Question

A crazy ant is standing on the origin. It begins by walking 1 unit in the +x direction and then turns 60 degrees counterclockwise and walks 1/2 units in that direction. The ant then turns another 60 degrees and walks 1/3 units in that direction. The ant keeps doing this endlessly. What is the ant's final position?

#Method by Which you can solve
1) Cartesian Method
2) Polar Coordinate method.

#Hint:- Use formula for displacement,
D = 1/n(cos∅+ isin∅)​

Answer 1

(basic question of calculus)

I had solved it using Polar coordinate method

Solution:-Displacement of ant =

$\frac{1}{n}( \cos \alpha + isin \alpha )$

where 1/n is the distance travelled.

The final position of the ant will be sum of all the consecutive displacements

so we can write

D=$1(cos0 + isin0) + \frac{1}{2}(sin \frac{\pi}{3} + isin \frac{\pi}{3}) + \frac{1}{3}(cos \frac{2\pi}{3} + isin \frac{2\pi}{3}) + .$

From Euler's rule,(which we had learnt in complex number)

we know

${e}^{i \alpha } = cos \alpha + isin \alpha$

$on \: comparing \: we \: can \: write\\ d = \sum\limits_{n=0}^{n= \infty } \frac{ {e}^{ \frac{i(n - 1)\pi}{3} } }{n}$

now we know that

$\frac{ln(1 - x)}{x } = - \sum\limits_{n=0}^{n= \infty } \frac{ {e}^{( n - 1)x} }{n} \\ comparing \: this \: with \: d \: we \: get \\ d = > \sum\limits_{n=0}^{n= \infty } \frac{ {e}^{( n - 1)x} }{n} = \sum\limits_{n=0}^{n= \infty } \frac{ {e}^{ \frac{i(n - 1)\pi}{3} } }{n} \\ from \: here \: we \: can \: say \: x = {e}^{i \frac{\pi}{3} } \\ now \\ d = \frac{ln(1 - {e}^{ \frac{i\pi}{3} }) }{ {e}^{ \frac{i\pi}{3} } } = \frac{i \frac{\pi}{3} }{cos \frac{\pi}{3} + isin \frac{\pi}{3} } = \frac{i \frac{\pi}{3} }{ \frac{1 + \sqrt{3}i }{2} } \\ = > \frac{2i \frac{\pi}{3} }{1 + \sqrt{3}i } = \frac{\pi}{2 \sqrt{3} } + \frac{i\pi}{6} \\ this \: shows \: that \: coordinates \: of \: final \: position \\ is \: ( \frac{\pi}{2 \sqrt{3} } \: \: \: \frac{\pi}{6} ) \\ distance \: from \: initial \: position = \sqrt{ {( \frac{\pi}{2 \sqrt{3} } })^{2} + ( { \frac{\pi}{6} })^{2} } \\ = > \sqrt{( { \frac{ \pi }{9} })^{2} } = \frac{\pi}{3} = 1.05$

hence the ant will be 1.05m away from it's initial position

Note:-Can also be solved by using Cartesian method,but will be a bit lengthy

Answer 2

Answer:

think I should let the origin be (0,0) and calculate the coordinates of the points that ant reaches each time it turns. I believe the x-coordinate at the nth step is ∑nk=1cos(π4(k−1))k and the y-coordinate at the nth step is ∑nk=1sin(π4(k−1))k.