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# Getting the slant

## Question

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

## 4 answers to this question

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Yes. Nicely done.

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

The star is twinkling. bona_gold_star.gif

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

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Well the first one is fairly simple.

He travels a total distance of 200 ft.

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

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Yes. Nicely done.

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

The star is twinkling.

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

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