Question

`\text{I was eating cookies and had some thoughts. If I wanted to cut out exactly }`
`(1)/(3)`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
`(1)/(n)`
`\text{ of the cookie?}`


`\text{Basically, the question is, find the value of }a\text{ given only n, and r}`


`\text{One way of finding this, is by finding the area of the shaded reigon, Q in terms of}`

`\text{r, a, and b, and equating it to the area of the fraction of the cookie then solving for a.}`


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


`\text{From the diagram, we can see that }r=a+b`


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


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


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


`\text{These 2 equations are equivalent, but annoying to solve.}`


`\text{To claim these points, please solve for a in terms of r and n, showing all work.}`

Answer 1

- the area of a circle

- the area of a circular segment

Answer 2

In the attachement, there is what I came up with so far. I think that finding 'a' is non-trivial, if possible at all.

`A_c` - the area of a circle

`A_(cs)` - the area of a circular segment