Getting the slant

[bonanova]

An ant walks East a distance of 100 feet. Next he walks north a distance of 50 feet. Then west for 25 feet, and so on. He keeps turning left and halving his previous distance. We know this geometric series converges. His path encloses a single point - the only point to which he will come arbitrarily close.

One question that might be asked involves an infinite series:

What is his total path length as he approaches the limit point?

A more interesting question can be answered without a lot of math.

What is the inclination [angle] from due east, of the line drawn back to his starting point?

There are several ways to determine the second question.

A coveted bonanova gold star will be awarded to the most elegant solution.

Enjoy

[Rob_Gandy]

Well the first one is fairly simple.

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He travels a total distance of 200 ft.

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If we think about how he is moving, his east and west movement will always be twice his north and south movement. So the inclination from due east will be arctan(1/2) or about 26.57 deg

[bonanova]

Yes. Nicely done.

And the obvious next question, what are the coordinates of that point?

The star is twinkling.

[Rob_Gandy]

  On 2/21/2013 at 9:21 PM, bonanova said:

Yes. Nicely done.

And the obvious next question, what are the coordinates of that point?

The star is twinkling. bona_gold_star.gif

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(80,40) assuming he starts at (0,0)

[bonanova]

Yes. Verified by starting at the first turn and noting 80/40 = 40/[100-80]