[tex]\text{I was eating cookies and had some thoughts. If I wanted to cut out exactly }[/tex][tex] \frac{1}{3} [/tex]of the cookie to share with someone, how far from one side would I have to make a straight cut to get that exact amount? How far would I have to cut if I wanted to cut off[tex] \frac{1}{n} [/tex][tex]\text{ of the cookie?}[/tex]

[tex]\text{Basically, the question is, find the value of }a\text{ given only n, and r}[/tex]

[tex]\text{One way of finding this, is by finding the area of the shaded reigon, Q in terms of}[/tex]
[tex]\text{r, a, and b, and equating it to the area of the fraction of the cookie then solving for a.}[/tex]

[tex]\text{In math, this means solving } \frac{1}{n}\pi r^2=Q \text{ for }f(r,n)=a.[/tex]

[tex]\text{From the diagram, we can see that }r=a+b[/tex]

[tex]\text{Eventually, by 2 different means, I found 2 equations that, if solved, would give the}[/tex][tex]\text{ relationship between r, n, and a.}[/tex][tex]\text{They are as follows:}[/tex]

[tex]\text{1. }\frac{1}{n}\pi r=r\theta-bsin(\theta) \text{ where }\theta=cos^{-1}(\frac{b}{r})[/tex]

[tex]\text{2. }\frac{1}{n}\pi=\theta-sin(2\theta)\text{ where }\theta=cos^{-1}(\frac{b}{r})[/tex]

[tex]\text{These 2 equations are equivalent, but annoying to solve.}[/tex]

[tex]\text{To claim these points, please solve for a in terms of r and n, showing all work.}[/tex]
[tex]\text{I would like an analytic solution if possible.}[\tex]
[tex]\text{All incorrect, spam, or no-work solutions will be reported.}[/tex]