Question 1162480
<font face="Times New Roman" size="+2">


 *[illustration Two_Paths1.jpg]


You need the distance between the two blue points.  Find the *[tex \Large x]-coordinate of the one on the positive side of the *[tex \Large x]-axis by solving:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sqrt{36\,-\,(x\,-\,10)^2}\ =\ \frac{\sqrt{3}}{3}x]


Note that the offset for the independent variable on the circle function is actually arbitrary because the two ends of the illuminated part of the unlit path will move proportionately as the actual placement of the lamps on the other path shift.  The particular configuration shown was chosen to simplify computation.


Note that the ratio of the long leg of a 30-60-90 right triangle is in proportion *[tex \Large 1:\frac{2\sqrt{3}}{3}], so once you have the *[tex \Large x]-coordinate of the circle/line intersection on the right side, multiply by *[tex \Large \frac{2\sqrt{3}}{3}] to get the measure of the slant path from the origin to the blue point intersection. Then just multiply by 2 to get the entire length.
								
								
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<img src="http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg">
*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  
								
{{n}\choose{r}}
</font>