the river bank. The camper and his reflection will always be the same distance from the river. In other words, if we choose a target point on the river that gives the camper the shortest distance, it also gives his reflection the shortest route. Therefore, choose the point that is on the line between the camper's reflection and the burning tent. The shortest distance is given in the comments of the question.

2 comments:

  1. So, I am teaching my precalculus students about the reflective properties of parabolas and am planning on starting them out with the burning tent problem. I Google "Burning Tent Problem" and this post by a Nate Burchell from 2009 popped up. I am guessing you posted it around the same time you were student teaching with me. The ironic thing is that I just asked my students about the off-center-cut pizza problem today and recalled your unbelievable sketch with with all of the sheered pieces. Do you still have that sketch? I would love to see it again. I hope all is well with you, Joie and the kids.

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  2. Hi there Rob, I have the pizza theorem sketch posted here: http://burchellmath.blogspot.co.ke/2009/07/geometry-pizza-theorem.html
    and I have shown it a few times in my geometry classes. Haven't looked at the burning tent one for awhile though. I was thinking about that pizza sketch recently because I was trying to make something about the haberdasher's dissection, with the pieces swinging around.

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