Brilli's faithfulness to Billi is challenged further

Geometry Level 5

Brilli the ant had his ant hill named $B_1$ which had coordinates $(8,10)$ .

Brilli had his girlfriend named Billi who had her ant hill named $B_2$ which had coordinates $(-2,30)$ . Once Billi was thirsty and she called Brilli to get her some water from a river named Bribill which travelled mainly in the same course as the line $B_1B_2$ but then travels in an arc of a circle with center $(8, 23)$ and radius 17. Thus to not displease his girlfriend, Brilli had to take the quickest path which would lead him to Bribill river first and then to Billi's ant hill. Brilli calculated a point $P$ on the Bribill river such that if he starts from his ant hill, goes to that point and then goes to Billi's ant hill, he would have taken the shortest possible path requiring the least effort and time.

Find the abscissa of the point $P$ that Brilli calculated.

Give your answer to 2 decimal places.

Details and Assumptions:

Assume the ant-hills are points and the river's arc is to the left of the ant hills.
Brilli was always perfect in his calculations.

The answer is 5.33.

1 solution

Brian Charlesworth
Jan 22, 2017

The brute force approach is to find $P(x,y)$ so as to minimize

$f(x,y) = \sqrt{(-2 - x)^{2} + (30 - y)^{2}} + \sqrt{(8 - x)^{2} + (10 - y)^{2}}$

given the constraint that $P$ lies on the circle $(x - 8)^{2} + (y - 23)^{2} = 17^{2}$ ,

This can be solved via substitution and differentiation or by Lagrange multipliers, but both methods are messy and require some "numerical methods" help, (i.e., Wolfram).

The more geometric approach, via Fermat's principle, is to find $P$ such that the angle of incidence the path joining $B1(8,10)$ and $P$ makes with the radius $OP$ is equal to the angle of reflection the path joining $P$ and $B2(-2,30)$ makes with the same radius. This solution is under construction, but I wanted to get something posted so that @Michael Mendrin could post his thoughts in the comments section.

@Guiseppi Butel Great question! Sorry for the incomplete solution, but as I mentioned I wanted to post something now so that Michael could have an opportunity to contribute.

Brian Charlesworth - 4 years, 4 months ago

Thanks for letting me in.

Brian, here's the plot of the function of angle from the center of the circle to the river, yielding the total distance that needs to be traveled

and the layout of the river, anthills, and routes

There is a "false optimal path" which is about where you'd think it'd be, and there's the actual shortest path which is to the right of it. Very sneaky. The difference is small. It's counter-intuitive because you'd think there'll be just a maximum and a minimum, not TWO "minimums".

Michael Mendrin - 4 years, 4 months ago

Great graph and diagram! I think I must have found the false minimum on my first attempt, and then when I started playing with counter-intuitive paths I found a deeper dip, so your graphics make sense of that mystery. Brilli is one smart ant to know that the optimal path involves going almost backward before going forward. Most humans would have instantly opted for something like the "false optimal". Of course realistically one would be slowed down somewhat while carrying the water, so as a "devious" variation we could ask what the fastest path is if Brilli's speed while carrying the water is $0 \lt p \lt 1$ times that of his speed before reaching the river. For some values of $p$ the double dip in your graph might become a single dip, (or who knows, maybe some other unexpected dips might show up). Guiseppi did indeed conjure up a sneaky-difficult problem here.

Brian Charlesworth - 4 years, 4 months ago

I think those optimum points are the ones that Brian has mentioned - points of reflection.

Maria Kozlowska - 4 years, 4 months ago

Brilli's faithfulness to Billi is challenged further

The answer is 5.33.

1 solution

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